Sketcher BSplineIncreaseKnotMultiplicity
Menu location |
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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 |
Contents |
Description
Increases the knot multiplicity of a B-spline curve knot (see B-spline).
B-splines are basically a combination of Bézier curves (nicely explained in this and 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^{n} continuity) whereas n=d-m.
Here is a cubic (degree 3) 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.
A multiplicity of 3 will change this spline so that even the first order derivatives are not equal (C^{0} 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 C^{0} 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.
Usage
- Select a B-spline knot.
- Either:
- Press the button B-spline increase knot multiplicity.
- Use the menu Sketch → Sketcher B-spline tools → Increase knot multiplicity.
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