Sketcher BSplineIncreaseKnotMultiplicity: Difference between revisions

Revision as of 16:36, 5 June 2021

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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 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 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 Cⁿ continuity) whereas $n=d-m$ .
Here is a cubic spline ( $d=3$ ) 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

Select a B-spline knot, either:
- Press the button B-spline increase knot multiplicity.
- Use the menu Sketch → Sketcher B-spline tools → Increase knot multiplicity.

Decrease B-spline degree

Decrease knot multiplicity

Sketcher

General: Create sketch, Edit sketch, Map sketch to face, Reorient sketch, Validate sketch, Merge sketches, Mirror sketch, Leave sketch, View sketch, View section, Toggle grid, Toggle snap, Configure rendering order, Stop operation

Sketcher geometries: Point, Line, Arc, Arc by 3 points, Circle, Circle by 3 points, Ellipse, Ellipse by 3 points, Arc of ellipse, Arc of hyperbola, Arc of parabola, B-spline by control points, Periodic B-spline by control points, B-spline by knots, Periodic B-spline by knots, Polyline, Rectangle, Centered rectangle, Rounded rectangle, Triangle, Square, Pentagon, Hexagon, Heptagon, Octagon, Regular polygon, Slot, Fillet, Corner-preserving fillet, Trim, Extend, Split, External geometry, Carbon copy, Toggle construction geometry

Sketcher constraints:
- Geometric constraints: Coincident, Point on object, Vertical, Horizontal, Parallel, Perpendicular, Tangent, Equal, Symmetric, Block
- Dimensional constraints: Lock, Horizontal distance, Vertical distance, Distance, Radius or weight, Diameter, Auto radius/diameter, Angle, Refraction (Snell's law)
- Constraint tools: Toggle driving/reference constraint, Activate/deactivate constraint

Sketcher tools: Select unconstrained DoF, Select associated constraints, Select associated geometry, Select redundant constraints, Select conflicting constraints, Show/hide internal geometry, Select origin, Select horizontal axis, Select vertical axis, Symmetry, Clone, Copy, Move, Rectangular array, Remove axes alignment, Delete all geometry, Delete all constraints

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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Additional: Sketcher Dialog, Preferences, Sketcher scripting

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@@ Line 7: / Line 7: @@
 |[[Sketcher_Workbench|Sketcher]]
 |IconL=Sketcher_BSplineDecreaseDegree.svg
-|IconC=Workbench_Sketcher.svg
 |IconR=Sketcher_BSplineDecreaseKnotMultiplicity.svg
+|IconC=Workbench_Sketcher.svg
 }}
@@ Line 17: / Line 17: @@
 |Workbenches=[[Sketcher_Workbench|Sketcher]]
 |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 23: @@
 <!--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 in FreeCAD are basically a combination of [https://en.wikipedia.org/wiki/Bezier_curve#Constructing_B%C3%A9zier_curves Bézier curves]. 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 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 <math>n=d-m</math>.<br/>
-Here is a spline whose knots have the multiplicity 1 (indicated by the number in parentheses, 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>
-[[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.}}
+<!--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 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>
-[[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|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-->
-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. 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:
-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:
-{{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_locality.png|400px]]
 <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-->
 <!--T:14-->
-# Select a B-spline knot.
+# Select a B-spline knot, either:
-# 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}}.
@@ Line 68: / Line 69: @@
 |[[Sketcher_Workbench|Sketcher]]
 |IconL=Sketcher_BSplineDecreaseDegree.svg
-|IconC=Workbench_Sketcher.svg
 |IconR=Sketcher_BSplineDecreaseKnotMultiplicity.svg
+|IconC=Workbench_Sketcher.svg
 }}