Sketcher BSplineIncreaseKnotMultiplicity: Difference between revisions

From FreeCAD Documentation
(Marked this version for translation)
 
Line 34: Line 34:
#* Select the {{MenuCommand|Sketch → Sketcher B-spline tools → [[Image:Sketcher_BSplineIncreaseKnotMultiplicity.svg|16px]] Increase knot multiplicity}} option from the menu.
#* Select the {{MenuCommand|Sketch → Sketcher B-spline tools → [[Image:Sketcher_BSplineIncreaseKnotMultiplicity.svg|16px]] Increase knot multiplicity}} option from the menu.


==Example==
==Example== <!--T:22-->


<!--T:15-->
<!--T:15-->
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 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 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''.


<!--T:23-->
In this cubic B-spline (degree 3) there are 3 segments, meaning 3 curves are connected at 2 knots. The 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.


<!--T:24-->
The multiplicity is indicated by the numbers in round brackets. See [[File:Sketcher_BSplineKnotMultiplicity.svg|16px]] [[Sketcher_BSplineKnotMultiplicity|Show/hide B-spline knot multiplicity]].
The multiplicity is indicated by the numbers in round brackets. See [[File:Sketcher_BSplineKnotMultiplicity.svg|16px]] [[Sketcher_BSplineKnotMultiplicity|Show/hide B-spline knot multiplicity]].


Line 61: Line 63:
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.
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.


<!--T:25-->
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:
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:


Line 67: Line 70:
<translate>
<translate>


<!--T:26-->
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:
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:


Line 73: Line 77:
<translate>
<translate>


==Notes==
==Notes== <!--T:27-->


<!--T:28-->
* Knot multiplicity can also be increased with [[Sketcher_BSplineInsertKnot|Sketcher BSplineInsertKnot]].
* Knot multiplicity can also be increased with [[Sketcher_BSplineInsertKnot|Sketcher BSplineInsertKnot]].



Latest revision as of 07:34, 22 April 2024

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 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 → 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 numbers 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 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