Sketcher BSplineDecreaseKnotMultiplicity/de: Difference between revisions
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{{Docnav/de |
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|[[Sketcher_BSplineIncreaseKnotMultiplicity/de|BSplineKnotenVielfachheitErhöhen]] |
|[[Sketcher_BSplineIncreaseKnotMultiplicity/de|BSplineKnotenVielfachheitErhöhen]] |
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<div class="mw-translate-fuzzy"> |
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{{GuiCommand/de |
{{GuiCommand/de |
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|Name=Sketcher BSplineDecreaseKnotMultiplicity |
|Name=Sketcher BSplineDecreaseKnotMultiplicity |
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|SeeAlso=[[Sketcher_BSplineKnotMultiplicity/de|BSplineKnotenVielfachheit]], [[Sketcher_BSplineIncreaseKnotMultiplicity/de|BSplineKnotenVielfachheitErhöhen]] |
|SeeAlso=[[Sketcher_BSplineKnotMultiplicity/de|BSplineKnotenVielfachheit]], [[Sketcher_BSplineIncreaseKnotMultiplicity/de|BSplineKnotenVielfachheitErhöhen]] |
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</div> |
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<span id="Description"></span> |
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==Beschreibung== |
==Beschreibung== |
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<div class="mw-translate-fuzzy"> |
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Verringert die Vielfachheit eines B-Spline-Knotens. (Siehe die Seite [[B-Splines|B-Splines]] für weitere Informationen über B-Splines). |
Verringert die Vielfachheit eines B-Spline-Knotens. (Siehe die Seite [[B-Splines|B-Splines]] für weitere Informationen über B-Splines). |
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</div> |
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<span id="Usage"></span> |
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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/> |
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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]]}}): |
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[[File:Sketcher_KnotMultiplicity_multiplicity1.png|400px]] |
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{{Caption|B-Spline-Kurve deren Knoten beide die Vielfachheit 1 besitzen.}} |
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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 auf 3 erhöhten Vielfachheit des linken Knotens: |
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[[File:Sketcher_KnotMultiplicity_multiplicity3.png|400px]] |
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{{Caption|B-spline from above with knot multiplicity 3. A control point was moved to show that the knot has ''C''<sup>0</sup> continuity.}} |
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A consequence of a higher multiplicity is that for the price of loosing continuity you gain local control. This means the change of one control point only affects the spline locally to this changed point. This can be seen in this example, where the spline from the first image above was taken and its second control point from the right side was moved up: |
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[[File:Sketcher_KnotMultiplicity_locality.png|400px]] |
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{{Caption|Effect of locality due to different multiplicity.}} |
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One can see that the spline with knot multiplicity 1 is completely changed while the one with multiplicity 2 kept its form at its left side. |
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⚫ | |||
==Anwendung== |
==Anwendung== |
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<div class="mw-translate-fuzzy"> |
<div class="mw-translate-fuzzy"> |
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# |
# Einen B-Spline-Knoten auswählen. |
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# Es gibt mehrere Möglichkeiten den Befehl aufzurufen: |
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# Rufe das Werkzeug mit mehreren Methoden auf: |
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#* |
#* Die Schaltfläche {{Button|[[File:Sketcher_BSplineDecreaseKnotMultiplicity.svg|16px]] [[Sketcher_BSplineDecreaseKnotMultiplicity/de|Vielfachheit eines B-Spline-Knotens verringern]]}} drücken. |
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#* |
#* Den Menüeintrag {{MenuCommand|Sketch → B-Spline-Werkzeuge → [[File:Sketcher_BSplineDecreaseKnotMultiplicity.svg|16px]] Vielfachheit eines B-Spline-Knotens verringern}} auswählen. |
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</div> |
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==Example== |
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'''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.) |
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See [[Sketcher_BSplineIncreaseKnotMultiplicity#Example|Sketcher_BSplineIncreaseKnotMultiplicity]] |
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==Notes== |
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If you decrease the multiplicity of a knot to zero, the knot vanishes. Mathematically it then appears 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 if you look at the multiplicity. For example a knot with multiplicity 0 on a B-spline with degree 3 |
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⚫ | 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 this means that both sides must be part of the same curve. There is then effectively no longer a knot connecting two Bézier curves. |
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Latest revision as of 07:36, 22 April 2024
Sketcher BSplineKnotenVielfachheitVerringern |
Menüeintrag |
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Sketch → B-Spline-Werkzeuge → Vielfachheit eines Spline-Knotens verringern |
Arbeitsbereich |
Sketcher |
Standardtastenkürzel |
Keiner |
Eingeführt in Version |
0.17 |
Siehe auch |
BSplineKnotenVielfachheit, BSplineKnotenVielfachheitErhöhen |
Beschreibung
Verringert die Vielfachheit eines B-Spline-Knotens. (Siehe die Seite B-Splines für weitere Informationen über B-Splines).
Anwendung
- Einen B-Spline-Knoten auswählen.
- Es gibt mehrere Möglichkeiten den Befehl aufzurufen:
- Die Schaltfläche Vielfachheit eines B-Spline-Knotens verringern drücken.
- Den Menüeintrag Sketch → B-Spline-Werkzeuge → Vielfachheit eines B-Spline-Knotens verringern auswählen.
Example
See Sketcher_BSplineIncreaseKnotMultiplicity
Notes
If you decrease the multiplicity of a knot to zero, the knot vanishes. Mathematically it then appears 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 if you look at the multiplicity. For example a knot with multiplicity 0 on a B-spline with degree 3 means that at the position of the knot two Bézier pieces are connected with C3 continuity. So the third derivative should be equal on both sides of the knot. However for a cubic Bézier curve this means that both sides must be part of the same curve. There is then effectively no longer a knot connecting two Bézier curves.
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