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
}}

<div class="mw-translate-fuzzy">
⚫
{{GuiCommand/de
|Name=Sketcher BSplineIncreaseKnotMultiplicity
|Name=Sketcher BSplineIncreaseKnotMultiplicity
|Name/de=Sketcher BSplineKnotenVielfachheitErhöhen
⚫
|Workbenches=[[Sketcher Workbench|Sketcher]]
|MenuLocation=Sketch → Sketcher B-spline tools → Increase knot multiplicity
|MenuLocation=Sketch → B-Spline-Werkzeuge → Vielfachheit 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]]
}}
}}
</div>


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


<div class="mw-translate-fuzzy">
Erhöht die Knotenvielfalt eines B-Spline Kurvenknotens (siehe [https://en.wikipedia.org/wiki/B-spline B-Spline]).
Erhöht die Vielfachheit eines B-Spline-Knotens. (Siehe die Seite [[B-Splines|B-Splines]] für weitere Informationen über B-Splines).
</div>


<span id="Usage"></span>
[[File:sketcher_SampleBSplineIncreaseKnotMultiplicity.png]]
==Anwendung==
⚫
{{Caption|B-spline curve showing increasing knot multiplicity.}}


<div class="mw-translate-fuzzy">
==How to use==
# Einen B-Spline-Knoten auswählen.
# Select a B-spline knot and apply.
# 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.
</div>


==Example==


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 with multiplicity ''m'' on a B-spline with degree ''d'' means the curve to the left and right of the knot has at least an equal ''n'' order derivative (called ''C<sup>n</sup>'' continuity) where ''n = d - m''.
{{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}}


In this cubic B-spline (degree 3) there are 3 segments, meaning 3 curves are connected at 2 knots. The knots have multiplicity 1.
{{Sketcher Tools navi}}


The multiplicity is indicated by the numbers in round brackets. See [[File:Sketcher_BSplineKnotMultiplicity.svg|16px]] [[Sketcher_BSplineKnotMultiplicity|Show/hide B-spline knot multiplicity]].
⚫
{{Userdocnavi}}

[[File:Sketcher_KnotMultiplicity_multiplicity1.png|400px]]
⚫
{{Caption|B-spline where both knots have multiplicity 1.}}

A multiplicity of 3 will change this B-spline so that even the first order derivatives are not equal (''C<sup>0</sup>'' continuity). Here is the same B-spline where the multiplicity of the knot on the left was increased to 3:

[[File:Sketcher_KnotMultiplicity_multiplicity3.png|400px]]
{{Caption|Same B-spline 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. Meaning changing one control point will only affect the B-spline locally.

This can be seen in this example, where the B-spline with knot multiplicity 1 from the first image above was taken, and the second control point from the right was moved up. As a result the complete shape of the B-spline has changed:

[[File:Sketcher_KnotMultiplicity_locality1.png|400px]]

After increasing the multiplicity of the knots to 2, moving the second control point from the right results in significant changes on the right side of the shape only:

[[File:Sketcher_KnotMultiplicity_locality2.png|400px]]

==Notes==

* Knot multiplicity can also be increased with [[Sketcher_BSplineInsertKnot|Sketcher BSplineInsertKnot]].


{{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 07:34, 22 April 2024

Other languages:

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).

Anwendung

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

Example

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 with multiplicity m on a B-spline with degree d means the curve to the left and right of the knot has at least an equal n order derivative (called Cn continuity) where n = d - m.

In this cubic B-spline (degree 3) there are 3 segments, meaning 3 curves are connected at 2 knots. The knots have multiplicity 1.

The multiplicity is indicated by the numbers in round brackets. See Show/hide B-spline knot multiplicity.

B-spline where both knots have multiplicity 1.

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

Same B-spline 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. Meaning changing one control point will only affect the B-spline locally.

This can be seen in this example, where the B-spline with knot multiplicity 1 from the first image above was taken, and the second control point from the right was moved up. As a result the complete shape of the B-spline has changed:

After increasing the multiplicity of the knots to 2, moving the second control point from the right results in significant changes on the right side of the shape only:

Notes


Sketcher

(FIXME)

Anwenderdokumentation
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