Sketcher BSplineIncreaseKnotMultiplicity

Sketcher BSplineIncreaseKnotMultiplicity
Menu location
Sketch → B-Spline Tools → Increase Knot Multiplicity
Workbenches
Sketcher
Default shortcut
None
Introduced in version
0.17
See also
Sketcher BSplineDecreaseKnotMultiplicity

Description

The Sketcher BSplineIncreaseKnotMultiplicity tool increases 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 Increase Knot Multiplicity button.
   • Select the Sketch → B-Spline Tools → Increase Knot Multiplicity option from the menu.

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 Toggle 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

• Knot multiplicity can also be increased with Sketcher BSplineInsertKnot.