Sketcher BSplineIncreaseKnotMultiplicity: Difference between revisions

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Here is a spline whose knots have the multiplicity 1 (indicated by the number in parentheses, indication can be changed using the toolbar button {{Button|[[File:Sketcher_BSplineKnotMultiplicity.svg|24px]] [[Sketcher_BSplineKnotMultiplicity|Show/hide B-spline knot multiplicity]]}}):
Here is a spline whose knots have the multiplicity 1 (indicated by the number in parentheses, 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]]
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{{Caption|B-spline where both knots have the multiplicity 1.}}


You can see that the second order derivative left and right to the knots is not equal. A multiplicity of 2 will change the spline so that the second order derivative becomes equal (''C''<sup>2</sup> continuity). Here is the same spline where the multiplicity was increased to 2:
You can see that the second order derivative left and right to the knots is not equal. A multiplicity of 2 will change the spline so that the second order derivative becomes equal (''C''<sup>2</sup> continuity). Here is the same spline where the multiplicity was increased to 2:

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[[File:Sketcher_KnotMultiplicity_multiplicity2.png]]
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{{Caption|B-spline where both knots have the multiplicity 2.}}


Note that the control points were added and changed to achieve this.
Note that the control points were added and changed to achieve this.
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[[File:Sketcher_KnotMultiplicity_locality.png]]
[[File:Sketcher_SampleBSplineIncreaseKnotMultiplicity_example.png]]
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Revision as of 01:02, 1 November 2020

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Sketcher BSplineIncreaseKnotMultiplicity

Menu location
Sketch → Sketcher B-spline tools → Increase knot multiplicity
Workbenches
Sketcher
Default shortcut
None
Introduced in version
0.17
See also
Show/hide B-spline knot multiplicity, Decrease knot multiplicity

Description

Increases the knot multiplicity of a B-spline curve knot (see B-spline).

B-splines in FreeCAD are basically a combination of Bézier curves. The points where two Bézier curves are connected to form the spline is called the knot. A knot with the multiplicity 1 means that the curve left and right to the knot has at least the same first order derivative (called C1 continuity). Here is a spline whose knots have the multiplicity 1 (indicated by the number in parentheses, indication can be changed using the toolbar button Show/hide B-spline knot multiplicity):

B-spline where both knots have the multiplicity 1.

You can see that the second order derivative left and right to the knots is not equal. A multiplicity of 2 will change the spline so that the second order derivative becomes equal (C2 continuity). Here is the same spline where the multiplicity was increased to 2:

File:Sketcher KnotMultiplicity multiplicity2.png

B-spline where both knots have the multiplicity 2.

Note that the control points were added and changed to achieve this.

Another consequence of a multiplicity of 2 is that you gain local control. This means the change of one control point only affects the splice locally to this changed point. This can be seen din this example:

Difference of locality due to different multiplicity.

The spline with knot multiplicity is completely changed while the one with multiplicity 2 is not due to the additions control points to keep the second order derivative equal.

B-spline curve showing increasing knot multiplicity.

Usage

  1. Select a B-spline knot.
  2. Either:
Sketcher
User documentation
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