Sketcher BSplineIncreaseKnotMultiplicity: Difference between revisions

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After increasing the multiplicity of the knots to 2, moving the second control point from the right results in changes on the right side of the shape, whereas the left side has not changed significantly:
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:


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Revision as of 09:57, 2 April 2024

This documentation is a work in progress. Please don't mark it as translatable since it will change in the next hours and days.

Sketcher BSplineIncreaseKnotMultiplicity

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

Description

The Sketcher BSplineIncreaseKnotMultiplicity tool increases the multiplicity of a B-spline knot. See this page for more information about B-splines.

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 → Sketcher 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 number in round brackets. See Show/hide 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 left's knot multiplicity 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: