Sketcher BSplineDecreaseKnotMultiplicity: Difference between revisions

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{{Docnav
{{Docnav
|[[Sketcher_BSplineIncreaseKnotMultiplicity|Increase knot multiplicity]]
|[[Sketcher_BSplineIncreaseKnotMultiplicity|Increase knot multiplicity]]
|[[Sketcher_BSplineInsertKnot|Insert knot]]
|[[Sketcher_SwitchVirtualSpace|Switch Virtual Space]]
|[[Sketcher_Workbench|Sketcher]]
|[[Sketcher_Workbench|Sketcher]]
|IconL=Sketcher_BSplineIncreaseKnotMultiplicity.svg
|IconL=Sketcher_BSplineIncreaseKnotMultiplicity.svg
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|Workbenches=[[Sketcher_Workbench|Sketcher]]
|Workbenches=[[Sketcher_Workbench|Sketcher]]
|Version=0.17
|Version=0.17
|SeeAlso=[[Sketcher_BSplineKnotMultiplicity|Sketcher Show/hide B-spline knot multiplicity]], [[Sketcher_BSplineIncreaseKnotMultiplicity|Sketcher BSpline Increase knot multiplicity]]
|SeeAlso=[[Sketcher_BSplineIncreaseKnotMultiplicity|Sketcher BSplineIncreaseKnotMultiplicity]]
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The [[Image:Sketcher_BSplineDecreaseKnotMultiplicity.svg|24px]] [[Sketcher_BSplineDecreaseKnotMultiplicity|Sketcher BSplineDecreaseKnotMultiplicity]] tool decreases the multiplicity of a [[B-Splines|B-spline]] knot.
Decreases the knot multiplicity of a B-spline curve knot (see [https://en.wikipedia.org/wiki/B-spline B-spline]).


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==Usage== <!--T:5-->
B-splines are basically a combination of [https://en.wikipedia.org/wiki/Bezier_curve#Constructing_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 n=d-m.<br/>
Here is a spline whose knots have the multiplicity 1 (indicated by the number in parentheses, <br/>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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# Select a B-spline knot.
[[File:Sketcher_KnotMultiplicity_multiplicity1.png|386px]]
# There are several ways to invoke the tool:
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#* Press the {{Button|[[File:Sketcher_BSplineDecreaseKnotMultiplicity.svg|16px]] [[Sketcher_BSplineDecreaseKnotMultiplicity|Decrease knot multiplicity]]}} button.
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#* Select the {{MenuCommand|Sketch → Sketcher B-spline tools → [[Image:Sketcher_BSplineDecreaseKnotMultiplicity.svg|16px]] Decrease knot multiplicity}} option from the menu.
{{Caption|B-spline where both knots have the multiplicity 1.}}


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==Example== <!--T:23-->
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:


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See [[Sketcher_BSplineIncreaseKnotMultiplicity#Example|Sketcher_BSplineIncreaseKnotMultiplicity]]
[[File:Sketcher_KnotMultiplicity_multiplicity3.png|386px]]
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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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==Notes== <!--T:25-->
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]]
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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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'''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.
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 ''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.

==Usage== <!--T:5-->

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# Select a B-spline knot
# Either:
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#* Press the button {{Button|[[File:Sketcher_BSplineDecreaseKnotMultiplicity.svg|24px]] [[Sketcher_BSplineDecreaseKnotMultiplicity|B-spline decrease knot multiplicity]]}}.
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#* Use the menu {{MenuCommand|Sketch → Sketcher B-spline tools → [[File:Sketcher_BSplineDecreaseKnotMultiplicity.svg|24px]] Decrease knot multiplicity}}.


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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 "edges", you can create two splines and connect them.)


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{{Docnav
{{Docnav
|[[Sketcher_BSplineIncreaseKnotMultiplicity|Increase knot multiplicity]]
|[[Sketcher_BSplineIncreaseKnotMultiplicity|Increase knot multiplicity]]
|[[Sketcher_BSplineInsertKnot|Insert knot]]
|[[Sketcher_SwitchVirtualSpace|Switch Virtual Space]]
|[[Sketcher_Workbench|Sketcher]]
|[[Sketcher_Workbench|Sketcher]]
|IconL=Sketcher_BSplineIncreaseKnotMultiplicity.svg
|IconL=Sketcher_BSplineIncreaseKnotMultiplicity.svg
|IconR=Sketcher_SwitchVirtualSpace.svg‎
|IconR=Sketcher_BSplineInsertKnot.svg‎
|IconC=Workbench_Sketcher.svg
|IconC=Workbench_Sketcher.svg
}}
}}


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Latest revision as of 07:35, 22 April 2024

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Sketcher BSplineDecreaseKnotMultiplicity

Menu location
Sketch → Sketcher B-spline tools → Decrease knot multiplicity
Workbenches
Sketcher
Default shortcut
None
Introduced in version
0.17
See also
Sketcher BSplineIncreaseKnotMultiplicity

Description

The Sketcher BSplineDecreaseKnotMultiplicity tool decreases the multiplicity of a B-spline knot.

Usage

  1. Select a B-spline knot.
  2. There are several ways to invoke the tool:
    • Press the Decrease knot multiplicity button.
    • Select the Sketch → Sketcher B-spline tools → Decrease knot multiplicity option from the menu.

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.


Sketcher
User documentation
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