Sketcher BSplineIncreaseKnotMultiplicity/de: Difference between revisions

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<languages/>
<languages/>
{{Docnav|[[Sketcher_BSplineIncreaseDegree|Increase degree]]|[[Sketcher_BSplineDecreaseKnotMultiplicity|Decrease knot multiplicity]]|[[Sketcher_Workbench|Sketcher]]|IconL=Sketcher_BSplineIncreaseDegree.svg|IconC=Workbench_Sketcher.svg|IconR=Sketcher_BSplineDecreaseKnotMultiplicity.svg}}


{{Docnav/de
{{GuiCommand
|[[Sketcher_BSplineDecreaseDegree/de|BSplineGradVerringern]]
|[[Sketcher_BSplineDecreaseKnotMultiplicity/de|BSplineKnotenVielfachheitVerringern]]
|[[Sketcher_Workbench/de|Sketcher]]
|IconL=Sketcher_BSplineDecreaseDegree.svg
|IconR=Sketcher_BSplineDecreaseKnotMultiplicity.svg
|IconC=Workbench_Sketcher.svg
}}

{{GuiCommand/de
|Name=Sketcher BSplineIncreaseKnotMultiplicity
|Name=Sketcher BSplineIncreaseKnotMultiplicity
|Name/de=Sketcher BSplineKnotenVielfachheitErhöhen
|Workbenches=[[Sketcher Workbench|Sketcher]]
|MenuLocation=Sketch → Sketcher B-spline toolsIncrease knot multiplicity
|MenuLocation=Sketch → B-Spline-WerkzeugeVielfachheit eines Spline-Knotens erhöhen
|Workbenches=[[Sketcher_Workbench/de|Sketcher]]
|Version=0.17
|Version=0.17
|SeeAlso=[[Sketcher_BSplineKnotMultiplicity/de|BSplineKnotenVielfachheit]], [[Sketcher_BSplineDecreaseKnotMultiplicity/de|BSplineKnotenVielfachheitVerringern]]
|SeeAlso=[[Sketcher CompCreateBSpline|Create B-spline]]
}}
}}


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


Erhöht die Vielfachheit eines B-Spline-Knotens. (Siehe die Seite [[B-Splines|B-Splines]] für weitere Informationen über B-Splines).
Increases the knot multiplicity of a B-spline curve knot (see [https://en.wikipedia.org/wiki/B-spline B-spline]).


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 pieces are connected are called knots. A knot on a spline with degree ''d'' and 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_SampleBSplineIncreaseKnotMultiplicity.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 showing increasing knot multiplicity.}}


[[File:Sketcher_KnotMultiplicity_multiplicity1.png|400px]]
==How to use==
{{Caption|B-spline where both knots have the multiplicity 1.}}
# Select a B-spline knot and apply.


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:


[[File:Sketcher_KnotMultiplicity_multiplicity3.png|400px]]
{{Docnav|[[Sketcher_BSplineIncreaseDegree|Increase degree]]|[[Sketcher_BSplineDecreaseKnotMultiplicity|Decrease knot multiplicity]]|[[Sketcher_Workbench|Sketcher]]|IconL=Sketcher_BSplineIncreaseDegree.svg|IconC=Workbench_Sketcher.svg|IconR=Sketcher_BSplineDecreaseKnotMultiplicity.svg}}
{{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:
{{Sketcher Tools navi}}


[[File:Sketcher_KnotMultiplicity_locality.png|400px]]
{{Userdocnavi}}
{{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.

<span id="Usage"></span>
==Anwendung==

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


{{Docnav/de
|[[Sketcher_BSplineDecreaseDegree/de|BSplineGradVerringern]]
|[[Sketcher_BSplineDecreaseKnotMultiplicity/de|BSplineKnotenVielfachheitVerringern]]
|[[Sketcher_Workbench/de|Sketcher]]
|IconL=Sketcher_BSplineDecreaseDegree.svg
|IconR=Sketcher_BSplineDecreaseKnotMultiplicity.svg
|IconC=Workbench_Sketcher.svg
}}


{{Sketcher_Tools_navi{{#translation:}}}}
{{clear}}
{{Userdocnavi{{#translation:}}}}

Latest revision as of 21:16, 18 January 2023

Sketcher BSplineKnotenVielfachheitErhöhen

Menüeintrag
Sketch → B-Spline-Werkzeuge → Vielfachheit eines Spline-Knotens erhöhen
Arbeitsbereich
Sketcher
Standardtastenkürzel
Keiner
Eingeführt in Version
0.17
Siehe auch
BSplineKnotenVielfachheit, BSplineKnotenVielfachheitVerringern

Beschreibung

Erhöht 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 pieces are connected are called knots. A knot on a spline with degree d and 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 where both knots have the multiplicity 1.

A multiplicity of 3 will change this spline so that even the first order derivatives are not equal (C0 continuity). Here is the same spline where the left's knot multiplicity was increased to 3:

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.

Anwendung

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