Sketcher BSplineIncreaseKnotMultiplicity: Difference between revisions

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Increases the knot multiplicity of a B-spline curve knot (see [https://en.wikipedia.org/wiki/B-spline B-spline]).
Increases the knot multiplicity of a B-spline curve knot (see [https://en.wikipedia.org/wiki/B-spline B-spline]).

B-splines in FreeCAD are basically a combination of [https://en.wikipedia.org/wiki/Bezier_curve#Constructing_B%C3%A9zier_curves 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 ''C''<sup>1</sup> 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 {{Button|[[File:Sketcher_BSplineKnotMultiplicity.svg|24px]] [[Sketcher_BSplineKnotMultiplicity|Show/hide B-spline knot multiplicity]]}}):


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:

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:
{{Caption|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.


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Revision as of 00:56, 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):


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:

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.

File:Sketcher SampleBSplineIncreaseKnotMultiplicity example.png

B-spline curve showing increasing knot multiplicity.

Usage

  1. Select a B-spline knot.
  2. Invoke the tool using several methods:
Sketcher
User documentation
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