Sketcher BSplineIncreaseKnotMultiplicity: Difference between revisions

Latest revision as of 07:34, 22 April 2024

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Decrease B-spline degree

Decrease knot multiplicity

Sketcher

Menu location
Sketcher BSplineIncreaseKnotMultiplicity
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

Select a B-spline knot.
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

Knot multiplicity can also be increased with Sketcher BSplineInsertKnot.

Decrease B-spline degree

Decrease knot multiplicity

Sketcher

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Sketcher B-spline tools: Show/hide B-spline degree, Show/hide B-spline control polygon, Show/hide B-spline curvature comb, Show/hide B-spline knot multiplicity, Show/hide B-spline control point weight, Convert geometry to B-spline, Increase B-spline degree, Decrease B-spline degree, Increase knot multiplicity, Decrease knot multiplicity, Insert knot, Join curves

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@@ Line 1: / Line 1: @@
 <languages/>
 <translate>
 <!--T:11-->
 {{Docnav
@@ Line 7: / Line 8: @@
 |[[Sketcher_Workbench|Sketcher]]
 |IconL=Sketcher_BSplineDecreaseDegree.svg
-|IconC=Workbench_Sketcher.svg
 |IconR=Sketcher_BSplineDecreaseKnotMultiplicity.svg
+|IconC=Workbench_Sketcher.svg
 }}
@@ Line 17: / Line 18: @@
 |Workbenches=[[Sketcher_Workbench|Sketcher]]
 |Version=0.17
-|SeeAlso=[[Sketcher_BSplineKnotMultiplicity|Show/hide B-spline knot multiplicity]], [[Sketcher_BSplineDecreaseKnotMultiplicity|Decrease knot multiplicity]]
+|SeeAlso=[[Sketcher_BSplineDecreaseKnotMultiplicity|Sketcher BSplineDecreaseKnotMultiplicity]]
 }}
@@ Line 23: / Line 24: @@
 <!--T:9-->
+The [[Image:Sketcher_BSplineIncreaseKnotMultiplicity.svg|24px]] [[Sketcher_BSplineIncreaseKnotMultiplicity|Sketcher BSplineIncreaseKnotMultiplicity]] tool increases the multiplicity of a [[B-Splines|B-spline]] knot.
-Increases the knot multiplicity of a B-spline curve knot (see [https://en.wikipedia.org/wiki/B-spline B-spline]).
+==Usage== <!--T:5-->
-B-splines are basically a combination of [https://en.wikipedia.org/wiki/Bezier_curve#Constructing_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 curves are connected to form the spline is called the knot. A knot with the multiplicity 1 means that the curve left and right to the knot has at least the same first order derivative (called ''C''<sup>1</sup> continuity).
-Here is a spline whose knots have the multiplicity 1 (indicated by the number in parentheses, <br/>indication can be changed using the toolbar button {{Button|[[File:Sketcher_BSplineKnotMultiplicity.svg|24px]] [[Sketcher_BSplineKnotMultiplicity|Show/hide B-spline knot multiplicity]]}}):
+<!--T:14-->
+# Select a B-spline knot.
+# There are several ways to invoke the tool:
+#* Press the {{Button|[[File:Sketcher_BSplineIncreaseKnotMultiplicity.svg|16px]] [[Sketcher_BSplineIncreaseKnotMultiplicity|Increase knot multiplicity]]}} button.
+#* Select the {{MenuCommand|Sketch → Sketcher B-spline tools → [[Image:Sketcher_BSplineIncreaseKnotMultiplicity.svg|16px]] Increase knot multiplicity}} option from the menu.
+==Example== <!--T:22-->
+<!--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''.
+<!--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.
+<!--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]].
 </translate>
-[[File:Sketcher_KnotMultiplicity_multiplicity1.png]]
+[[File:Sketcher_KnotMultiplicity_multiplicity1.png|400px]]
 <translate>
-<!--T:10-->
+<!--T:16-->
-{{Caption|B-spline where both knots have the multiplicity 1.}}
+{{Caption|B-spline where both knots have multiplicity 1.}}
+<!--T:17-->
-You can see that the second order derivative left and right to the knots is not equal. A multiplicity of 2 will change the spline so that the second order derivative becomes equal (''C''<sup>2</sup> continuity). Here is the same spline where the multiplicity was increased to 2:
+A multiplicity of 3 will change this B-spline so that even the first order derivatives are not equal (''C<sup>0</sup>'' continuity). Here is the same B-spline where the multiplicity of the knot on the left was increased to 3:
 </translate>
-[[File:Sketcher_KnotMultiplicity_multiplicity2.png]]
+[[File:Sketcher_KnotMultiplicity_multiplicity3.png|400px]]
 <translate>
-<!--T:10-->
+<!--T:18-->
-{{Caption|B-spline where both knots have the multiplicity 2.}}
+{{Caption|Same B-spline with knot multiplicity 3. A control point was moved to show that the knot has ''C<sup>0</sup>'' continuity.}}
+<!--T:19-->
-Note that the control points were added and changed to achieve this.
+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-->
-Another consequence of a multiplicity of 2 is that you gain local control. This means the change of one control point only affects the splice locally to this changed point. This can be seen din this example:
+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:
-{{Caption|Difference of locality due to different multiplicity.}}
-The spline with knot multiplicity is completely changed while the one with multiplicity 2 is not due to the additions control points to keep the second order derivative equal.
 </translate>
-[[File:Sketcher_KnotMultiplicity_locality.png]]
+[[File:Sketcher_KnotMultiplicity_locality1.png|400px]]
 <translate>
-<!--T:10-->
-{{Caption|B-spline curve showing increasing knot multiplicity.}}
-==Usage== <!--T:5-->
+<!--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:
+</translate>
+[[File:Sketcher_KnotMultiplicity_locality2.png|400px]]
+<translate>
+==Notes== <!--T:27-->
+<!--T:28-->
+* Knot multiplicity can also be increased with [[Sketcher_BSplineInsertKnot|Sketcher BSplineInsertKnot]].
-<!--T:14-->
-# Select a B-spline knot.
-# Either:
-#* Press the button {{Button|[[File:Sketcher_BSplineIncreaseKnotMultiplicity.svg|24px]] [[Sketcher_BSplineIncreaseKnotMultiplicity|B-spline increase knot multiplicity]]}}.
-#* Use the menu {{MenuCommand|Sketch → Sketcher B-spline tools → [[File:Sketcher_BSplineIncreaseKnotMultiplicity.svg|24px]] Increase knot multiplicity}}.
 <!--T:8-->
@@ Line 68: / Line 89: @@
 |[[Sketcher_Workbench|Sketcher]]
 |IconL=Sketcher_BSplineDecreaseDegree.svg
-|IconC=Workbench_Sketcher.svg
 |IconR=Sketcher_BSplineDecreaseKnotMultiplicity.svg
+|IconC=Workbench_Sketcher.svg
 }}
 </translate>
-{{Sketcher Tools navi{{#translation:}}}}
+{{Sketcher_Tools_navi{{#translation:}}}}
 {{Userdocnavi{{#translation:}}}}
-{{clear}}