Difference between revisions of "Sketcher BSplineDecreaseKnotMultiplicity"
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Decreases the knot multiplicity of a B-spline curve knot (see [https://en.wikipedia.org/wiki/B-spline B-spline]). | Decreases the knot multiplicity of a B-spline curve knot (see [https://en.wikipedia.org/wiki/B-spline B-spline]). | ||
− | 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 | + | 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 are called knots. A knot on a degree ''d'' spline 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 n=d-m.<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 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]]}}): | ||
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− | {{Caption|B-spline where both knots have the multiplicity 1 | + | {{Caption|B-spline where both knots have the multiplicity 1.}} |
− | A multiplicity of | + | 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: | + | [[File:Sketcher_KnotMultiplicity_multiplicity3.png|386px]] |
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− | + | {{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.}} | |
− | {{Caption|B-spline | + | |
+ | 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: | ||
− | |||
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[[File:Sketcher_KnotMultiplicity_locality.png]] | [[File:Sketcher_KnotMultiplicity_locality.png]] | ||
<translate> | <translate> | ||
{{Caption|Effect of locality due to different multiplicity.}} | {{Caption|Effect of locality due to different multiplicity.}} | ||
− | One can see that the spline with knot 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== <!--T:5--> | ==Usage== <!--T:5--> |
Revision as of 01:44, 2 November 2020
Menu location |
---|
Sketch → Sketcher B-spline tools → Decrease knot multiplicity |
Workbenches |
Sketcher |
Default shortcut |
None |
Introduced in version |
0.17 |
See also |
Show/hide B-spline knot multiplicity, Increase knot multiplicity |
Contents |
Description
Decreases the knot multiplicity of a B-spline curve knot (see B-spline).
B-splines are basically a combination of Bézier curves (nicely explained in this and this video). The points where two Bézier curves are connected to form the spline are called knots. A knot on a degree d spline with the multiplicity m means that the curve left and right to the knot has at least an equal n order derivative (called C^{n} continuity) whereas n=d-m.
Here is a spline whose knots have the multiplicity 1 (indicated by the number in parentheses,
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 (C^{0} 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 C^{0} 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
- Select a B-spline knot
- Either:
- Press the button B-spline decrease knot multiplicity.
- Use the menu Sketch → Sketcher B-spline tools → Decrease knot multiplicity.
Note: Decreasing the multiplicity from 1 to 0 will remove the knot since the result would be a curve with an "edge" at the knot position (C^{0} continuity) and this is not supported. (To create curves with an "edges", you can create two splines and connect them.)
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