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Sketcher BSplineDecreaseKnotMultiplicity/de - Revision history
2024-03-28T09:18:45Z
Revision history for this page on the wiki
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FuzzyBot: Updating to match new version of source page
2023-01-18T21:16:29Z
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<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>
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<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>B-splines are basically a combination of [[B-Splines#B.C3.A9zier_curves|Bézier curves]] (nicely explained in [https://www.youtube.com/watch?v=bE1MrrqBAl8 this] and [https://www.youtube.com/watch?v=xXJylM2S72s this] video). The points where two Bézier curves are connected to form the spline are called knots. A knot on a degree ''d'' spline with the multiplicity ''m'' means that the curve left and right to the knot has at least an equal ''n'' order derivative (called ''C''<sup>''n''</sup> continuity) whereas <math>n=d-m</math>.<br/></div></td>
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<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>B-splines are basically a combination of [[B-Splines#B.C3.A9zier_curves|Bézier curves]] (nicely explained in [https://www.youtube.com/watch?v=bE1MrrqBAl8 this] and [https://www.youtube.com/watch?v=xXJylM2S72s this] video). The points where two Bézier curves are connected to form the spline are called knots. A knot on a degree ''d'' spline with the multiplicity ''m'' means that the curve left and right to the knot has at least an equal ''n'' order derivative (called ''C''<sup>''n''</sup> continuity) whereas <math>n=d-m</math>.<br/></div></td>
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<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>Here is a cubic spline (<math>d=3</math>) whose knots have the multiplicity 1. The multiplicity is indicated by the number in parentheses. The indication can be changed using the toolbar button {{Button|[[File:Sketcher_BSplineKnotMultiplicity.svg|24px]] [[Sketcher_BSplineKnotMultiplicity|Show/hide B-spline knot multiplicity]]}}<del style="font-weight: bold; text-decoration: none;">)</del>:</div></td>
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<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>Here is a cubic spline (<math>d=3</math>) whose knots have the multiplicity 1. The multiplicity is indicated by the number in parentheses. The indication can be changed using the toolbar button {{Button|[[File:Sketcher_BSplineKnotMultiplicity.svg|24px]] [[Sketcher_BSplineKnotMultiplicity|Show/hide B-spline knot multiplicity]]}}:</div></td>
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<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>[[File:Sketcher_KnotMultiplicity_multiplicity1.png|400px]]</div></td>
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<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>'''Note:''' If you decrease the multiplicity, the knot vanishes, because mathematically it appears then zero times in the knot vector, meaning there is no longer a basis function. Understanding this, requires some math, but it will also be clear when you look at the multiplicity: For example degree = 3 then multiplicity = 0 means that at the position of the knot two Bézier pieces are connected with ''C''<sup>3</sup> continuity. So the third derivative should be equal on both sides of the knot. However for a cubic Bézier curve (that is a polynom with degree 3) , this means both sides must be part of the same curve. So there is then actually no longer a knot connecting 2 different Bézier curves, the former knot is then simply a point onto one Bézier curve.</div></td>
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<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>'''Note:''' If you decrease the multiplicity, the knot vanishes, because mathematically it appears then zero times in the knot vector, meaning there is no longer a basis function. Understanding this, requires some math, but it will also be clear when you look at the multiplicity: For example degree = 3 then multiplicity = 0 means that at the position of the knot two Bézier pieces are connected with ''C''<sup>3</sup> continuity. So the third derivative should be equal on both sides of the knot. However for a cubic Bézier curve (that is a polynom with degree 3) , this means both sides must be part of the same curve. So there is then actually no longer a knot connecting 2 different Bézier curves, the former knot is then simply a point onto one Bézier curve.</div></td>
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FuzzyBot
https://wiki.freecad.org/index.php?title=Sketcher_BSplineDecreaseKnotMultiplicity/de&diff=1198006&oldid=prev
FBXL5 at 15:24, 9 October 2022
2022-10-09T15:24:41Z
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<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># Rufe das Werkzeug mit mehreren Methoden auf: </div></td>
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<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;">Drücke</del> <del style="font-weight: bold; text-decoration: none;">die</del> {{<del style="font-weight: bold; text-decoration: none;">Button</del>|[[File:Sketcher_BSplineDecreaseKnotMultiplicity.svg|16px]] <del style="font-weight: bold; text-decoration: none;">[[Sketcher_BSplineDecreaseKnotMultiplicity/de|</del>B-<del style="font-weight: bold; text-decoration: none;">spline Knotenvervielfalt</del> <del style="font-weight: bold; text-decoration: none;">Verringern]]</del>}} <del style="font-weight: bold; text-decoration: none;">Schaltfläche</del>.</div></td>
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<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;">Den</ins> <ins style="font-weight: bold; text-decoration: none;">Menüeintrag</ins> {{<ins style="font-weight: bold; text-decoration: none;">MenuCommand</ins>|<ins style="font-weight: bold; text-decoration: none;">Sketch → B-Spline-Werkzeuge → </ins>[[File:Sketcher_BSplineDecreaseKnotMultiplicity.svg|16px]] <ins style="font-weight: bold; text-decoration: none;">Vielfachheit eines</ins> B-<ins style="font-weight: bold; text-decoration: none;">Spline-Knotens</ins> <ins style="font-weight: bold; text-decoration: none;">verringern</ins>}} <ins style="font-weight: bold; text-decoration: none;">auswählen</ins>.</div></td>
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<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>#* Verwende den {{MenuCommand|Skizze → Skizzierer B-Spline Werkzeuge → [[File:Sketcher_BSplineDecreaseKnotMultiplicity.svg|16px]] Knotenvervielfalt Verringern}} Eintrag im oberen Menü.</div></td>
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<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>
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<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>'''Note:''' Decreasing the multiplicity from 1 to 0 will remove the knot since the result would be a curve with an "edge" at the knot position (''C''<sup>0</sup> continuity) and this is not supported. (To create curves with an "edge", you can create two splines and connect them.)</div></td>
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<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>'''Note:''' Decreasing the multiplicity from 1 to 0 will remove the knot since the result would be a curve with an "edge" at the knot position (''C''<sup>0</sup> continuity) and this is not supported. (To create curves with an "edge", you can create two splines and connect them.)</div></td>
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FBXL5
https://wiki.freecad.org/index.php?title=Sketcher_BSplineDecreaseKnotMultiplicity/de&diff=1198004&oldid=prev
FBXL5: Created page with "Eine Vielfachheit von 3 ändert diesen Spline so, dass sogar die ersten Ableitungen nicht gleich sind (''C''<sup>0</sup>-Stetigkeit). Hier ist dieselbe Spline-Kurve mit einer..."
2022-10-09T15:16:41Z
<p>Created page with "Eine Vielfachheit von 3 ändert diesen Spline so, dass sogar die ersten Ableitungen nicht gleich sind (''C''<sup>0</sup>-Stetigkeit). Hier ist dieselbe Spline-Kurve mit einer..."</p>
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<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>{{Caption|B-Spline-Kurve deren Knoten beide die Vielfachheit 1 besitzen.}}</div></td>
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<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>
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<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>
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<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;">A</del> <del style="font-weight: bold; text-decoration: none;">multiplicity</del> <del style="font-weight: bold; text-decoration: none;">of</del> 3 <del style="font-weight: bold; text-decoration: none;">will</del> <del style="font-weight: bold; text-decoration: none;">change</del> <del style="font-weight: bold; text-decoration: none;">this spline</del> so <del style="font-weight: bold; text-decoration: none;">that</del> <del style="font-weight: bold; text-decoration: none;">even</del> <del style="font-weight: bold; text-decoration: none;">the</del> <del style="font-weight: bold; text-decoration: none;">first</del> <del style="font-weight: bold; text-decoration: none;">order</del> <del style="font-weight: bold; text-decoration: none;">derivatives</del> <del style="font-weight: bold; text-decoration: none;">are</del> <del style="font-weight: bold; text-decoration: none;">not equal</del> (''C''<sup>0</sup><del style="font-weight: bold; text-decoration: none;"> continuity</del>). <del style="font-weight: bold; text-decoration: none;">Here</del> <del style="font-weight: bold; text-decoration: none;">is</del> <del style="font-weight: bold; text-decoration: none;">the</del> <del style="font-weight: bold; text-decoration: none;">same</del> <del style="font-weight: bold; text-decoration: none;">spline</del> <del style="font-weight: bold; text-decoration: none;">where</del> <del style="font-weight: bold; text-decoration: none;">the</del> <del style="font-weight: bold; text-decoration: none;">left's</del> <del style="font-weight: bold; text-decoration: none;">knot</del> <del style="font-weight: bold; text-decoration: none;">multiplicity</del> <del style="font-weight: bold; text-decoration: none;">was</del> <del style="font-weight: bold; text-decoration: none;">increased</del> <del style="font-weight: bold; text-decoration: none;">to 3</del>:</div></td>
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<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;">Eine</ins> <ins style="font-weight: bold; text-decoration: none;">Vielfachheit</ins> <ins style="font-weight: bold; text-decoration: none;">von</ins> 3 <ins style="font-weight: bold; text-decoration: none;">ändert</ins> <ins style="font-weight: bold; text-decoration: none;">diesen</ins> <ins style="font-weight: bold; text-decoration: none;">Spline</ins> so<ins style="font-weight: bold; text-decoration: none;">,</ins> <ins style="font-weight: bold; text-decoration: none;">dass</ins> <ins style="font-weight: bold; text-decoration: none;">sogar</ins> <ins style="font-weight: bold; text-decoration: none;">die</ins> <ins style="font-weight: bold; text-decoration: none;">ersten</ins> <ins style="font-weight: bold; text-decoration: none;">Ableitungen</ins> <ins style="font-weight: bold; text-decoration: none;">nicht</ins> <ins style="font-weight: bold; text-decoration: none;">gleich</ins> <ins style="font-weight: bold; text-decoration: none;">sind</ins> (''C''<sup>0</sup><ins style="font-weight: bold; text-decoration: none;">-Stetigkeit</ins>). <ins style="font-weight: bold; text-decoration: none;">Hier</ins> <ins style="font-weight: bold; text-decoration: none;">ist</ins> <ins style="font-weight: bold; text-decoration: none;">dieselbe</ins> <ins style="font-weight: bold; text-decoration: none;">Spline-Kurve</ins> <ins style="font-weight: bold; text-decoration: none;">mit</ins> <ins style="font-weight: bold; text-decoration: none;">einer</ins> <ins style="font-weight: bold; text-decoration: none;">auf</ins> <ins style="font-weight: bold; text-decoration: none;">3</ins> <ins style="font-weight: bold; text-decoration: none;">erhöhten</ins> <ins style="font-weight: bold; text-decoration: none;">Vielfachheit</ins> <ins style="font-weight: bold; text-decoration: none;">des</ins> <ins style="font-weight: bold; text-decoration: none;">linken</ins> <ins style="font-weight: bold; text-decoration: none;">Knotens</ins>:</div></td>
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<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>[[File:Sketcher_KnotMultiplicity_multiplicity3.png|400px]]</div></td>
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<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>[[File:Sketcher_KnotMultiplicity_multiplicity3.png|400px]]</div></td>
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FBXL5
https://wiki.freecad.org/index.php?title=Sketcher_BSplineDecreaseKnotMultiplicity/de&diff=1198002&oldid=prev
FBXL5 at 15:09, 9 October 2022
2022-10-09T15:09:57Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 15:09, 9 October 2022</td>
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<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>[[File:Sketcher_KnotMultiplicity_multiplicity1.png|400px]]</div></td>
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<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>[[File:Sketcher_KnotMultiplicity_multiplicity1.png|400px]]</div></td>
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<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>{{Caption|B-Spline-Kurve deren Knoten beide die Vielfachheit 1 besitzen.}}</div></td>
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<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 class="mw-translate-fuzzy"></div></td>
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<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>{{Caption|B-spline Kurve zeigt abnehmende Knotenvielfalt.}}</div></td>
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<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></div></td>
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<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>
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<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>
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<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>A multiplicity of 3 will change this spline so that even the first order derivatives are not equal (''C''<sup>0</sup> continuity). Here is the same spline where the left's knot multiplicity was increased to 3:</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>A multiplicity of 3 will change this spline so that even the first order derivatives are not equal (''C''<sup>0</sup> continuity). Here is the same spline where the left's knot multiplicity was increased to 3:</div></td>
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FBXL5
https://wiki.freecad.org/index.php?title=Sketcher_BSplineDecreaseKnotMultiplicity/de&diff=1198000&oldid=prev
FBXL5 at 15:07, 9 October 2022
2022-10-09T15:07:35Z
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<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>
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<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>
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<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>Verringert die Vielfachheit eines B-Spline-Knotens. (Siehe die Seite [[B-Splines|B-Splines]] für weitere Informationen über B-Splines).</div></td>
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<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 class="mw-translate-fuzzy"></div></td>
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<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>Verringert die Knotenvielfalt eines B-Spline Kurvenknotens (siehe [https://en.wikipedia.org/wiki/B-spline B-Spline]).</div></td>
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<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></div></td>
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<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>
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<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>
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<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>B-splines are basically a combination of [[B-Splines#B.C3.A9zier_curves|Bézier curves]] (nicely explained in [https://www.youtube.com/watch?v=bE1MrrqBAl8 this] and [https://www.youtube.com/watch?v=xXJylM2S72s this] video). The points where two Bézier curves are connected to form the spline are called knots. A knot on a degree ''d'' spline with the multiplicity ''m'' means that the curve left and right to the knot has at least an equal ''n'' order derivative (called ''C''<sup>''n''</sup> continuity) whereas <math>n=d-m</math>.<br/></div></td>
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<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>B-splines are basically a combination of [[B-Splines#B.C3.A9zier_curves|Bézier curves]] (nicely explained in [https://www.youtube.com/watch?v=bE1MrrqBAl8 this] and [https://www.youtube.com/watch?v=xXJylM2S72s this] video). The points where two Bézier curves are connected to form the spline are called knots. A knot on a degree ''d'' spline with the multiplicity ''m'' means that the curve left and right to the knot has at least an equal ''n'' order derivative (called ''C''<sup>''n''</sup> continuity) whereas <math>n=d-m</math>.<br/></div></td>
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FBXL5
https://wiki.freecad.org/index.php?title=Sketcher_BSplineDecreaseKnotMultiplicity/de&diff=1172059&oldid=prev
FBXL5 at 21:27, 14 August 2022
2022-08-14T21:27:13Z
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<td class="diff-marker"><a class="mw-diff-movedpara-right" title="Paragraph was moved. Click to jump to old location." href="#movedpara_8_0_lhs">⚫</a></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><a name="movedpara_5_0_rhs"></a>|MenuLocation=<ins style="font-weight: bold; text-decoration: none;">Sketch</ins> → B-<ins style="font-weight: bold; text-decoration: none;">Spline-</ins>Werkzeuge → <ins style="font-weight: bold; text-decoration: none;">Vielfachheit eines Spline-Knotens</ins> verringern</div></td>
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<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>|Version=0.17</div></td>
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FBXL5
https://wiki.freecad.org/index.php?title=Sketcher_BSplineDecreaseKnotMultiplicity/de&diff=1172054&oldid=prev
FBXL5 at 21:21, 14 August 2022
2022-08-14T21:21:48Z
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FBXL5
https://wiki.freecad.org/index.php?title=Sketcher_BSplineDecreaseKnotMultiplicity/de&diff=1172052&oldid=prev
FBXL5 at 21:21, 14 August 2022
2022-08-14T21:21:39Z
<p></p>
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FBXL5
https://wiki.freecad.org/index.php?title=Sketcher_BSplineDecreaseKnotMultiplicity/de&diff=1118778&oldid=prev
FuzzyBot: Updating to match new version of source page
2022-03-21T11:14:21Z
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<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>
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<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>'''Note:''' Decreasing the multiplicity from 1 to 0 will remove the knot since the result would be a curve with an "edge" at the knot position (''C''<sup>0</sup> continuity) and this is not supported. (To create curves with an "<del style="font-weight: bold; text-decoration: none;">edges</del>", you can create two splines and connect them.)</div></td>
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<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>'''Note:''' Decreasing the multiplicity from 1 to 0 will remove the knot since the result would be a curve with an "edge" at the knot position (''C''<sup>0</sup> continuity) and this is not supported. (To create curves with an "<ins style="font-weight: bold; text-decoration: none;">edge</ins>", you can create two splines and connect them.)</div></td>
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FuzzyBot
https://wiki.freecad.org/index.php?title=Sketcher_BSplineDecreaseKnotMultiplicity/de&diff=910775&oldid=prev
FuzzyBot: Updating to match new version of source page
2021-07-02T10:28:32Z
<p>Updating to match new version of source page</p>
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<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><div class="mw-translate-fuzzy"></div></td>
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<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>Verringert die Knotenvielfalt eines B-Spline Kurvenknotens (siehe [https://en.wikipedia.org/wiki/B-spline B-Spline]).</div></td>
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<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>Verringert die Knotenvielfalt eines B-Spline Kurvenknotens (siehe [https://en.wikipedia.org/wiki/B-spline B-Spline]).</div></td>
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<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>
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<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>B-splines are basically a combination of [<del style="font-weight: bold; text-decoration: none;">https://en</del>.<del style="font-weight: bold; text-decoration: none;">wikipedia</del>.<del style="font-weight: bold; text-decoration: none;">org/wiki/Bezier_curve#Constructing_B%C3%</del>A9zier_curves<del style="font-weight: bold; text-decoration: none;"> </del>Bézier curves] (nicely explained in [https://www.youtube.com/watch?v=bE1MrrqBAl8 this] and [https://www.youtube.com/watch?v=xXJylM2S72s this] video). The points where two Bézier curves are connected to form the spline are called knots. A knot on a degree ''d'' spline with the multiplicity ''m'' means that the curve left and right to the knot has at least an equal ''n'' order derivative (called ''C''<sup>''n''</sup> continuity) whereas <math>n=d-m</math>.<br/></div></td>
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<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>B-splines are basically a combination of [<ins style="font-weight: bold; text-decoration: none;">[B-Splines#B</ins>.<ins style="font-weight: bold; text-decoration: none;">C3</ins>.A9zier_curves<ins style="font-weight: bold; text-decoration: none;">|</ins>Bézier curves<ins style="font-weight: bold; text-decoration: none;">]</ins>] (nicely explained in [https://www.youtube.com/watch?v=bE1MrrqBAl8 this] and [https://www.youtube.com/watch?v=xXJylM2S72s this] video). The points where two Bézier curves are connected to form the spline are called knots. A knot on a degree ''d'' spline with the multiplicity ''m'' means that the curve left and right to the knot has at least an equal ''n'' order derivative (called ''C''<sup>''n''</sup> continuity) whereas <math>n=d-m</math>.<br/></div></td>
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<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>Here is a cubic spline (<math>d=3</math>) whose knots have the multiplicity 1. The multiplicity is indicated by the number in parentheses. The indication can be changed using the toolbar button {{Button|[[File:Sketcher_BSplineKnotMultiplicity.svg|24px]] [[Sketcher_BSplineKnotMultiplicity|Show/hide B-spline knot multiplicity]]}}):</div></td>
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<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>Here is a cubic spline (<math>d=3</math>) whose knots have the multiplicity 1. The multiplicity is indicated by the number in parentheses. The indication can be changed using the toolbar button {{Button|[[File:Sketcher_BSplineKnotMultiplicity.svg|24px]] [[Sketcher_BSplineKnotMultiplicity|Show/hide B-spline knot multiplicity]]}}):</div></td>
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