Sketcher BSplineDecreaseKnotMultiplicity/de: Difference between revisions

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{{Docnav/de
{{Docnav|[[Sketcher_BSplineIncreaseKnotMultiplicity|Increase knot multiplicity]]|[[Sketcher_SwitchVirtualSpace|Switch Virtual Space]]|[[Sketcher_Workbench|Sketcher]]|IconL=Sketcher_BSplineIncreaseKnotMultiplicity.svg|IconC=Workbench_Sketcher.svg|IconR=Sketcher SwitchVirtualSpace.png‎}}
|[[Sketcher_BSplineIncreaseKnotMultiplicity/de|BSplineKnotenVielfachheitErhöhen]]
|[[Sketcher_BSplineInsertKnot/de|BSplineKnotenEinfügen]]
|[[Sketcher_Workbench/de|Sketcher]]
|IconL=Sketcher_BSplineIncreaseKnotMultiplicity.svg
|IconR=Sketcher_BSplineInsertKnot.svg‎
|IconC=Workbench_Sketcher.svg
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{{GuiCommand
{{GuiCommand/de
|Name=Sketcher BSplineDecreaseKnotMultiplicity
|Name=Sketcher BSplineDecreaseKnotMultiplicity
|Name/de=Sketcher BSplineKnotenVielfachheitVerringern
|Workbenches=[[Sketcher Workbench|Sketcher]]
|MenuLocation=Sketch → Sketcher B-spline toolsDecrease knot multiplicity
|MenuLocation=Sketch → B-Spline-WerkzeugeVielfachheit eines Spline-Knotens verringern
|Workbenches=[[Sketcher_Workbench/de|Sketcher]]
|Version=0.17
|Version=0.17
|SeeAlso=[[Sketcher_BSplineKnotMultiplicity/de|BSplineKnotenVielfachheit]], [[Sketcher_BSplineIncreaseKnotMultiplicity/de|BSplineKnotenVielfachheitErhöhen]]
|SeeAlso=[[Sketcher CompCreateBSpline|Create B-spline]]
}}
}}


<span id="Description"></span>
==Beschreibung==
==Beschreibung==


Verringert die Knotenvielfalt eines B-Spline-Kurvenknotens (siehe [https://en.wikipedia.org/wiki/B-spline B-Spline]).
Verringert die Vielfachheit eines B-Spline-Knotens. (Siehe die Seite [[B-Splines|B-Splines]] für weitere Informationen über B-Splines).


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/>
[[File:sketcher_SampleBSplineDecreaseKnotMultiplicity.png]]
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]]}}:
{{Caption|B-spline curve zeigt abnehmende Knoten Vielzahl.}}


[[File:Sketcher_KnotMultiplicity_multiplicity1.png|400px]]
{{Caption|B-Spline-Kurve deren Knoten beide die Vielfachheit 1 besitzen.}}

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:

[[File:Sketcher_KnotMultiplicity_multiplicity3.png|400px]]
{{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.}}

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:

[[File:Sketcher_KnotMultiplicity_locality.png|400px]]
{{Caption|Effect of locality due to different multiplicity.}}

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.

'''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.

<span id="Usage"></span>
==Anwendung==
==Anwendung==
# Wähle einen B-Spline Knoten aus und wende sie an.


# Einen B-Spline-Knoten auswählen.
# Es gibt mehrere Möglichkeiten den Befehl aufzurufen:
#* Die Schaltfläche {{Button|[[File:Sketcher_BSplineDecreaseKnotMultiplicity.svg|16px]] [[Sketcher_BSplineDecreaseKnotMultiplicity/de|Vielfachheit eines B-Spline-Knotens verringern]]}} drücken.
#* Den Menüeintrag {{MenuCommand|Sketch → B-Spline-Werkzeuge → [[File:Sketcher_BSplineDecreaseKnotMultiplicity.svg|16px]] Vielfachheit eines B-Spline-Knotens verringern}} auswählen.


'''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.)
{{Docnav|[[Sketcher_BSplineIncreaseKnotMultiplicity|Increase knot multiplicity]]|[[Sketcher_SwitchVirtualSpace|Switch Virtual Space]]|[[Sketcher_Workbench|Sketcher]]|IconL=Sketcher_BSplineIncreaseKnotMultiplicity.svg|IconC=Workbench_Sketcher.svg|IconR=Sketcher SwitchVirtualSpace.png‎}}


{{Sketcher Tools navi/de}}


{{Docnav/de
{{Userdocnavi}}
|[[Sketcher_BSplineIncreaseKnotMultiplicity/de|BSplineKnotenVielfachheitErhöhen]]
|[[Sketcher_BSplineInsertKnot/de|BSplineKnotenEinfügen]]
|[[Sketcher_Workbench/de|Sketcher]]
|IconL=Sketcher_BSplineIncreaseKnotMultiplicity.svg
|IconR=Sketcher_BSplineInsertKnot.svg‎
|IconC=Workbench_Sketcher.svg
}}


{{Sketcher_Tools_navi{{#translation:}}}}
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Latest revision as of 21:16, 18 January 2023

Sketcher BSplineKnotenVielfachheitVerringern

Menüeintrag
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).

B-splines are basically a combination of Bézier curves (nicely explained in this and 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 Cn continuity) whereas .
Here is a cubic spline () whose knots have the multiplicity 1. The multiplicity is indicated by the number in parentheses. The indication can be changed using the toolbar button Show/hide B-spline knot multiplicity:

B-Spline-Kurve deren Knoten beide die Vielfachheit 1 besitzen.

Eine Vielfachheit von 3 ändert diesen Spline so, dass sogar die ersten Ableitungen nicht gleich sind (C0-Stetigkeit). Hier ist dieselbe Spline-Kurve mit einer auf 3 erhöhten Vielfachheit des linken Knotens:

B-spline from above with knot multiplicity 3. A control point was moved to show that the knot has C0 continuity.

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:

Effect of locality due to different multiplicity.

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.

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 C3 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.

Anwendung

  1. Einen B-Spline-Knoten auswählen.
  2. Es gibt mehrere Möglichkeiten den Befehl aufzurufen:

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 (C0 continuity) and this is not supported. (To create curves with an "edge", you can create two splines and connect them.)