Sketcher BSplineIncreaseKnotMultiplicity: Difference between revisions

Latest revision as of 11:16, 2 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 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 left's knot multiplicity 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:

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

Sketcher virtual space: Switch virtual space

Additional: Sketcher Dialog, Preferences, Sketcher scripting

User documentation

Getting started
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Hubs: User hub, Power users hub, Developer hub

@@ Line 1: / Line 1: @@
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+{{Page_in_progress}}
 <translate>
 <!--T:11-->
 {{Docnav
@@ Line 7: / Line 9: @@
 |[[Sketcher_Workbench|Sketcher]]
 |IconL=Sketcher_BSplineDecreaseDegree.svg
-|IconC=Workbench_Sketcher.svg
 |IconR=Sketcher_BSplineDecreaseKnotMultiplicity.svg
+|IconC=Workbench_Sketcher.svg
 }}
@@ Line 17: / Line 19: @@
 |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 25: @@
 <!--T:9-->
+The [[Image:Sketcher_BSplineIncreaseKnotMultiplicity.svg|24px]] [[Sketcher_BSplineIncreaseKnotMultiplicity|Sketcher BSplineIncreaseKnotMultiplicity]] tool increases the multiplicity of [[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 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).
-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]]}}):
+<!--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==
-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:
+<!--T:15-->
-Note that the control points were added and changed to achieve this.
+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''.
+In this cubic B-spline (degree 3) there are 3 segments, meaning 3 curves are connected at 2 knots. The knots have multiplicity 1.
-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 multiplicity is indicated by the numbers in round brackets. See [[File:Sketcher_BSplineKnotMultiplicity.svg|16px]] [[Sketcher_BSplineKnotMultiplicity|Show/hide B-spline knot 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_multiplicity1.png|400px]]
-[[File:Sketcher_SampleBSplineIncreaseKnotMultiplicity_example.png]]
 <translate>
-<!--T:10-->
+<!--T:16-->
-{{Caption|B-spline curve showing increasing knot multiplicity.}}
+{{Caption|B-spline where both knots have multiplicity 1.}}
-==Usage== <!--T:5-->
+<!--T:17-->
+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 left's knot multiplicity was increased to 3:
+</translate>
+[[File:Sketcher_KnotMultiplicity_multiplicity3.png|400px]]
+<translate>
+<!--T:18-->
+{{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-->
+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:
+</translate>
+[[File:Sketcher_KnotMultiplicity_locality1.png|400px]]
+<translate>
+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>
-<!--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-->
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 |[[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}}