http://geometrie.idea-sketch.com/index.php?action=history&feed=atom&title=Bresenham-Algorithmus_%28in_Arbeit%29 Bresenham-Algorithmus (in Arbeit) - Versionsgeschichte 2026-08-01T08:32:06Z Versionsgeschichte dieser Seite in Geometrie-Wiki MediaWiki 1.43.9 http://geometrie.idea-sketch.com/index.php?title=Bresenham-Algorithmus_(in_Arbeit)&diff=22107&oldid=prev Peterpummel: /* Zusammenfassung */ 2013-03-11T17:00:20Z <p><span class="autocomment">Zusammenfassung</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="de"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Nächstältere Version</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Version vom 11. März 2013, 17:00 Uhr</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l167">Zeile 167:</td> <td colspan="2" class="diff-lineno">Zeile 167:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Zusammenfassung ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Zusammenfassung ==</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Durch die beiden Fallunterscheidungen in 2.1 und 2.2 kann nun im ersten Quadranten jede Linie  mit positiver Steigung gezeichnet werden. Wollen wir nun <del style="font-weight: bold; text-decoration: none;">in der Lage sein, </del>auch die negativen Steigungen <del style="font-weight: bold; text-decoration: none;">zu </del>zulassen, <del style="font-weight: bold; text-decoration: none;">Spiegeln </del>wir den Endpunkt der Strecke in den ersten Quadranten<del style="font-weight: bold; text-decoration: none;">, </del>indem wir in die passende Gleichung die Absolutbeträge des Punktes und des Normalenvektors einsetzten. &lt;br&gt;</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Durch die beiden Fallunterscheidungen in 2.1 und 2.2 kann nun im ersten Quadranten jede Linie  mit positiver Steigung gezeichnet werden. Wollen wir nun auch die negativen Steigungen zulassen, <ins style="font-weight: bold; text-decoration: none;">spiegeln </ins>wir den Endpunkt der Strecke in den ersten Quadranten<ins style="font-weight: bold; text-decoration: none;">. Dies geschieht </ins>indem wir in die passende Gleichung die Absolutbeträge des Punktes und des Normalenvektors einsetzten. &lt;br&gt;</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Also für Steigungen mit &lt;math&gt; |m| &lt;= 1 &lt;/math&gt;&lt;br&gt;&lt;br&gt;  </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Also für Steigungen mit &lt;math&gt; |m| &lt;= 1 &lt;/math&gt;&lt;br&gt;&lt;br&gt;  </div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>&lt;math&gt;= - ( <del style="font-weight: bold; text-decoration: none;">|</del>n_x<del style="font-weight: bold; text-decoration: none;">| </del>\cdot 2 \cdot |p_x| \ + 2 \cdot <del style="font-weight: bold; text-decoration: none;">|</del>n_x<del style="font-weight: bold; text-decoration: none;">| </del>+ <del style="font-weight: bold; text-decoration: none;">|</del>n_y <del style="font-weight: bold; text-decoration: none;">|</del>\cdot 2 \cdot |p_y| + <del style="font-weight: bold; text-decoration: none;">|</del>n_y<del style="font-weight: bold; text-decoration: none;">| </del>) =: d &lt;/math&gt;&lt;br&gt;&lt;br&gt;</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>&lt;math&gt;= - ( n_x \cdot 2 \cdot |p_x| \ + 2 \cdot n_x + n_y \cdot 2 \cdot |p_y| + n_y ) =: d &lt;/math&gt;&lt;br&gt;&lt;br&gt;</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Also für Steigungen mit &lt;math&gt; |m| &gt; 1 &lt;/math&gt;&lt;br&gt;&lt;br&gt;</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Also für Steigungen mit &lt;math&gt; |m| &gt; 1 &lt;/math&gt;&lt;br&gt;&lt;br&gt;</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>&lt;math&gt;= <del style="font-weight: bold; text-decoration: none;">|</del>n_x<del style="font-weight: bold; text-decoration: none;">| </del>\cdot 2 \cdot |p_x| + <del style="font-weight: bold; text-decoration: none;">|</del>n_x<del style="font-weight: bold; text-decoration: none;">| </del>+ 2 \cdot <del style="font-weight: bold; text-decoration: none;">|</del>n_y<del style="font-weight: bold; text-decoration: none;">| </del>\cdot |p_y| + 2 \cdot <del style="font-weight: bold; text-decoration: none;">|</del>n_y<del style="font-weight: bold; text-decoration: none;">| </del>=: d&lt;/math&gt;&lt;br&gt;&lt;br&gt;</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>&lt;math&gt;= n_x \cdot 2 \cdot |p_x| + n_x + 2 \cdot n_y \cdot |p_y| + 2 \cdot n_y =: d&lt;/math&gt;&lt;br&gt;&lt;br&gt;</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=Literaturnachweise=</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=Literaturnachweise=</div></td></tr> <!-- diff cache key geometrie:diff:1.41:old-22106:rev-22107:php=table --> </table> Peterpummel http://geometrie.idea-sketch.com/index.php?title=Bresenham-Algorithmus_(in_Arbeit)&diff=22106&oldid=prev Peterpummel: /* Algorithmus für Gerade mit der Steigung m <= 1 */ 2013-03-11T16:50:57Z <p><span class="autocomment">Algorithmus für Gerade mit der Steigung m &lt;= 1</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="de"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Nächstältere Version</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Version vom 11. März 2013, 16:50 Uhr</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l126">Zeile 126:</td> <td colspan="2" class="diff-lineno">Zeile 126:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&lt;math&gt; T = \begin{pmatrix} p_x + 1 \\ p_y \end{pmatrix} &lt;/math&gt; und der Punkt &lt;math&gt; S = \begin{pmatrix} p_x + 1 \\ p_y + 1 \end{pmatrix} &lt;/math&gt;</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&lt;math&gt; T = \begin{pmatrix} p_x + 1 \\ p_y \end{pmatrix} &lt;/math&gt; und der Punkt &lt;math&gt; S = \begin{pmatrix} p_x + 1 \\ p_y + 1 \end{pmatrix} &lt;/math&gt;</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>&lt;ggb_applet width="1260" height="880"  version="4.2" ggbBase64="<del style="font-weight: bold; text-decoration: none;">UEsDBBQACAgIAJRlaEIAAAAAAAAAAAAAAAAWAAAAZ2VvZ2VicmFfamF2YXNjcmlwdC5qc0srzUsuyczPU0hPT</del>/<del style="font-weight: bold; text-decoration: none;">LP88zLLNHQVKiu5QIAUEsHCEXM3l0aAAAAGAAAAFBLAwQUAAgICACUZWhCAAAAAAAAAAAAAAAADAAAAGdlb2dlYnJhLnhtbOVb23Lbxhm</del>+<del style="font-weight: bold; text-decoration: none;">dp5ihxeddCpCez6kVDKJUzdOlMQTuZ20Nx4QWJKIQIABQB08eZy</del>+<del style="font-weight: bold; text-decoration: none;">QPMKvc8z9d9dgCREUTElWVbVEcnFLvb0f</del>/<del style="font-weight: bold; text-decoration: none;">95AY0+u5jn6MxWdVYWRwMS4QGyRVKmWTE9GiybyVAPPvv0o9HUllM7rmI0Kat53BwNeEQH63FQiwhzg7MUZknGdkIJGcp4LIY8JvFwPMF4KJhVE</del>/<del style="font-weight: bold; text-decoration: none;">hyxckAoYs6</del>+<del style="font-weight: bold; text-decoration: none;">6Qov4vntl7EiT1JZnYeH5dJ3Pg5Z02z</del>+<del style="font-weight: bold; text-decoration: none;">OTw8Pz8POpWj8pqejidjqOLOh0g2HlRHw3ai09gut6gc</del>+<del style="font-weight: bold; text-decoration: none;">a7U4zJ4Y</del>/<del style="font-weight: bold; text-decoration: none;">fHofph1lRN3GR2AFyVC2zTz96NjrPirQ8R</del>+<del style="font-weight: bold; text-decoration: none;">dZ2sxg91SZAZrZbDoDOo0Gog5drwUQu7BJk53ZGsZuVD3RzXwx8N3iwt1</del>/<del style="font-weight: bold; text-decoration: none;">Fq5QvqJngNLsLEttdTTAEWVKaiENMVxqxrgeoLLKbNG0nUm76GE33egss+dhXnfll+TYKGBCVmfj3B4NJnFeA11ZMakAU9hRtYRq3VzmdhxXXX29IXIAf9Ahe2vdXEBnAOJoQLU4AP4dKIwPhGgB2Fx4gJqyzP2sGAmDfvkFUUwxOnAFCQWFQspwC4c2zEJBQ8FDIUIfHobz0JWHPjz04ewGOtv6mtC2oUdpRyfbpJMAfe4r4esBuEKn3qCTOCJ</del>+QcTt3hcMuX0Tv39X8LYqQ1X5guBQkPamdj8eL3lHititKCIbqwZ52L3olrx0KxIqN5ZkHB9Qs3tJug+hV9fcoJKvlxRE+CWp2EHlHcFdEcrFxqKgC+<del style="font-weight: bold; text-decoration: none;">7jv1tLsjuRuV5R4HddUfK76P4tFlS4p</del>/<del style="font-weight: bold; text-decoration: none;">adzoeStOVNMNzbpkaHnTUctRtC9cz1bUW6sfPabZEZb5wQQQKUVyqwJQIRA4VySkwREYgLqBKNpCsVYk5vOWJII9ePMORNkNDww71OSyRgLteognIjxpFgiHjDxRGggLzxA0wogx5CIAGD3OrELcsk4hIqTCMOG3RmTznTwmAc1GFxihhBzI0lClGJJEXKmU7CnUWV2u0dJqVIYiTdULCdYDeDzYQRGjFHDWjBoqyzFbgzmy9WXPE4ZsVi2bTYte3JPO1wbMor3dMyOf3iCtg2rpvuGjqBx1o7xuDBen7z2SiPxzaH8OLEyQFCZ3Hu1NzPPymLBq2MTGibVvFiliX1iW0aGFWjn</del>+<del style="font-weight: bold; text-decoration: none;">Kz</del>+<del style="font-weight: bold; text-decoration: none;">Dhu7MUL6F13G</del>/RLe38+<del style="font-weight: bold; text-decoration: none;">ssskz9IsLv4OQuKmcBOitXt3xqtz71q0yyRlWaUnlzWIDrr4p63Ko8GQy4gQTgkzAjPJKBiDy3DLMBFJTZWmgjJBOLjhOomdzBMlI6y5dg4CnLsCJb3ccUuFle3ZirT4wq4IQtPKKd1G5WX9RZmvmxZlVjTP40WzrHywBqayckR9Xkxz68H1thfCnuR0XF6cBFRZmOv15QJqOOxgPH1e5mWFQCWpACKnbTkOpe/jtrbqhX0f7Hvgjk1ZurpPDPU9fDkOpe8FfA9ba0klHZkEd8tktTc2MHlPLL3UuCBqWWTNcVdpsuS0JZWEAd8t52MQuFYi</del>+<del style="font-weight: bold; text-decoration: none;">3OS</del>+<del style="font-weight: bold; text-decoration: none;">5pzdHhFxkantipsHgSpAGYuy2UdRHslns9Gy9q</del>+<del style="font-weight: bold; text-decoration: none;">ipvZ50X6g52CUr6KnV1sYOrQdb3l1CbZHAaG9ha82DH2b7DV0JraaWU7EnMfHwdo</del>/<del style="font-weight: bold; text-decoration: none;">V28KdZbzX6qF1U5f1mcvQapubLV0WFHz6hOqmzhpBONwVCf2rX8pVkdg5lPN8cB8TVQkTiTA0A2DsQBipfNrKx8BAx6C44Eff2ffxWFrcBUgkA6lb1YVLZ2uUTHlNf2ogH44cbR4A8</del>/<del style="font-weight: bold; text-decoration: none;">L8vmz6</del>/<del style="font-weight: bold; text-decoration: none;">Qxz</del>++<del style="font-weight: bold; text-decoration: none;">yQ7QP95kfwwtfkmb2znEzajxEg1moRn05</del>/<del style="font-weight: bold; text-decoration: none;">AWALiIyvFPYIZWbif02WAP3N8h4SjOF7PYxe0tfnl8CbvfRNRP922Z9nH2Bq22VTZZeT9Qw2</del>+<del style="font-weight: bold; text-decoration: none;">BpS5XWhm64BNXoyBDqZpXTr0hSXJ9DebwgUhHGqwpBmtz6XI1JQiWRBitlRYcrP3bkMEFCQ247ASX9sA98eCiPyHiAXYXe4BMHw3IXaS1P8o8MoIDnkxSA8Bq6kGGxBZLaQxhSjCBjdgPZNYD</del>+<del style="font-weight: bold; text-decoration: none;">XUf5D0AZk8DYKMMpQCy4kIq7xq9GHPDmYSMTzPONdY7Ie5h5N3fCoDP7w6Qd3m3hSguwF57qwcRxCJ4joW1wemEDcPFAqbzvnpjO95C1w4gf</del>/<del style="font-weight: bold; text-decoration: none;">jiZQ5vQvBs5El1</del>/<del style="font-weight: bold; text-decoration: none;">rsXLIXWK6b9XfH64gngxR4Qr</del>+<del style="font-weight: bold; text-decoration: none;">dPAC</del>/<del style="font-weight: bold; text-decoration: none;">xgHh9+QTw6vSRPgBef3kCeLEHxOvFE8BL3Dde14YgvBeCfI8</del>+<del style="font-weight: bold; text-decoration: none;">xgd4j8iDP4HIA0dGaS6pkoopTCGa6yIPzQnnQhkB4bM7Y94</del>/<del style="font-weight: bold; text-decoration: none;">8PjrExBE9YCK</del>+<del style="font-weight: bold; text-decoration: none;">9UTwushHOnLbbz6pxfrQ4NHi5eOjMRGaq2pxpDASo</del>+<del style="font-weight: bold; text-decoration: none;">eiJgkhkjCDTNSSfPesfz6CWApI2a0lFgwYrTBlHksWQQwcow5oZS5j7wXMJNyPo</del>+<del style="font-weight: bold; text-decoration: none;">LFBX</del>++<del style="font-weight: bold; text-decoration: none;">P44K</del>+<del style="font-weight: bold; text-decoration: none;">xgfWocYxfmoZg4YANoy6a7EYep2gm2</del>+<del style="font-weight: bold; text-decoration: none;">OLOAVeoxx</del>/<del style="font-weight: bold; text-decoration: none;">UxayhHZJrUAyCehVywHZoIqKpoVgKAvmsJvzqEWczy5LTAjyyP4dt2hNXf</del>/<del style="font-weight: bold; text-decoration: none;">FVlqbWH8r7MfbnIgxp3Vg2X</del>+<del style="font-weight: bold; text-decoration: none;">RZkjV78</del>+<del style="font-weight: bold; text-decoration: none;">N54MeLLX6M9</del>+<del style="font-weight: bold; text-decoration: none;">DH</del>+<del style="font-weight: bold; text-decoration: none;">PfU5KEYQls7i1s7OyTryR4M5ZeFO6AFFK5AHQeox1tQnySzImugWpw2N6Pet069cbfjgNSeBa4Yh</del>+<del style="font-weight: bold; text-decoration: none;">LuTBAsus7g8Ehgo7ngFESfdediOFJGMeqO0SDkwlzeo</del>/<del style="font-weight: bold; text-decoration: none;">k5sVPXfoUNPdQ8R55vcSR9U9</del>/<del style="font-weight: bold; text-decoration: none;">MiLqduoPXDbiVTeoenwQlEORezBKONGZaEAI</del>/<del style="font-weight: bold; text-decoration: none;">GMJUyntaAVEIMbCwhB</del>+<del style="font-weight: bold; text-decoration: none;">AXlJxg4qIm1UEQMud81kJPTir7ecYp9Yu3AOk74vXVVzU7pWg</del>/<del style="font-weight: bold; text-decoration: none;">gOMuzNx24Klb35Hm7aZeI0avQsTCQ5Pr6h75eC</del>+<del style="font-weight: bold; text-decoration: none;">bBmJiAHvorWPdwTEO33TJkDLsOKcwY9UmqnHxcXrc0vRyy1Ba949rxT3oWAfPLPkkcKEElBOITBhUrXRGPAX4gf3zAAiW7</del>/<del style="font-weight: bold; text-decoration: none;">nfZ4ayCuwNu8Oq7xLLNU99P3gsIoIG</del>+<del style="font-weight: bold; text-decoration: none;">weaWGlOBPatOmpZOBxjCCaK6aVUrtg7duc76tmVk7LIs6viZheBHsTb9mbZI</del>+IKXksEdPwmlj1ckdk+<del style="font-weight: bold; text-decoration: none;">9a9VBVRCt6acQ13jdCPJbhKdnHlm30iqm9ul</del>++<del style="font-weight: bold; text-decoration: none;">9jygKLH</del>/<del style="font-weight: bold; text-decoration: none;">EGFgDEGFDjZCBMQZagVtcAXsUZThINPOPdjXl0CIVAW7eYxh1ozY834V7uoc2pP</del>+<del style="font-weight: bold; text-decoration: none;">r2sAiDkEtBh+steKYP3hGt0sd0l1sOd5HHY4fkTpIESlInKUw8NHCOR53IovBdSotKTfgNyG3kO9fH66PSI935RJ2vyDUfuAwp5dGKAbYalAGJbmQPJyCD50PwBLCF0GwwIKC323zCqGIgXRDMgwMYXfQhveeVvh33q5P0O0WC3</del>/<del style="font-weight: bold; text-decoration: none;">798089C9HrTgEvd142MiyhZYAMFgZCcgow7WRrdu8txMs8o48JrfgSPcuWZVsKE53Ppnn5fkPdpLbCw</del>/<del style="font-weight: bold; text-decoration: none;">o3ZK6b3ZlcpP9VGjygUPa3gNhA3x3r3xiCE8laVM4RjWDrFyCVkHIT4ITH8qIcMkNISArAoNvvymje4wKNNnldX77dS8F+vX3FEj93+nP9Xmf6uV9eyTT6kkk08IdqGNIpDmRoGSGd482uHtIxImgSlJCd2Z914Oqe6DukUrrJ5FK80hqcPgQzBKlIILibSrNDTQYzYSgVJKd744ebr7R61</del>+<del style="font-weight: bold; text-decoration: none;">xb</del>/<del style="font-weight: bold; text-decoration: none;">8V79P</del>/<del style="font-weight: bold; text-decoration: none;">AlBLBwhKml4UngoAADo4AABQSwECFAAUAAgICACUZWhCRczeXRoAAAAYAAAAFgAAAAAAAAAAAAAAAAAAAAAAZ2VvZ2VicmFfamF2YXNjcmlwdC5qc1BLAQIUABQACAgIAJRlaEJKml4UngoAADo4AAAMAAAAAAAAAAAAAAAAAF4AAABnZW9nZWJyYS54bWxQSwUGAAAAAAIAAgB</del>+AAAANgsAAAAA" showResetIcon = "false" enableRightClick = "false" errorDialogsActive = "true" enableLabelDrags = "false" showMenuBar = "false" showToolBar = "false" showToolBarHelp = "false" showAlgebraInput = "false" useBrowserForJS = "true" allowRescaling = "true" /&gt;</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>&lt;ggb_applet width="1260" height="880"  version="4.2" ggbBase64="<ins style="font-weight: bold; text-decoration: none;">UEsDBBQACAgIAEOOa0IAAAAAAAAAAAAAAAAWAAAAZ2VvZ2VicmFfamF2YXNjcmlwdC5qc0srzUsuyczPU0hPT</ins>/<ins style="font-weight: bold; text-decoration: none;">LP88zLLNHQVKiu5QIAUEsHCEXM3l0aAAAAGAAAAFBLAwQUAAgICABDjmtCAAAAAAAAAAAAAAAADAAAAGdlb2dlYnJhLnhtbOVb23Lbxhm</ins>+<ins style="font-weight: bold; text-decoration: none;">dp5ihxeddCpCez64VDKJUzdOlNgTuZ1ObzwgsCQRgQADgDpk8jh9geYVep9n6r</ins>+<ins style="font-weight: bold; text-decoration: none;">7AEmIomJKsqyqI5KLXezp</ins>//<ins style="font-weight: bold; text-decoration: none;">7zAhp9fjHP0Zmt6qwsjgYkwgNki6RMs2J6NFg2k6EefP7ZJ6OpLad2XMVoUlbzuDka8IgO1uOgFhHmBmcpzJKM7YQSMpTxWAx5TOLheILxUDCrJvDlipMBQhd19rwov4/ntl7EiT1JZnYeH5dJ3Pg5Z02zeH54eH5</ins>+<ins style="font-weight: bold; text-decoration: none;">HnWrR2U1PZxOx9FFnQ4Q7LyojwbtxXOYrjfonPnuFGNy</ins>+<ins style="font-weight: bold; text-decoration: none;">I/vjsP0w6yom7hI7AA5qpbZZ588G51nRVqeo/MsbWawe6rMAM1sNp0BnUYDUYeu1wKIXdikyc5sDWM3qp7oZr4Y</ins>+<ins style="font-weight: bold; text-decoration: none;">G5x4e4</ins>/<ins style="font-weight: bold; text-decoration: none;">C1coX9EzQGl2lqW2OhrgiDIltZCGGC41Y1wPUFlltmjazqRd9LCbbnSW2fMwr7vyS3JsFDAhq7Nxbo8Gkzivga6smFSAKeyoWkK1bi5zO46rrr7eEDmAP</ins>+<ins style="font-weight: bold; text-decoration: none;">iQ</ins>/<ins style="font-weight: bold; text-decoration: none;">WzdXEBnAOJoQLU4AP4dKIwPhGgB2Fx4gJqyzP2sGAmDfvkFUUwxOnAFCQWFQspwC4c2zEJBQ8FDIUIfHobz0JWHPjz04ewGOtv6mtC2oUdpRyfbpJMAfe4r4esBuEKn3qCTOCJ</ins>+QcTt3hcMuX0Tv39X8LYqQ1X5guBQkPamdj8eL3lHititKCIbqwZ52L3olrx0KxIqN5ZkHB9Qs3tJug+hV9fcoJKvlxRE+CWp2EHlHcFdEcrFxqKgC+<ins style="font-weight: bold; text-decoration: none;">7jv1tLsjuRuV5R4PddUfK76P4tFlS4p</ins>/<ins style="font-weight: bold; text-decoration: none;">adzoeStOVNMNzbpkaHnTUctRtC9cz1bUW6sfPabZEZb5wQQQKUVyqwJQIRA4VySkwREYgLqBKNpCsVYk5vOWJII9ePMORNkNDww71OSyRgLteognIjxpFgiHjDxRGggLzxA0wogx5CIAGD3OrELcsk4hIqTCMOG3RmTznTwmAc1GFxihhBzI0lClGJJEXKmU7CnUWV2u0dJqVIYiTdULCdYDeDzYQRGjFHDWjBoqyzFbgzmy9WXPE4ZsVi2bTYte3JPO1wbMor3dMyOf3yCtg2rpvuGjqBx1o7xuDBen7z2SiPxzaH8OLEyQFCZ3Hu1NzPPymLBq2MTGibVvFiliX1iW0aGFWjH</ins>+<ins style="font-weight: bold; text-decoration: none;">Oz</ins>+<ins style="font-weight: bold; text-decoration: none;">Dhu7MVL6F13G</ins>/RLe38+<ins style="font-weight: bold; text-decoration: none;">ssskz9IsLv4OQuKmcBOitXt3xqtz71q0yyRlWaUnlzWIDrr4p63Ko8GQy4gQTgkzAjPJKBiDy3DLMBFJTZWmgjJBnPuuk9jJPFEywppr5yDAuStQ0ssdt+CeX9qerWiLL+yKIjStnNZtVF7VX5b5umlRZkXzIl40y8pHa2ArK0fVF8U0tx5db3wh7klOx</ins>+<ins style="font-weight: bold; text-decoration: none;">XFSYCVhbneXi6ghsMOxtMXZV5WCHSSCqBy2pbjUPo</ins>+<ins style="font-weight: bold; text-decoration: none;">bmurXtj3wb4H7viUpav7xFDfw5fjUPpewPiwtZZU0pFJcLdMVntrA5P35NKLjYuilkXWHHeVJktOW1JJGPD9cj4GiWtFsj8nua85R4dXhGx0aqvC5kGSCmDmslzWQbZX8vlstKztm7iZfVGkP9gpaOWb2BnGBqYOXddbTm2SzWFgaG/Bix1j/wZbDa2pnVa2IzH3AXKA1t/Fm3K91eynelmV81fF2VuQmitbHR129IzqpMoWTjrRGCz1qV3LX5rVMdj5dHMcEF8DFYmzOQBk40AcoHjZzMrKh8CguOBJ0Df/</ins>+<ins style="font-weight: bold; text-decoration: none;">VdR2ApsJQik09mLRWVrl0x0THlrLxqAH24cDf7w07Js</ins>/<ins style="font-weight: bold; text-decoration: none;">vwGfbp4d3GAFu8u</ins>/<ins style="font-weight: bold; text-decoration: none;">xha</ins>/<ins style="font-weight: bold; text-decoration: none;">JI2t3MInFHjJRrsQjPoz</ins>+<ins style="font-weight: bold; text-decoration: none;">FNAHARleMfwQ6t/E7os8EeuL9DwlGcL2axC9xb/PL4Ena/iaif7rsy7ePsLVptq2yycn</ins>+<ins style="font-weight: bold; text-decoration: none;">ght8BS12ytLJ0wSmuRkGKUjVvnHpDluT6GszhA6GONFhTDFH</ins>/<ins style="font-weight: bold; text-decoration: none;">pUvWlCBYEmG0VlpwMPc/hxQuSGjAZSe4tAfuiQcX/QkRD7C72ANk</ins>+<ins style="font-weight: bold; text-decoration: none;">mhA7kKt/VHmkREc8GSSGgBWUw8yZLZYSmMIU4IJbMR</ins>+<ins style="font-weight: bold; text-decoration: none;">ILMeyG</ins>/<ins style="font-weight: bold; text-decoration: none;">7IO8BMHsaABtlKAWQFRdSed</ins>/<ins style="font-weight: bold; text-decoration: none;">oxZgbziSkfJpxrrHeCXEPI</ins>+/+<ins style="font-weight: bold; text-decoration: none;">VgB8cXeAvMu7LURxAfbaWz0IIRbBcyysDU4nbBguFjCd99Ub2</ins>/<ins style="font-weight: bold; text-decoration: none;">EWunYA</ins>+<ins style="font-weight: bold; text-decoration: none;">dMXL3N4E4JnI0</ins>+<ins style="font-weight: bold; text-decoration: none;">q89</ins>+<ins style="font-weight: bold; text-decoration: none;">9aCm0XjHt74vXl08AL/aAeL14AniJB8TrqyeAV6eP9AHw</ins>+<ins style="font-weight: bold; text-decoration: none;">ssTwIs9IF4vnwBe4r7xujYE4b0Q5DX6FB/gPSIP/gQiDxwZpbmkSiqmMIVoros8NCecC2UEhM/ukHn/wOOvT0AQ1QMq7tdPCK</ins>+<ins style="font-weight: bold; text-decoration: none;">HcKSvtvHqn16sDw0eLV46MhIbqbWmGkMCKz16ImKSGCIJN8xIJc0Hx/KbJ4CljJjRUmLBiNEGU</ins>+<ins style="font-weight: bold; text-decoration: none;">axZBHAyDHmhFLmPvJewEzK</ins>+<ins style="font-weight: bold; text-decoration: none;">TwuUlT48</ins>/<ins style="font-weight: bold; text-decoration: none;">vjrLCD9bFxjF2Yh2LigA2gLZvuRhymaifY4os7B1yhHn9UF7OGdkiuQTEI6lXIAduhiYimhmIpCOSzmvCrR5zNLEtOC</ins>/<ins style="font-weight: bold; text-decoration: none;">DI/hy2aU9c/cXXWZpafyrvx9ifijCkdWPZfJFnSdbszY8XgR8vt/gx3oMf499Tk4diCG3tLG7t7JCsJ3swlF8V7oAWULgCdRygHm9BfZLMiqyBanHa3Ix63zr1xt2OA1J7FrhiHIq7M0Gw6DqDwyOBjeaCUxB91p2L4UgZxag7RoOQC3N5j</ins>+<ins style="font-weight: bold; text-decoration: none;">bnxE5d</ins>+<ins style="font-weight: bold; text-decoration: none;">xU29FDzHHmxxZH0XX0zI</ins>+<ins style="font-weight: bold; text-decoration: none;">p26g5eN</ins>+<ins style="font-weight: bold; text-decoration: none;">BWNql7fBKUQJB7MUs40phpQQj8YAhTKe9pBUQhxMDCEn4AeknFDSoiblYRAC13zmcl9OCstp9jnFq7cA</ins>+<ins style="font-weight: bold; text-decoration: none;">QXhdvq7io3TtB</ins>/<ins style="font-weight: bold; text-decoration: none;">QcYd2fitgVL3</ins>/<ins style="font-weight: bold; text-decoration: none;">2ONm0z8Ro1eh8mEhyeXlH3zsF92TISEQPeRWsf7wiId</ins>/<ins style="font-weight: bold; text-decoration: none;">qmTYCWYcU5gx</ins>+<ins style="font-weight: bold; text-decoration: none;">pNFOPi4vX55ail1uC1rx</ins>/<ins style="font-weight: bold; text-decoration: none;">XinuQ8E</ins>+<ins style="font-weight: bold; text-decoration: none;">embJI4UJJaCcQmDCpGqjMeAvxA</ins>/<ins style="font-weight: bold; text-decoration: none;">umQFEtn7P</ins>+<ins style="font-weight: bold; text-decoration: none;">zw1kFdgbd4fVnmXWKp76PvRYRURNtg90sJKcSa0adNTycDjGEE0V0wrpXbB2rc5r6tmVk7LIs6viZheBnsTb9mbZI</ins>+IKXksEdPwmlj1ckdk+<ins style="font-weight: bold; text-decoration: none;">7N7qyqiFLw14xruGqEfS3CV7OLKt/tEVN/eLt/7EFEUWP6IMbAGIMKGGiEDYwy0Are4AvYoynCQaOYf7WrKoUUqAty8xzDqRm14sQv3dA9tSP9XtYFFHIJaDD5Ya8Uxf/CMbpc6pLvYcryPOhw/InWQIlKQOEth4KOFczzuRBaD61RaUm7Ab0JuIT+8PlwfkR7vyiXsfkGo/chhTi+NUAyw1aAMSnIheTgFHzofgCWEL4JggQUFv9vmFUIRA</ins>+<ins style="font-weight: bold; text-decoration: none;">mGZBgYwu6gDR88rfDvvF2foNstFv7275t56F</ins>+<ins style="font-weight: bold; text-decoration: none;">OWnEIervxsJFlCy0BYLAyEpBRhmsjW7d5bydY5D15TG7Bke5dsirZUJzufDLPy</ins>/<ins style="font-weight: bold; text-decoration: none;">Mf7CS3Fx7QuyV13</ins>+<ins style="font-weight: bold; text-decoration: none;">7K5Cb7qdDkI4e0vQfCBvju3vnEEJ5K0qZwjGoGWbkErYKQnwQnPpQR4ZIbQkBWBAbfflNG9xgVaLLL6</ins>/<ins style="font-weight: bold; text-decoration: none;">z2614K9OvvKZD6v9Of6</ins>/<ins style="font-weight: bold; text-decoration: none;">M</ins>+<ins style="font-weight: bold; text-decoration: none;">1cv79kim1ZNIpoU7UMeQSHMiQckM7x5tcPeQiBNBlaSE7sz6rgdV90DdI5XWTyKV5pHU4PAhmCVKQQTF21SaG2gwmglBqSQ73x093Hyj179j3</ins>/<ins style="font-weight: bold; text-decoration: none;">4v3mf</ins>/<ins style="font-weight: bold; text-decoration: none;">BVBLBwicO7LbngoAADs4AABQSwECFAAUAAgICABDjmtCRczeXRoAAAAYAAAAFgAAAAAAAAAAAAAAAAAAAAAAZ2VvZ2VicmFfamF2YXNjcmlwdC5qc1BLAQIUABQACAgIAEOOa0KcO7LbngoAADs4AAAMAAAAAAAAAAAAAAAAAF4AAABnZW9nZWJyYS54bWxQSwUGAAAAAAIAAgB</ins>+AAAANgsAAAAA" showResetIcon = "false" enableRightClick = "false" errorDialogsActive = "true" enableLabelDrags = "false" showMenuBar = "false" showToolBar = "false" showToolBarHelp = "false" showAlgebraInput = "false" useBrowserForJS = "true" allowRescaling = "true" /&gt;</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> </table> Peterpummel http://geometrie.idea-sketch.com/index.php?title=Bresenham-Algorithmus_(in_Arbeit)&diff=22105&oldid=prev Peterpummel: /* Algorithmus für Gerade mit der Steigung m > 1 */ 2013-03-11T16:49:15Z <p><span class="autocomment">Algorithmus für Gerade mit der Steigung m &gt; 1</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="de"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Nächstältere Version</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Version vom 11. März 2013, 16:49 Uhr</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l152">Zeile 152:</td> <td colspan="2" class="diff-lineno">Zeile 152:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Wir wollen nun den Fall für Geraden betrachten deren Steigung größer ist als Eins. Die Vorausetzungen seien wie in dem Fall für die Steigung kleiner Eins. Im Fall der Steigung kleiner gleich Eins musste die Stiftposition bei jedem Zeichenschritt in x-Achsenrichtung inkrementiert werden und es wurde anhand einer Gleichung entschieden  ob wir auch in y-Richtung inkrementieren müssen. Im Falle der Steigung größer Eins ist es nun genau umgekehrt, wir müssen bei jedem Zeichenschritt in y-Richtung inkrementieren und an Hand der Abstandsgleichung entscheiden ob wir dies auch in x-Richtung tun müssen.&lt;br&gt;&lt;br&gt;</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Wir wollen nun den Fall für Geraden betrachten deren Steigung größer ist als Eins. Die Vorausetzungen seien wie in dem Fall für die Steigung kleiner Eins. Im Fall der Steigung kleiner gleich Eins musste die Stiftposition bei jedem Zeichenschritt in x-Achsenrichtung inkrementiert werden und es wurde anhand einer Gleichung entschieden  ob wir auch in y-Richtung inkrementieren müssen. Im Falle der Steigung größer Eins ist es nun genau umgekehrt, wir müssen bei jedem Zeichenschritt in y-Richtung inkrementieren und an Hand der Abstandsgleichung entscheiden ob wir dies auch in x-Richtung tun müssen.&lt;br&gt;&lt;br&gt;</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>&lt;ggb_applet width="<del style="font-weight: bold; text-decoration: none;">1260</del>" height="<del style="font-weight: bold; text-decoration: none;">880</del>"  version="4.2" ggbBase64="<del style="font-weight: bold; text-decoration: none;">UEsDBBQACAgIANN7a0IAAAAAAAAAAAAAAAAWAAAAZ2VvZ2VicmFfamF2YXNjcmlwdC5qc0srzUsuyczPU0hPT</del>/<del style="font-weight: bold; text-decoration: none;">LP88zLLNHQVKiu5QIAUEsHCEXM3l0aAAAAGAAAAFBLAwQUAAgICADTe2tCAAAAAAAAAAAAAAAADAAAAGdlb2dlYnJhLnhtbN1b23LbuBm</del>+<del style="font-weight: bold; text-decoration: none;">zj4FRhed7dSmcT6kcnaSbdNNaseZddrZ9iZDSZDENUVqScqHTB6nL9B9hd73mfoDIHW0vKbtOK4msUGAOP3ffwbo7neXkxSd26JM8uywQyLcQTbr54MkGx12ZtVwX3e</del>+<del style="font-weight: bold; text-decoration: none;">e</del>/<del style="font-weight: bold; text-decoration: none;">FNd2Tzke0VMRrmxSSuDjs8op3FOKhFhLnByQBm6ffskBKyL</del>+<del style="font-weight: bold; text-decoration: none;">Oe2Ocxifd7Q4z3BbNqCD9ccdJB6LJMnmf5u3hiy2nct6f9sZ3ER3k</del>/<del style="font-weight: bold; text-decoration: none;">rvyc46qaPj84uLi4iJrVo7wYHYxGveiyHHQQ7DwrDzv1w3OYbmXQBfPdKcbk4KfjozD9fpKVVZz1bQc5qmbJi2</del>+edS+<del style="font-weight: bold; text-decoration: none;">SbJBfoItkUI1h91SZDhrbZDQGOo0Gog5crykQO7X9Kjm3JYxdqnqiq8m047vFmXv</del>/<del style="font-weight: bold; text-decoration: none;">LDyhdE5PBw2S82Rgi8MOjihTUgtpiOFSM8Z1B</del>+<del style="font-weight: bold; text-decoration: none;">VFYrOq7kzqRQ</del>+<del style="font-weight: bold; text-decoration: none;">a6brnib0I87onvyTHRgETkjLppfawM4zTEuhKsmEBmMKOihlUy</del>+<del style="font-weight: bold; text-decoration: none;">oqtb24aOqLDZE9</del>+<del style="font-weight: bold; text-decoration: none;">Acdkk/WzQV0BiAOO9TgPUrEnsJ4T4gagOWFO6jK89TPipEw6PNnRDHFaM8VJBQUCinDKxzaMAsFDQUPhQh9eBjOQ1ce</del>+<del style="font-weight: bold; text-decoration: none;">vDQh7Mb6KzrC0LrhhVKGzrZMp0E6HM/En48AGt06iU6iSPiMyJu975gyO2b</del>+<del style="font-weight: bold; text-decoration: none;">P27gtdVGarKFwSHgtQvtfvl8ZL3pIjdiSKytGqQh</del>+<del style="font-weight: bold; text-decoration: none;">2LbshLsyKhcmlJxkFYzPYlaRtC19dcopIvlhQgm15GxRYq7wnunFAulhYFXXD//c</del>/<del style="font-weight: bold; text-decoration: none;">GkuxeZC5WFPi2K0p</del>+<del style="font-weight: bold; text-decoration: none;">H92/w4IKr6h9o</del>/<del style="font-weight: bold; text-decoration: none;">OhJHV5EwwPtqnuQWMNu</del>/<del style="font-weight: bold; text-decoration: none;">WGUDl2fWuRruykdFtkxhsnRJAA5ZUKbIlAxEChnBJTRATiAqpEI</del>+<del style="font-weight: bold; text-decoration: none;">lKhZjTW44Y0sj1Iwx5EyQ0</del>/<del style="font-weight: bold; text-decoration: none;">OJepyUSMJdrVEG5EeNIMES84eIIUEDe</del>+<del style="font-weight: bold; text-decoration: none;">AEmlEEPIZCAQW514pZlEnEJFaYRhw06s6ecaWEwDuqwOEWMIObGEoWoRJIi5Uwn4c6iSu32DpNSJDGSbijYTrCbwWbCCI2Yowa0YJqXyRzcsU2nc654HJNsOqtq7Or2/mTQ4Fjla90Hef/s1RrYNi6r5hk6gcdaOMbgwVb85rNuGvdsCuHFqZMDhM7j1Km5n3</del>+<del style="font-weight: bold; text-decoration: none;">YZxWaG5nQNiri6Tjpl6e2qmBUiX6Oz</del>+<del style="font-weight: bold; text-decoration: none;">OjuLKXr6F32WzQL</del>+<del style="font-weight: bold; text-decoration: none;">39edfO</del>+<del style="font-weight: bold; text-decoration: none;">mkySOLs7yAkbgo3IVq4d2e8GveuRb1MP8</del>+<del style="font-weight: bold; text-decoration: none;">LwelVCaKDLv9pixwMFxaRVBQzhiU3WDJQuKvwSjMaCamYEYRwKkHoy36ceslVkVKUY0GMIuDgYcj1r5TmYWV7PictvrRzgtCocEq3VHlTvsrTRdM0T7Lq</del>+<del style="font-weight: bold; text-decoration: none;">3hazQofrIGpLBxRL7NRaj243vZC2NM</del>/<del style="font-weight: bold; text-decoration: none;">6+WXpwFVFub6cDWFGg476I2+z9O8QKCSVIDFG9VlL5S+j9vavBf2fbDvgRs2JYP5e2Ko7</del>+<del style="font-weight: bold; text-decoration: none;">HLXih9L</del>+<del style="font-weight: bold; text-decoration: none;">B72FpNKmnIJLhZJim9sYHJV8TSS40LomZZUh01lSrpn9WkkjDg3WzSA4GrJXJ1TvJQc3YP1mSse2aLzKZBkDJg5iyflUG05</del>+<del style="font-weight: bold; text-decoration: none;">L5rDsr7fu4Gr</del>/<del style="font-weight: bold; text-decoration: none;">MBj</del>/<del style="font-weight: bold; text-decoration: none;">aESjl+9jZxQqmDl0XWx7YfjKBgaG9Bi92jP0bbDW0DuyosA2JqY+PA7T+LV4W641mP9XrIp+8yc4</del>/<del style="font-weight: bold; text-decoration: none;">gNSsbbV70NDTLftFMnXSiXpgqM</del>/<del style="font-weight: bold; text-decoration: none;">sQv4GSRmDmR8sjwPiS6Ci70wOAFk5EDsonlXjvPARMOgtOBL09j</del>//<del style="font-weight: bold; text-decoration: none;">yjJbgKkEgXQqezktbOlyiYYpH</del>+<del style="font-weight: bold; text-decoration: none;">xlBfDDi8PO736Z5dUf36Nvf</del>/<del style="font-weight: bold; text-decoration: none;">qY7KF</del>/<del style="font-weight: bold; text-decoration: none;">fEx+H1r8kja1E4ibUeUlGsxC1Vmdw1sA4CLKez</del>+<del style="font-weight: bold; text-decoration: none;">DGZq7ndBniT3wfouEozidjmMXt9f4pfEV7H4ZUT</del>/<del style="font-weight: bold; text-decoration: none;">dcT5YxdkbtNIWyXDu</del>/<del style="font-weight: bold; text-decoration: none;">UANj4GlLleaG7rgE</del>+<del style="font-weight: bold; text-decoration: none;">ejIEMpqvdOvSFJ8kkZRD8SlIhRITVxOnzlp1DCELAkVFNFXHz0KWRwQUIDLlvBpSvgni7ARX9ApAXA9MkA3ERZ7RGGZcHIK0g6GBaKMC2Mh5gDxFRpKSDOkYYa0g5itgLxBw</del>+<del style="font-weight: bold; text-decoration: none;">xQ</del>/<del style="font-weight: bold; text-decoration: none;">cuMLMdgJlFmHJwmBC</del>+<del style="font-weight: bold; text-decoration: none;">Y0iboGhQ5gZEWAimDBGabkV5BSLv</del>/+<del style="font-weight: bold; text-decoration: none;">b0v7w</del>/<del style="font-weight: bold; text-decoration: none;">Pt7n3RWhOAOD7c0ehBDT4Dqm1gavEzYMD1OYzjvrpe14E13Wit4AsgzBs64n1TnwlWgptK7Z9tvi9WoH8KKRYQIEihnGNcFSN</del>+<del style="font-weight: bold; text-decoration: none;">gRpqRhUivBIUb74lh</del>+<del style="font-weight: bold; text-decoration: none;">vwNYikeUvT</del>/<del style="font-weight: bold; text-decoration: none;">tAF64xgtH4mHwutaD8BUPcoK</del>+<del style="font-weight: bold; text-decoration: none;">xXu4hdPgO</del>+<del style="font-weight: bold; text-decoration: none;">A0YFlqGIW0jQkmCCfMpVseeSrBlRCtteSYcGXaeWexgu3gY3l7XMXdcG2SqyeCLIkIpLgac80UddFlYz</del>+<del style="font-weight: bold; text-decoration: none;">pJi7soYwzSahuB6tcg7W6PazyPuLaZKxfHVYaQZSuCDeCEkkUNtLDyiKpBcecQ3ApGePbY8kbLOefd8ByNlGOfARP83oH8GKPiNdfdgAv8Yh4</del>/<del style="font-weight: bold; text-decoration: none;">bADeDXyRR8BrzebeK2exS2OwJ4yXtIIIYxzmRgzE7JYFUESAm4UY60huTUPk3b088kkzgYo8xcoR0lmPYLh3D7GLpZGMXHABtBmVfMiDlPVE2zwxZ3EzlGPv2qcuIB2X0Z6HcSr6xEHbPdJJBhXShnwqERgYlS9zPyUuRon</del>/<del style="font-weight: bold; text-decoration: none;">bMMghR</del>/<del style="font-weight: bold; text-decoration: none;">FF7Vh97</del>+<del style="font-weight: bold; text-decoration: none;">4YdkMLD</del>+<del style="font-weight: bold; text-decoration: none;">XsSPsb9kYUjt25PJNE36SdWaIS8DQ15tMKTXgiG939KTR</del>+<del style="font-weight: bold; text-decoration: none;">MIXsumtZRB3slm8u14oiIXOioJkSUBpjH22Cx5k7kDdYBsjS</del>+<del style="font-weight: bold; text-decoration: none;">9wJd4gy9vb</del>+<del style="font-weight: bold; text-decoration: none;">bLqgF7e0fGOHhGoeiF4v68oThSAkPozowhggoV4ndBI4MJFhDZUwqBfH0uzCIKuZIQFJgIT0w</del>+<del style="font-weight: bold; text-decoration: none;">oF06tSPXfr0mvN1AvH8z4mU9WwNp/ytnW6vagDnYJWEUU9pQzGptwJFQWIKmcEIVwxD4B3WAZEBpMFhMKkUY4fwGdeA3qwMAmDq3NBdwcGObd0xn1k7d5d5J9qGIs9J9rrV6udSWi2</del>+<del style="font-weight: bold; text-decoration: none;">32bNBOy4OvnJyt8ZFA4mZY4kSmmhaOxoSMcmVUVJQijVYsNqoiYhzwzWGRJkLybR</del>+<del style="font-weight: bold; text-decoration: none;">Uly8yV79dQcCVPLAAeqq1J8U1Tgf5Vmcbnfmm07DtnDm9sk48y2h63Vh1yf3yVXEJeWMGME5gwf52L78Rt682sabYQveDP9feSMjbpztgsiLaqrNTY7lUeMsu40tR23irKMnFGdpCKiMkJwqSgWX9fEzhF0RKAUHbjFBlcHKM0ZIaMXUuRamlMbsAU3VNsiH2yA</del>/<del style="font-weight: bold; text-decoration: none;">bgP58ROCnGDI7cAFKwxxLeaC1ZgLHknh7o8h7TO0vin</del>+<del style="font-weight: bold; text-decoration: none;">kpBfHxPVpud4A</del>/<del style="font-weight: bold; text-decoration: none;">Bxu5ho</del>/<del style="font-weight: bold; text-decoration: none;">JRiIkgllHJXLVwLbOqICHJswqiGoEhr7i4LZMBcRhhCJwKGSkqIbfE90ryvFNfWrv1og4ejdjwcPZnsBEOGR6mR4A4g74NEj8g6N4GUD7gEOQukiUq7j</del>+<del style="font-weight: bold; text-decoration: none;">lDbuLSds4VKBqFvJ7cw4d8ibD22usltXK91OLOTu3KnR0Rxn09A6m</del>/<del style="font-weight: bold; text-decoration: none;">Ac9TH5BBsgLclYZgykjLGzu9AmmL</del>+<del style="font-weight: bold; text-decoration: none;">zq9I</del>/<del style="font-weight: bold; text-decoration: none;">d1jFLQD4ElU4SAMtTXoNCsiYL4l3ChNdsG6qqx8fOumZpNx</del>/<del style="font-weight: bold; text-decoration: none;">yujWN</del>+<del style="font-weight: bold; text-decoration: none;">d7dDc0KD5PryKeRx1EW4khkM1gkQ5Tw4dQEirZghXGItwB49zPcUd</del>+<del style="font-weight: bold; text-decoration: none;">HKSRuunOwIV5yDN1iC0wcHoJhQzZE7J9R5B</del>+<del style="font-weight: bold; text-decoration: none;">HOuQz</del>/<del style="font-weight: bold; text-decoration: none;">AkzxH6pvi6xQTGs1WWHRf</del>/<del style="font-weight: bold; text-decoration: none;">99M4</del>/<del style="font-weight: bold; text-decoration: none;">8h81zFkBvNx42M6u3RNw5nQIppFIZro2sc9o7GjGCN</del>/<del style="font-weight: bold; text-decoration: none;">0CuSUPm35tvG3zJXjRXwK</del>/<del style="font-weight: bold; text-decoration: none;">EYY0zS9</del>+<del style="font-weight: bold; text-decoration: none;">tMPUXnpkbxseXceGeXDk2XCyyYZfW7Hh199ig</del>+<del style="font-weight: bold; text-decoration: none;">B3ZMN29</del>/<del style="font-weight: bold; text-decoration: none;">z02XCTdXm</del>/<del style="font-weight: bold; text-decoration: none;">A</del>+<del style="font-weight: bold; text-decoration: none;">d24oHP7Q6W</del>/<del style="font-weight: bold; text-decoration: none;">7rA</del>/<del style="font-weight: bold; text-decoration: none;">7lP</del>/<del style="font-weight: bold; text-decoration: none;">WfBL/4HUEsHCLkXsPkhCwAAxjwAAFBLAQIUABQACAgIANN7a0JFzN5dGgAAABgAAAAWAAAAAAAAAAAAAAAAAAAAAABnZW9nZWJyYV9qYXZhc2NyaXB0LmpzUEsBAhQAFAAICAgA03trQrkXsPkhCwAAxjwAAAwAAAAAAAAAAAAAAAAAXgAAAGdlb2dlYnJhLnhtbFBLBQYAAAAAAgACAH4AAAC5CwAAAAA=</del>" showResetIcon = "false" enableRightClick = "false" errorDialogsActive = "true" enableLabelDrags = "false" showMenuBar = "false" showToolBar = "false" showToolBarHelp = "false" showAlgebraInput = "false" useBrowserForJS = "true" allowRescaling = "true" /&gt;</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>&lt;ggb_applet width="<ins style="font-weight: bold; text-decoration: none;">1269</ins>" height="<ins style="font-weight: bold; text-decoration: none;">888</ins>"  version="4.2" ggbBase64="<ins style="font-weight: bold; text-decoration: none;">UEsDBBQACAgIABOOa0IAAAAAAAAAAAAAAAAWAAAAZ2VvZ2VicmFfamF2YXNjcmlwdC5qc0srzUsuyczPU0hPT</ins>/<ins style="font-weight: bold; text-decoration: none;">LP88zLLNHQVKiu5QIAUEsHCEXM3l0aAAAAGAAAAFBLAwQUAAgICAATjmtCAAAAAAAAAAAAAAAADAAAAGdlb2dlYnJhLnhtbN1b23LbuBm</ins>+<ins style="font-weight: bold; text-decoration: none;">zj4FRhed7dSmcT6kcnaStOkmtePMOu10epOhJEjimiK1JOVDJo/TF+i</ins>+<ins style="font-weight: bold; text-decoration: none;">Qu/7TP0BkDpaXtN2HFeTyBBAnP7vPwNU94fLSYrObVEmeXbYIRHuIJv180GSjQ47s2q4rzs</ins>/<ins style="font-weight: bold; text-decoration: none;">vPiuO7L5yPaKGA3zYhJXhx0e0c5iHNQiwtzgZACz9Ht2SAnZl3FP7POYxPu9Icb7glk1hA9XnHQQuiyT51n</ins>+<ins style="font-weight: bold; text-decoration: none;">Pp7Ychr37Wl/bCfxUd6PKz/nuKqmzw8OLi4uomb1KC9GB6NRL7osBx0EO8/Kw0795TlMtzLogvnuFGNy8I</ins>/<ins style="font-weight: bold; text-decoration: none;">jozD9fpKVVZz1bQc5qmbJi+</ins>+edS+<ins style="font-weight: bold; text-decoration: none;">SbJBfoItkUI1h91TrDhrbZDQGOo2rHLheUyB2avtVcm5LGLtU9URXk2nHd4sz9</ins>/<ins style="font-weight: bold; text-decoration: none;">xZ</ins>+<ins style="font-weight: bold; text-decoration: none;">IbSOT0dNEjOk4EtDjs4okxiyjXnimJupOmgvEhsVtV9Sb3mQTNb9zyxF2Fa982vyLFRwIOkTHqpPewM47QEspJsWACksKFiBtWyukptLy6a</ins>+<ins style="font-weight: bold; text-decoration: none;">mI/ZA/</ins>+<ins style="font-weight: bold; text-decoration: none;">QYfks3VzAe8CDocdavAeJWJPYbwnBA57WV64g6o8T/2sGAmDvnxBFFOM9lxBQkGhkDI8wqENs1DQUPBQiNCHh</ins>+<ins style="font-weight: bold; text-decoration: none;">E8dOWhDw99OLuBzrq</ins>+<ins style="font-weight: bold; text-decoration: none;">ILRuWKG0oZMt00mAPveR8PEArNGpl</ins>+<ins style="font-weight: bold; text-decoration: none;">gkjogviLjd</ins>+<ins style="font-weight: bold; text-decoration: none;">4Iht2</ins>/<ins style="font-weight: bold; text-decoration: none;">i9</ins>+<ins style="font-weight: bold; text-decoration: none;">8KXldlqCpfEBwKUj</ins>/<ins style="font-weight: bold; text-decoration: none;">U7o</ins>/<ins style="font-weight: bold; text-decoration: none;">HS96TInYnisjSqkEeti</ins>+<ins style="font-weight: bold; text-decoration: none;">6IS</ins>/<ins style="font-weight: bold; text-decoration: none;">NioQ6eW2WZByExWxfkrYhdH3NJSr5YkkBsullVGyh8p7gzgnlYmlR0AX33382lmT3InOxosC3XVHy</ins>++<ins style="font-weight: bold; text-decoration: none;">j</ins>+<ins style="font-weight: bold; text-decoration: none;">HRZUeEXtG50PJanLm2B4sE11Dxpr2K03hMqx61uLdGUnpdsiM944IYIEKK9UYEsEIgYK5ZSYIiIQF1AlGklXKsSc3nLEkEauH2HImyCh4Q/3Oi2RgLlcowrKjRhHgiHiDRdHgALyxg8woQx6CIEEDHKrE7csk4hLqDCNOGzQmT3lTAuDcVCHxSliBDE3lihEJZIUKWc6CXcWVWq3d5iUIomRdEPBdoLdDDYTRmjEHDWgBdO8TObgjm06nXPF45hk01lVY1e39yeDBscqX</ins>+<ins style="font-weight: bold; text-decoration: none;">s</ins>+<ins style="font-weight: bold; text-decoration: none;">yPtnr9bAtnFZNd</ins>+<ins style="font-weight: bold; text-decoration: none;">hE3ishV8MHmzFbT7rpnHPphBdnDo5QOg8Tp2a</ins>+/<ins style="font-weight: bold; text-decoration: none;">mHeVahuZEJbaMino6TfnlqqwpGlejn</ins>+<ins style="font-weight: bold; text-decoration: none;">Dw</ins>+<ins style="font-weight: bold; text-decoration: none;">iit7</ins>+<ins style="font-weight: bold; text-decoration: none;">QZ6l80G</ins>/<ins style="font-weight: bold; text-decoration: none;">dLenXftrJ8mgyTO</ins>/<ins style="font-weight: bold; text-decoration: none;">g5C4qZwE6KFd3fGq</ins>/<ins style="font-weight: bold; text-decoration: none;">HuWuKwTD</ins>/<ins style="font-weight: bold; text-decoration: none;">Pi8HpVQmigy7</ins>/<ins style="font-weight: bold; text-decoration: none;">aYscDBcWkQT3zBiW3GDJwNlehUea0UhIxYwghFMJQl</ins>/<ins style="font-weight: bold; text-decoration: none;">249RLroqUohwLYhQR0oABu7r</ins>+<ins style="font-weight: bold; text-decoration: none;">mdI8LG3P57TFl3ZOERoVTuuWKm</ins>/<ins style="font-weight: bold; text-decoration: none;">LV3m6aJrmSVa9jqfVrPDBGixVOKpeZqPUenS98YWwp3</ins>/<ins style="font-weight: bold; text-decoration: none;">Wyy9PA6wszPXxagq1mvje6HWe5gUCnaQCTN6oLnuh9H3c1ua9sO</ins>+<ins style="font-weight: bold; text-decoration: none;">DfQ</ins>/<ins style="font-weight: bold; text-decoration: none;">c8CkZzJ8TQ30PX</ins>/<ins style="font-weight: bold; text-decoration: none;">ZC6XsB48PWalJJQybBzTJJ6a0NTL4il15sXBA1y5LqqKlUSf</ins>+<ins style="font-weight: bold; text-decoration: none;">sJpWEAe9nkx5IXC2Sq3OSh5qze7AmZN0zW2Q2DZKUATNn</ins>+<ins style="font-weight: bold; text-decoration: none;">awMsj2Xz2fdWWk/xNX4ZTb4yY5AKz</ins>/<ins style="font-weight: bold; text-decoration: none;">EzjBWMHXoutjywPaTCQwM7TV4sWPs32CroXVgR4VtSEx9fByg9U9X5Hqj2U/1psgnb7PzjyA1a1vtHjT0dMt</ins>+<ins style="font-weight: bold; text-decoration: none;">kUyddKIeWOozu5C</ins>/<ins style="font-weight: bold; text-decoration: none;">QVLGYOcHy</ins>+<ins style="font-weight: bold; text-decoration: none;">OA+BKo6DubA0BWDsQOimfVOC98BAyKC54EvfvPv7LMFmArQSCdzl5OC1u6XKJhykd7WQH88OCw87tfZnn1xw</ins>/<ins style="font-weight: bold; text-decoration: none;">o</ins>++<ins style="font-weight: bold; text-decoration: none;">mnyz00</ins>/<ins style="font-weight: bold; text-decoration: none;">XT1</ins>+<ins style="font-weight: bold; text-decoration: none;">9Dil7SpnUDgjCov0WAXqs7qHN4EABdR3vsZ7NDc74Q</ins>+<ins style="font-weight: bold; text-decoration: none;">S</ins>+<ins style="font-weight: bold; text-decoration: none;">yB51skHMXpdBy7uL3GL42vYPfLiPrpjvPBKs7eopW2SIZz9wdqeAwsdbnS3NIFpzgfBRlKUX1w6g1Jkk</ins>/<ins style="font-weight: bold; text-decoration: none;">KIPyRoESMCqmJ0+ErP4UShoAloZoq4gKkzyGDCxIacNkKLl0B93QBLvoDIi0Apk8G4CbMao8wLAtWXkHWwbBQhGlhPMQcIKZKSwGBjjTUkHYQsxWIP3qIHbp3gZntAMwsgpwXPCbE7xjyJigalLkBERaCKUOEpltRXoHI+785</ins>/<ins style="font-weight: bold; text-decoration: none;">S</ins>/<ins style="font-weight: bold; text-decoration: none;">vj4/3eXdFKM7AYHuzBzHENLiOqbXB64QNw5cpTOed9dJ2vIkua0VvAFmG4FnXk+oc</ins>+<ins style="font-weight: bold; text-decoration: none;">Eq4FFrXbPtt8Xq1A3jRyDABAsUM45pgqRv0CFMQekmtBIcg7atj</ins>+<ins style="font-weight: bold; text-decoration: none;">XoHsBSPKHt/2gG8cI0XjsTD4HWtB</ins>+<ins style="font-weight: bold; text-decoration: none;">ErHuQEfY/3cAunwXfAacCy1DAKeRsTTBBOmMu3PPJUgishWmvJMeHKtPPOYgXbwafy9riKu</ins>+<ins style="font-weight: bold; text-decoration: none;">HaJFdPBFkSEchxNeaaKeqiy8Z</ins>+<ins style="font-weight: bold; text-decoration: none;">Uk1c2EMZZ5JQ3Q5WuQZrdXtY5X3EtclYvzmsNIIoXRFuBCWSKGykh5VFUguOOYfgUjLGt8eSN1jOP</ins>++<ins style="font-weight: bold; text-decoration: none;">A5WyiHPkInubNDuDFHhGvv</ins>+<ins style="font-weight: bold; text-decoration: none;">wAXuIR8fpxB</ins>/<ins style="font-weight: bold; text-decoration: none;">Bq5Is</ins>+<ins style="font-weight: bold; text-decoration: none;">Al5vN/FaPYtbHIE9ZbykEUIY5zIxZiZksSqCJATcKMZaQ3JrHibt6OeTSZwNUOZvUI6SzHoEw8F9jF0sjWLigA2gzarmQRymqifY4Is7iZ2jHn</ins>/<ins style="font-weight: bold; text-decoration: none;">TOHEB7b6M9DqIV9cjDtjuk0gwrpQy4FGJwMSoepn5KXM1TvpnGQQp</ins>/<ins style="font-weight: bold; text-decoration: none;">ii8qg</ins>+<ins style="font-weight: bold; text-decoration: none;">9</ins>/<ins style="font-weight: bold; text-decoration: none;">Zcfk8HA+osRP8b</ins>+<ins style="font-weight: bold; text-decoration: none;">koUhtW9PJtM06SdVa4a8DAx5tcGQXguG9H5LTx6NI3gtm9ZSBnknm8m344mKXOioJESWBJjG2GOz5G3mDtQBsjW+9AJf4g2+vLuZL6sG7N0dGePgGYWiF4r784biSAkMoTszhggqVIjfBY0MJlhAZE8pBPL1uTCLKORKQlBgInxj8gHt0qkdufbrNeHdBuL9mxEv69kaSPvfONta1QbMwS4Jo5jShmJWawOOhMISNIUTqhiGwD+oAyQDSoPBYlIpwgjnN6gDv1kdAMDUuaW5gIMb27xjOrN26i73TrKPRZyV7nWt1cultlx8t82eDdpxcfCNk7s1LhpIzBxLlNBE09rRkIhJroySglKswYLVRk1EnBuuMSTKXEim9ZPi4k326q87EKCSBw5QV6X+pKjG+SjP4nS7M990GraFM7dPxplvCV2vC7s+u3euIi4pZ8QIzhl8kY/ty2/kzattvBm24M3w</ins>/<ins style="font-weight: bold; text-decoration: none;">5U3MuLG2S6IvKim2tzkWB41zrLb2HLUJs46ekJxloaAygjJqaJUcFkfP0PYFYFScOAWE1QZrDxjhIRWTJ1rYUppzB7QVG2DfLgN8uM2kB8</ins>/<ins style="font-weight: bold; text-decoration: none;">IcgJhtwOXLDCENdiLliNueCRFO7+GNI+Q+ub4q8J</ins>+<ins style="font-weight: bold; text-decoration: none;">fUxUW16jjcAH7eLicZPKSaCVEIpd9XCtcCmjoggxyaMagiKtObuskAGzGWEIXQiYKikhNgW3yPN</ins>+<ins style="font-weight: bold; text-decoration: none;">0Zxbe3ajzZ4OGrHw9GTyU4wZHiUGgnuAPI</ins>+<ins style="font-weight: bold; text-decoration: none;">SPSIrHMTSPmAS5CzQJqotHubPuQmLm33P0GAIVKSe</ins>/<ins style="font-weight: bold; text-decoration: none;">iQrxHWXnu9pFaul1rc2aldubMjwri3ZyD1N+B56gMySFaAu9IQTBlpeWOnVyBtcV+nd+S+jlEK+iHcK76EgDLU16DQrImC+JdwoTXbBuqqsfHzrpmaTcf8vo1jfn+3Q3NCg+T68inkcdRFuJIZDNYJEOU8OHUBIq2YIVxiLcAePcz7FHfhykkbrpzsCFecgzdYgtMHB6CYUM2ROyfUeQfhzrkM</ins>/<ins style="font-weight: bold; text-decoration: none;">wpM8S+qb4usUExrNVlh0X</ins>//<ins style="font-weight: bold; text-decoration: none;">fTOP</ins>/<ins style="font-weight: bold; text-decoration: none;">IvNcxZAbzceNjOrt0TcOZ0CKaRSGa6NrHPaOxoxgjf9ArklD5t</ins>+<ins style="font-weight: bold; text-decoration: none;">bbxt8yZ40V8CvxGGNM0vfrLD1F56ZG8bHl3Hhnlw5NlwssmGX1ux4dffYoPgd2TDdvf89Nlwk3X5sAPnduKBz</ins>+<ins style="font-weight: bold; text-decoration: none;">0Oln9d4H</ins>/<ins style="font-weight: bold; text-decoration: none;">vU</ins>/<ins style="font-weight: bold; text-decoration: none;">8s</ins>+<ins style="font-weight: bold; text-decoration: none;">MX</ins>/<ins style="font-weight: bold; text-decoration: none;">AFBLBwgE5F</ins>/<ins style="font-weight: bold; text-decoration: none;">XIgsAAMY8AABQSwECFAAUAAgICAATjmtCRczeXRoAAAAYAAAAFgAAAAAAAAAAAAAAAAAAAAAAZ2VvZ2VicmFfamF2YXNjcmlwdC5qc1BLAQIUABQACAgIABOOa0IE5F</ins>/<ins style="font-weight: bold; text-decoration: none;">XIgsAAMY8AAAMAAAAAAAAAAAAAAAAAF4AAABnZW9nZWJyYS54bWxQSwUGAAAAAAIAAgB+AAAAugsAAAAA</ins>" showResetIcon = "false" enableRightClick = "false" errorDialogsActive = "true" enableLabelDrags = "false" showMenuBar = "false" showToolBar = "false" showToolBarHelp = "false" showAlgebraInput = "false" useBrowserForJS = "true" allowRescaling = "true" /&gt;</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&lt;br&gt;&lt;br&gt;</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&lt;br&gt;&lt;br&gt;</div></td></tr> </table> Peterpummel http://geometrie.idea-sketch.com/index.php?title=Bresenham-Algorithmus_(in_Arbeit)&diff=22104&oldid=prev Peterpummel: /* Zusammenfassung */ 2013-03-11T16:47:45Z <p><span class="autocomment">Zusammenfassung</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="de"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Nächstältere Version</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Version vom 11. März 2013, 16:47 Uhr</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l171">Zeile 171:</td> <td colspan="2" class="diff-lineno">Zeile 171:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Also für Steigungen mit &lt;math&gt; |m| &lt;= 1 &lt;/math&gt;&lt;br&gt;&lt;br&gt;  </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Also für Steigungen mit &lt;math&gt; |m| &lt;= 1 &lt;/math&gt;&lt;br&gt;&lt;br&gt;  </div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>&lt;math&gt;= - ( n_x \cdot 2 \cdot p_x \ + 2 \cdot n_x + n_y \cdot 2 \cdot p_y + n_y ) =: d &lt;/math&gt;&lt;br&gt;&lt;br&gt;</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>&lt;math&gt;= - ( <ins style="font-weight: bold; text-decoration: none;">|</ins>n_x<ins style="font-weight: bold; text-decoration: none;">| </ins>\cdot 2 \cdot <ins style="font-weight: bold; text-decoration: none;">|</ins>p_x<ins style="font-weight: bold; text-decoration: none;">| </ins>\ + 2 \cdot <ins style="font-weight: bold; text-decoration: none;">|</ins>n_x<ins style="font-weight: bold; text-decoration: none;">| </ins>+ <ins style="font-weight: bold; text-decoration: none;">|</ins>n_y <ins style="font-weight: bold; text-decoration: none;">|</ins>\cdot 2 \cdot <ins style="font-weight: bold; text-decoration: none;">|</ins>p_y<ins style="font-weight: bold; text-decoration: none;">| </ins>+ <ins style="font-weight: bold; text-decoration: none;">|</ins>n_y<ins style="font-weight: bold; text-decoration: none;">| </ins>) =: d &lt;/math&gt;&lt;br&gt;&lt;br&gt;</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Also für Steigungen mit &lt;math&gt; |m| &gt; 1 &lt;/math&gt;&lt;br&gt;&lt;br&gt;</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Also für Steigungen mit &lt;math&gt; |m| &gt; 1 &lt;/math&gt;&lt;br&gt;&lt;br&gt;</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>&lt;math&gt;= n_x \cdot 2 \cdot <del style="font-weight: bold; text-decoration: none;">x_i </del>+ n_x + 2 \cdot n_y \cdot <del style="font-weight: bold; text-decoration: none;">y_i </del>+ 2 \cdot n_y =: d&lt;/math&gt;&lt;br&gt;&lt;br&gt;</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>&lt;math&gt;= <ins style="font-weight: bold; text-decoration: none;">|</ins>n_x<ins style="font-weight: bold; text-decoration: none;">| </ins>\cdot 2 \cdot <ins style="font-weight: bold; text-decoration: none;">|p_x| </ins>+ <ins style="font-weight: bold; text-decoration: none;">|</ins>n_x<ins style="font-weight: bold; text-decoration: none;">| </ins>+ 2 \cdot <ins style="font-weight: bold; text-decoration: none;">|</ins>n_y<ins style="font-weight: bold; text-decoration: none;">| </ins>\cdot <ins style="font-weight: bold; text-decoration: none;">|p_y| </ins>+ 2 \cdot <ins style="font-weight: bold; text-decoration: none;">|</ins>n_y<ins style="font-weight: bold; text-decoration: none;">| </ins>=: d&lt;/math&gt;&lt;br&gt;&lt;br&gt;</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=Literaturnachweise=</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=Literaturnachweise=</div></td></tr> </table> Peterpummel http://geometrie.idea-sketch.com/index.php?title=Bresenham-Algorithmus_(in_Arbeit)&diff=22103&oldid=prev Peterpummel: /* Algorithmus für Gerade mit der Steigung m > 1 */ 2013-03-11T16:45:44Z <p><span class="autocomment">Algorithmus für Gerade mit der Steigung m &gt; 1</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="de"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Nächstältere Version</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Version vom 11. März 2013, 16:45 Uhr</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l160">Zeile 160:</td> <td colspan="2" class="diff-lineno">Zeile 160:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&lt;math&gt; |s| - |t| = s + t &lt;/math&gt;&lt;br&gt;&lt;br&gt;</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&lt;math&gt; |s| - |t| = s + t &lt;/math&gt;&lt;br&gt;&lt;br&gt;</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&lt;math&gt;=\vec{n_n} \cdot (\vec{s} - \vec{t}) ) &lt;/math&gt;&lt;br&gt;&lt;br&gt;  </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&lt;math&gt;=\vec{n_n} \cdot (\vec{s} - \vec{t}) ) &lt;/math&gt;&lt;br&gt;&lt;br&gt;  </div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>&lt;math&gt;= \begin{pmatrix} n_x \\ n_y \end{pmatrix}  \cdot  \begin{pmatrix} (<del style="font-weight: bold; text-decoration: none;">x_i</del>) + (<del style="font-weight: bold; text-decoration: none;">x_i </del>+ 1) \\ (<del style="font-weight: bold; text-decoration: none;">y_i </del>+ 1) + (<del style="font-weight: bold; text-decoration: none;">y_i </del>+ 1) \end{pmatrix} &lt;/math&gt;&lt;br&gt;&lt;br&gt;</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>&lt;math&gt;= \begin{pmatrix} n_x \\ n_y \end{pmatrix}  \cdot  \begin{pmatrix} (<ins style="font-weight: bold; text-decoration: none;">p_x</ins>) + (<ins style="font-weight: bold; text-decoration: none;">p_x </ins>+ 1) \\ (<ins style="font-weight: bold; text-decoration: none;">p_y </ins>+ 1) + (<ins style="font-weight: bold; text-decoration: none;">p_y </ins>+ 1) \end{pmatrix} &lt;/math&gt;&lt;br&gt;&lt;br&gt;</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>&lt;math&gt;= n_x \cdot 2 \cdot <del style="font-weight: bold; text-decoration: none;">x_i </del>+ n_x + 2 \cdot n_y \cdot <del style="font-weight: bold; text-decoration: none;">y_i </del>+ 2 \cdot n_y =: d&lt;/math&gt;&lt;br&gt;&lt;br&gt;</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>&lt;math&gt;= n_x \cdot 2 \cdot <ins style="font-weight: bold; text-decoration: none;">p_x </ins>+ n_x + 2 \cdot n_y \cdot <ins style="font-weight: bold; text-decoration: none;">p_y </ins>+ 2 \cdot n_y =: d&lt;/math&gt;&lt;br&gt;&lt;br&gt;</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Auch hier entscheidet nun wieder die Differenz &lt;math&gt; d &lt;/math&gt; welcher Punkt angesteuert werden soll.&lt;br&gt;</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Auch hier entscheidet nun wieder die Differenz &lt;math&gt; d &lt;/math&gt; welcher Punkt angesteuert werden soll.&lt;br&gt;</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> </table> Peterpummel http://geometrie.idea-sketch.com/index.php?title=Bresenham-Algorithmus_(in_Arbeit)&diff=22102&oldid=prev Peterpummel: /* Zusammenfassung */ 2013-03-11T16:43:46Z <p><span class="autocomment">Zusammenfassung</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="de"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Nächstältere Version</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Version vom 11. März 2013, 16:43 Uhr</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l170">Zeile 170:</td> <td colspan="2" class="diff-lineno">Zeile 170:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Durch die beiden Fallunterscheidungen in 2.1 und 2.2 kann nun im ersten Quadranten jede Linie  mit positiver Steigung gezeichnet werden. Wollen wir nun in der Lage sein, auch die negativen Steigungen zu zulassen, Spiegeln wir den Endpunkt der Strecke in den ersten Quadranten, indem wir in die passende Gleichung die Absolutbeträge des Punktes und des Normalenvektors einsetzten. &lt;br&gt;</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Durch die beiden Fallunterscheidungen in 2.1 und 2.2 kann nun im ersten Quadranten jede Linie  mit positiver Steigung gezeichnet werden. Wollen wir nun in der Lage sein, auch die negativen Steigungen zu zulassen, Spiegeln wir den Endpunkt der Strecke in den ersten Quadranten, indem wir in die passende Gleichung die Absolutbeträge des Punktes und des Normalenvektors einsetzten. &lt;br&gt;</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Also für Steigungen mit &lt;math&gt; |m| &lt;= 1 &lt;\math&gt;&lt;br&gt;&lt;br&gt;  </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Also für Steigungen mit &lt;math&gt; |m| &lt;= 1 &lt;<ins style="font-weight: bold; text-decoration: none;">/math&gt;&lt;br&gt;&lt;br&gt; </ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">&lt;math&gt;= - ( n_x </ins>\<ins style="font-weight: bold; text-decoration: none;">cdot 2 \cdot p_x \ + 2 \cdot n_x + n_y \cdot 2 \cdot p_y + n_y ) =: d &lt;/</ins>math&gt;&lt;br&gt;&lt;br&gt;</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Also für Steigungen mit &lt;math&gt; |m| &gt; 1 &lt;\math&gt;&lt;br&gt;&lt;br&gt;</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Also für Steigungen mit &lt;math&gt; |m| &gt; 1 &lt;<ins style="font-weight: bold; text-decoration: none;">/math&gt;&lt;br&gt;&lt;br&gt;</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">&lt;math&gt;= n_x </ins>\<ins style="font-weight: bold; text-decoration: none;">cdot 2 \cdot x_i + n_x + 2 \cdot n_y \cdot y_i + 2 \cdot n_y =: d&lt;/</ins>math&gt;&lt;br&gt;&lt;br&gt;</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=Literaturnachweise=</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=Literaturnachweise=</div></td></tr> </table> Peterpummel http://geometrie.idea-sketch.com/index.php?title=Bresenham-Algorithmus_(in_Arbeit)&diff=22101&oldid=prev Peterpummel: /* Zusammenfassung */ 2013-03-11T15:10:28Z <p><span class="autocomment">Zusammenfassung</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="de"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Nächstältere Version</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Version vom 11. März 2013, 15:10 Uhr</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l168">Zeile 168:</td> <td colspan="2" class="diff-lineno">Zeile 168:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Zusammenfassung ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Zusammenfassung ==</div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Durch die beiden Fallunterscheidungen in 2.1 und 2.2 kann nun im ersten Quadranten jede Linie  mit positiver Steigung gezeichnet werden. Wollen wir nun in der Lage sein, auch die negativen Steigungen zu zulassen, Spiegeln wir den Endpunkt der Strecke in den ersten Quadranten, indem wir in die passende Gleichung die Absolutbeträge des Punktes und des Normalenvektors einsetzten. &lt;br&gt;</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Also für Steigungen mit &lt;math&gt; |m| &lt;= 1 &lt;\math&gt;&lt;br&gt;&lt;br&gt; </ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Also für Steigungen mit &lt;math&gt; |m| &gt; 1 &lt;\math&gt;&lt;br&gt;&lt;br&gt;</ins></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=Literaturnachweise=</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=Literaturnachweise=</div></td></tr> </table> Peterpummel http://geometrie.idea-sketch.com/index.php?title=Bresenham-Algorithmus_(in_Arbeit)&diff=22100&oldid=prev Peterpummel: /* Bresenham-Algortihmus für Geraden */ 2013-03-11T14:51:28Z <p><span class="autocomment">Bresenham-Algortihmus für Geraden</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="de"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Nächstältere Version</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Version vom 11. März 2013, 14:51 Uhr</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l167">Zeile 167:</td> <td colspan="2" class="diff-lineno">Zeile 167:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Für &lt;math&gt; d &lt; 0 &lt;/math&gt; wird der Punkt &lt;math&gt;S&lt;/math&gt; angesteuert.&lt;br&gt;</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Für &lt;math&gt; d &lt; 0 &lt;/math&gt; wird der Punkt &lt;math&gt;S&lt;/math&gt; angesteuert.&lt;br&gt;</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Zusammenfassung ==</ins></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=Literaturnachweise=</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=Literaturnachweise=</div></td></tr> </table> Peterpummel http://geometrie.idea-sketch.com/index.php?title=Bresenham-Algorithmus_(in_Arbeit)&diff=22099&oldid=prev Peterpummel: /* Algorithmus für Gerade mit der Steigung m <= 1 */ 2013-03-11T14:50:47Z <p><span class="autocomment">Algorithmus für Gerade mit der Steigung m &lt;= 1</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="de"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Nächstältere Version</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Version vom 11. März 2013, 14:50 Uhr</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l93">Zeile 93:</td> <td colspan="2" class="diff-lineno">Zeile 93:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=Bresenham-Algortihmus für Geraden=</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=Bresenham-Algortihmus für Geraden=</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Algorithmus für Gerade mit der Steigung m &lt;= 1 ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Algorithmus für Gerade mit der Steigung m &lt;= 1 ==</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Wir <del style="font-weight: bold; text-decoration: none;">wollen uns nun mit der folgenden Problemstellung beschäftigen, wir </del>wollen eine Gerade, welche in Hesseform gegeben ist, mit einem Plotter (z.B. Monitor) zeichnen lassen. Wir wollen für den ersten Teil nur Geraden mit einer Steigung &lt;math&gt; m &lt;= 1 &lt;/math&gt; betrachten.&lt;br&gt;&lt;br&gt;</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Betrachten wir die folgende Problemstellung: </ins>Wir wollen eine Gerade, welche in Hesseform gegeben ist, mit einem Plotter (z.B. <ins style="font-weight: bold; text-decoration: none;">auf einem </ins>Monitor) zeichnen lassen. Wir wollen für den ersten Teil nur Geraden mit einer Steigung &lt;math&gt; m &lt;= 1 &lt;/math&gt; betrachten.&lt;br&gt;&lt;br&gt;</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Sei die Gerade &lt;math&gt; g : \vec{n_n} \cdot ( \vec{r} - \vec{a} ) = 0 &lt;/math&gt; gegeben. Hierbei ist &lt;math&gt; n_n = \begin{pmatrix} n_x \\ n_y \end{pmatrix} &lt;/math&gt; der normierte Normalenvektor, &lt;math&gt; \vec{a}&lt;/math&gt; ein gegebener Ortsvektor eines Punktes der Geraden.&lt;br&gt;</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Sei die Gerade &lt;math&gt; g : \vec{n_n} \cdot ( \vec{r} - \vec{a} ) = 0 &lt;/math&gt; gegeben. Hierbei ist &lt;math&gt; n_n = \begin{pmatrix} n_x \\ n_y \end{pmatrix} &lt;/math&gt; der normierte Normalenvektor, &lt;math&gt; \vec{a}&lt;/math&gt; ein gegebener Ortsvektor eines Punktes der Geraden.&lt;br&gt;</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Wir geben den Startpunkt der Geraden auf unserem Plotter die Koordinaten (0,0), d.h. jede Gerade, die wir zeichnen erhält damit ein eigenes Koordinatensystem.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Wir geben den Startpunkt der Geraden auf unserem Plotter die Koordinaten (0,0), d.h. jede Gerade, die wir zeichnen erhält damit ein eigenes Koordinatensystem.</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Den ersten </del>Punkt den wir zeichnen ist natürlich der Punkt &lt;math&gt;O (0,0)&lt;/math&gt;, denn dies haben wir so festgelegt. Nun müssen wir für den nächsten Punkt entscheiden, ob wir nur ein Pixel (allgemein: einen Plotterschritt) nach rechts zum Punkt &lt;math&gt; T &lt;/math&gt; gehen oder ein Pixel nach rechts und nach oben zum Punkt &lt;math&gt; S &lt;/math&gt;. Um dies zu entscheiden müssen wir die Abstände, die die jeweiligen Punkte zu der idealen Geraden haben, vergleichen. Der Punkt mit dem kleineren Abstand wird dann angesteuert.&lt;br&gt;</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Der erste </ins>Punkt den wir zeichnen ist natürlich der Punkt &lt;math&gt;O (0,0)&lt;/math&gt;, denn dies haben wir so festgelegt. Nun müssen wir für den nächsten Punkt entscheiden, ob wir nur ein Pixel (allgemein: einen Plotterschritt) nach rechts zum Punkt &lt;math&gt; T &lt;/math&gt; gehen oder ein Pixel nach rechts und nach oben zum Punkt &lt;math&gt; S &lt;/math&gt;. Um dies zu entscheiden müssen wir die Abstände, die die jeweiligen Punkte zu der idealen Geraden haben, vergleichen. Der Punkt mit dem kleineren Abstand wird dann angesteuert.&lt;br&gt;</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Hierfür betrachten wir die Differenz der Abstände &lt;math&gt; |d_t|-|d_s|&lt;/math&gt;.&lt;br&gt;</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Hierfür betrachten wir die Differenz der Abstände &lt;math&gt; |d_t|-|d_s|&lt;/math&gt;.&lt;br&gt;</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Da wir nur die y-Achsenabweichung der gezeichneten Geraden mit dem Auge wahrnehmen, aber der folgende Zusammenhang besteht, können wir mit den reinen Abständen, die wir aus der Hesseform gewinnen rechnen:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Da wir nur die y-Achsenabweichung der gezeichneten Geraden mit dem Auge wahrnehmen, aber der folgende Zusammenhang besteht, können wir mit den reinen Abständen, die wir aus der Hesseform gewinnen rechnen:</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l134">Zeile 134:</td> <td colspan="2" class="diff-lineno">Zeile 134:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&#039;&#039;&#039;Achtung:&#039;&#039;&#039;&lt;br&gt;</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&#039;&#039;&#039;Achtung:&#039;&#039;&#039;&lt;br&gt;</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Wie wir unter dem <del style="font-weight: bold; text-decoration: none;">Punkt </del>Hesseform gesehen haben, kommt es bei dem Vorzeichen der Abständen der Punkte auf die Lage der Punkte im Bezug zur Geraden und dem Ursprung an. Da der Punkt &lt;math&gt; S &lt;/math&gt; nicht zwischen Gerade und Ursprung liegt <del style="font-weight: bold; text-decoration: none;">hat</del>, ist der Abstand &lt;math&gt; d_s &lt;/math&gt; positiv, analog ist der Abstand &lt;math&gt; d_t &lt;/math&gt; negativ.&lt;br&gt;&lt;br&gt;</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Wie wir unter dem <ins style="font-weight: bold; text-decoration: none;">Abschnitt </ins>Hesseform gesehen haben, kommt es bei dem Vorzeichen der Abständen der Punkte auf die Lage der Punkte im Bezug zur Geraden und dem Ursprung an. Da der Punkt &lt;math&gt; S &lt;/math&gt; nicht zwischen Gerade und Ursprung liegt, ist der Abstand &lt;math&gt; d_s &lt;/math&gt; positiv, analog ist der Abstand &lt;math&gt; d_t &lt;/math&gt; negativ.&lt;br&gt;&lt;br&gt;</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Also folgt mit:&lt;br&gt;</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Also folgt mit:&lt;br&gt;</div></td></tr> </table> Peterpummel http://geometrie.idea-sketch.com/index.php?title=Bresenham-Algorithmus_(in_Arbeit)&diff=22098&oldid=prev Peterpummel: /* Effektivität des Algorithmus */ 2013-03-11T14:46:40Z <p><span class="autocomment">Effektivität des Algorithmus</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="de"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Nächstältere Version</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Version vom 11. März 2013, 14:46 Uhr</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l22">Zeile 22:</td> <td colspan="2" class="diff-lineno">Zeile 22:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Effektivität des Algorithmus==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Effektivität des Algorithmus==</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Dieser Algorithmus ist auch in heutiger Zeit in seiner Geschwindigkeit und Einfachheit ungeschlagen, da die Rechenoperationen <del style="font-weight: bold; text-decoration: none;">sich </del>auf Addition und die Multiplikation mit <del style="font-weight: bold; text-decoration: none;">der </del>Zwei beschränken <del style="font-weight: bold; text-decoration: none;">lässt</del>. Dadurch kann dieser Rechenalgorithmus direkt in die Hardware der Grafikkarten implementiert werden.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Dieser Algorithmus ist auch in heutiger Zeit in seiner Geschwindigkeit und Einfachheit ungeschlagen, da <ins style="font-weight: bold; text-decoration: none;">sich </ins>die Rechenoperationen <ins style="font-weight: bold; text-decoration: none;">h </ins>auf Addition und die Multiplikation mit Zwei beschränken. Dadurch kann dieser Rechenalgorithmus direkt in die Hardware der Grafikkarten implementiert werden.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&#039;&#039;&#039;Potenzen:&#039;&#039;&#039;</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&#039;&#039;&#039;Potenzen:&#039;&#039;&#039;</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Obwohl in den Rechentermen Potenzen auftreten, kann man diese umgehen.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Obwohl in den Rechentermen Potenzen auftreten, kann man diese umgehen.</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l32">Zeile 32:</td> <td colspan="2" class="diff-lineno">Zeile 32:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&#039;&#039;&#039;Multiplikation mit Zwei&#039;&#039;&#039;:&lt;br\&gt;</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&#039;&#039;&#039;Multiplikation mit Zwei&#039;&#039;&#039;:&lt;br\&gt;</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Eine Multiplikation mit Zwei ist für einen Computer eine elementare Rechenoperation, da die sogenannte Shift Operation (Verrückungsoperation) durchgeführt <del style="font-weight: bold; text-decoration: none;">werden kann</del>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Eine Multiplikation mit Zwei ist für einen Computer eine elementare Rechenoperation, da die sogenannte Shift Operation (Verrückungsoperation) durchgeführt <ins style="font-weight: bold; text-decoration: none;">wird</ins>.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Wir erinnern uns an die Darstellung von Zahlen im Binärsystem.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Wir erinnern uns an die Darstellung von Zahlen im Binärsystem.</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l61">Zeile 61:</td> <td colspan="2" class="diff-lineno">Zeile 61:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Hier wird die Zahl (im Dezimalsystem) dargestellt&lt;br\&gt;&lt;br\&gt;</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Hier wird die Zahl (im Dezimalsystem) dargestellt&lt;br\&gt;&lt;br\&gt;</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&lt;math&gt;  0 \cdot 2^7 + 1 \cdot 2^6 + 0 \cdot 2^5 + 0 \cdot 2^4 + 1 \cdot 2^3 +  1 \cdot 2^2 + 0\cdot 2^1 + 1 \cdot 2^0 = 77&lt;/math&gt;&lt;br\&gt;&lt;br\&gt;</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&lt;math&gt;  0 \cdot 2^7 + 1 \cdot 2^6 + 0 \cdot 2^5 + 0 \cdot 2^4 + 1 \cdot 2^3 +  1 \cdot 2^2 + 0\cdot 2^1 + 1 \cdot 2^0 = 77&lt;/math&gt;&lt;br\&gt;&lt;br\&gt;</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Multiplizerien wir nun mit <del style="font-weight: bold; text-decoration: none;">zwei</del>, so folgt:&lt;br\&gt;&lt;br\&gt;</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Multiplizerien wir nun mit <ins style="font-weight: bold; text-decoration: none;">Zwei</ins>, so folgt:&lt;br\&gt;&lt;br\&gt;</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&lt;math&gt;2 \cdot ( 0 \cdot 2^7 + 1 \cdot 2^6 + 0 \cdot 2^5 + 0 \cdot 2^4 + 1 \cdot 2^3 +  1 \cdot 2^2 + 0\cdot 2^1 + 1 \cdot 2^0 )  = 154&lt;/math&gt;&lt;br\&gt;&lt;br\&gt;</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&lt;math&gt;2 \cdot ( 0 \cdot 2^7 + 1 \cdot 2^6 + 0 \cdot 2^5 + 0 \cdot 2^4 + 1 \cdot 2^3 +  1 \cdot 2^2 + 0\cdot 2^1 + 1 \cdot 2^0 )  = 154&lt;/math&gt;&lt;br\&gt;&lt;br\&gt;</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&lt;math&gt; \Leftrightarrow &lt;/math&gt;&lt;br\&gt;&lt;br\&gt;</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&lt;math&gt; \Leftrightarrow &lt;/math&gt;&lt;br\&gt;&lt;br\&gt;</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l89">Zeile 89:</td> <td colspan="2" class="diff-lineno">Zeile 89:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| 0</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| 0</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|}</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Vergleichen wir nun die Darstellung der Zahl 77 mit der Darstellung der Zahl 154, so fällt auf, dass durch die Multiplikation mit Zwei jedes Bit um eine Stelle nach links verschoben wurde.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Vergleichen wir nun die Darstellung der Zahl 77 mit der Darstellung der Zahl 154, so fällt auf, dass durch die Multiplikation mit Zwei<ins style="font-weight: bold; text-decoration: none;">, </ins>jedes Bit um eine Stelle nach links verschoben wurde.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=Bresenham-Algortihmus für Geraden=</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=Bresenham-Algortihmus für Geraden=</div></td></tr> </table> Peterpummel