Sketcher BSplineIncreaseKnotMultiplicity: Difference between revisions

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B-splines are basically a combination of [[B-Splines#B.C3.A9zier_curves|Bézier curves]] (nicely explained in [https://www.youtube.com/watch?v=bE1MrrqBAl8 this] and [https://www.youtube.com/watch?v=xXJylM2S72s 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 that the curve to the left and right of the knot has at least an equal ''n'' order derivative (called ''C''<sup>''n''</sup> continuity) where ''n = d - m''.
B-splines are basically a combination of [[B-Splines#B.C3.A9zier_curves|Bézier curves]] (nicely explained in [https://www.youtube.com/watch?v=bE1MrrqBAl8 this] and [https://www.youtube.com/watch?v=xXJylM2S72s 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 that the curve to the left and right of the knot has at least an equal ''n'' order derivative (called ''C''<sup>''n''</sup> continuity) where ''n = d - m''.


Here is a cubic B-spline (degree 3) whose knots have multiplicity 1.
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 [[File:Sketcher_BSplineKnotMultiplicity.svg|16px]] [[Sketcher_BSplineKnotMultiplicity|Show/hide B-spline knot multiplicity]].
The multiplicity is indicated by the number in round brackets. See [[File:Sketcher_BSplineKnotMultiplicity.svg|16px]] [[Sketcher_BSplineKnotMultiplicity|Show/hide B-spline knot multiplicity]].

Revision as of 09:17, 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 that 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:

B-spline from above 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. This means changing one control point only affects the B-spline locally. This can be seen in this example, where the B-spline from the first image was taken and the second control point from the right was moved up:

Effect of locality due to different multiplicity.

The B-spline with knot multiplicity 1 has changed completely while the one with multiplicity 2 has kept its form on the left side.