Sketcher BSplineIncreaseKnotMultiplicity: Difference between revisions

From FreeCAD Documentation
(step 4)
No edit summary
 
(18 intermediate revisions by 4 users not shown)
Line 1: Line 1:
<languages/>
<languages/>
<translate>
<translate>

<!--T:11-->
<!--T:11-->
{{Docnav
{{Docnav
Line 7: Line 8:
|[[Sketcher_Workbench|Sketcher]]
|[[Sketcher_Workbench|Sketcher]]
|IconL=Sketcher_BSplineDecreaseDegree.svg
|IconL=Sketcher_BSplineDecreaseDegree.svg
|IconC=Workbench_Sketcher.svg
|IconR=Sketcher_BSplineDecreaseKnotMultiplicity.svg
|IconR=Sketcher_BSplineDecreaseKnotMultiplicity.svg
|IconC=Workbench_Sketcher.svg
}}
}}


Line 17: Line 18:
|Workbenches=[[Sketcher_Workbench|Sketcher]]
|Workbenches=[[Sketcher_Workbench|Sketcher]]
|Version=0.17
|Version=0.17
|SeeAlso=[[Sketcher_BSplineKnotMultiplicity|Show/hide B-spline knot multiplicity]], [[Sketcher_BSplineDecreaseKnotMultiplicity|Decrease knot multiplicity]]
|SeeAlso=[[Sketcher_BSplineKnotMultiplicity|Sketcher Show/hide B-spline knot multiplicity]], [[Sketcher_BSplineDecreaseKnotMultiplicity|Sketcher Decrease knot multiplicity]]
}}
}}


Line 23: Line 24:


<!--T:9-->
<!--T:9-->
Increases the knot multiplicity of a B-spline curve knot (see [https://en.wikipedia.org/wiki/B-spline B-spline]).
Increases the multiplicity of a B-spline knot. (See [[B-Splines|this page]] for more info about B-splines).


<!--T:15-->
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).
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 on a spline with degree ''d'' and with the multiplicity ''m'' means that the curve left and right to the knot has at least an equal ''n'' order derivative (called ''C''<sup>''n''</sup> continuity) whereas <math>n=d-m</math>.<br/>
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]]}}):
Here is a cubic spline (<math>d=3</math>) whose knots have the multiplicity 1. The multiplicity is indicated by the number in parentheses. The indication can be changed using the toolbar button {{Button|[[File:Sketcher_BSplineKnotMultiplicity.svg|24px]] [[Sketcher_BSplineKnotMultiplicity|Show/hide B-spline knot multiplicity]]}}:


</translate>
</translate>
[[File:Sketcher_KnotMultiplicity_multiplicity1.png]]
[[File:Sketcher_KnotMultiplicity_multiplicity1.png|400px]]
<translate>
<translate>
<!--T:10-->
<!--T:16-->
{{Caption|B-spline where both knots have the multiplicity 1. You can see that the second order derivatives left and right to the knots are not equal.}}
{{Caption|B-spline where both knots have the multiplicity 1.}}


<!--T:17-->
A multiplicity of 2 will change the spline so that the second order derivatives 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 spline so that even the first order derivatives are not equal (''C''<sup>0</sup> continuity). Here is the same spline where the left's knot multiplicity was increased to 3:


</translate>
</translate>
[[File:Sketcher_KnotMultiplicity_multiplicity2.png]]
[[File:Sketcher_KnotMultiplicity_multiplicity3.png|400px]]
<translate>
<translate>
<!--T:10-->
<!--T:18-->
{{Caption|B-spline where both knots have now the multiplicity 2. Note that control points were added and changed to achieve this.}}
{{Caption|B-spline from above with knot multiplicity 3. A control point was moved to show that the knot has ''C''<sup>0</sup> continuity.}}


<!--T:19-->
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:
A consequence of a higher multiplicity is that for the price of loosing continuity you gain local control. This means the change of one control point only affects the spline locally to this changed point. This can be seen in this example, where the spline from the first image above was taken and its second control point from the right side was moved up:
{{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>
</translate>
[[File:Sketcher_KnotMultiplicity_locality.png]]
[[File:Sketcher_KnotMultiplicity_locality.png|400px]]
<translate>
<translate>
<!--T:10-->
<!--T:20-->
{{Caption|B-spline curve showing increasing knot multiplicity.}}
{{Caption|Effect of locality due to different multiplicity.}}

<!--T:21-->
One can see that the spline with knot multiplicity 1 is completely changed while the one with multiplicity 2 kept its form at its left side.


==Usage== <!--T:5-->
==Usage== <!--T:5-->


<!--T:14-->
<!--T:14-->
# Select a B-spline knot.
# Select a B-spline knot, either:
#* Press the button {{Button|[[File:Sketcher_BSplineIncreaseKnotMultiplicity.svg|16px]] [[Sketcher_BSplineIncreaseKnotMultiplicity|Increase knot multiplicity]]}}.
# 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|16px]] Increase knot multiplicity}}.

#* Use the menu {{MenuCommand|Sketch → Sketcher B-spline tools → [[File:Sketcher_BSplineIncreaseKnotMultiplicity.svg|24px]] Increase knot multiplicity}}.


<!--T:8-->
<!--T:8-->
Line 66: Line 71:
|[[Sketcher_Workbench|Sketcher]]
|[[Sketcher_Workbench|Sketcher]]
|IconL=Sketcher_BSplineDecreaseDegree.svg
|IconL=Sketcher_BSplineDecreaseDegree.svg
|IconC=Workbench_Sketcher.svg
|IconR=Sketcher_BSplineDecreaseKnotMultiplicity.svg
|IconR=Sketcher_BSplineDecreaseKnotMultiplicity.svg
|IconC=Workbench_Sketcher.svg
}}
}}


</translate>
</translate>
{{Sketcher Tools navi{{#translation:}}}}
{{Sketcher_Tools_navi{{#translation:}}}}
{{Userdocnavi{{#translation:}}}}
{{Userdocnavi{{#translation:}}}}
{{clear}}

Latest revision as of 15:50, 16 January 2023

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

Increases the multiplicity of a B-spline knot. (See this page for more info about B-splines).

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 on a spline with degree d and with the multiplicity m means that the curve left and right to the knot has at least an equal n order derivative (called Cn continuity) whereas .
Here is a cubic spline () whose knots have the multiplicity 1. The multiplicity is indicated by the number in parentheses. The indication can be changed using the toolbar button Show/hide B-spline knot multiplicity:

B-spline where both knots have the multiplicity 1.

A multiplicity of 3 will change this spline so that even the first order derivatives are not equal (C0 continuity). Here is the same 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 the change of one control point only affects the spline locally to this changed point. This can be seen in this example, where the spline from the first image above was taken and its second control point from the right side was moved up:

Effect of locality due to different multiplicity.

One can see that the spline with knot multiplicity 1 is completely changed while the one with multiplicity 2 kept its form at its left side.

Usage

  1. Select a B-spline knot, either: