Difference between revisions of "Sketcher BSplineDecreaseKnotMultiplicity"
(update see also) |
(update page according to the increase page) |
||
| Line 24: | Line 24: | ||
<!--T:9--> | <!--T:9--> | ||
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 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]]}}): | ||
| + | |||
| + | </translate> | ||
| + | [[File:Sketcher_KnotMultiplicity_multiplicity1.png|386px]] | ||
| + | <translate> | ||
| + | <!--T:10--> | ||
| + | {{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.}} | ||
| + | |||
| + | 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: | ||
</translate> | </translate> | ||
| − | [[File: | + | [[File:Sketcher_KnotMultiplicity_multiplicity2.png|386px]] |
<translate> | <translate> | ||
<!--T:10--> | <!--T:10--> | ||
| − | {{Caption|B-spline | + | {{Caption|B-spline where both knots have now the multiplicity 2. Note that control points were added and changed to achieve this.}} |
| + | |||
| + | 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 in this example, where the splines from the images above were taken and their the second control point from the right side was moved up: | ||
| + | </translate> | ||
| + | [[File:Sketcher_KnotMultiplicity_locality.png]] | ||
| + | <translate> | ||
| + | {{Caption|Effect of locality due to different multiplicity.}} | ||
| + | One can see that the spline with knot multiplicity 1is completely changed while the one with multiplicity 2 kept its form at its left side. | ||
==Usage== <!--T:5--> | ==Usage== <!--T:5--> | ||
| Line 35: | Line 53: | ||
<!--T:14--> | <!--T:14--> | ||
# Select a B-spline knot | # Select a B-spline knot | ||
| − | # | + | # Either: |
| − | #* Press the {{Button|[[File:Sketcher_BSplineDecreaseKnotMultiplicity.svg| | + | #* Press the button {{Button|[[File:Sketcher_BSplineDecreaseKnotMultiplicity.svg|24px]] [[Sketcher_BSplineDecreaseKnotMultiplicity|B-spline decrease knot multiplicity]]}}. |
| − | #* Use the {{MenuCommand|Sketch → Sketcher B-spline tools → [[File:Sketcher_BSplineDecreaseKnotMultiplicity.svg| | + | #* Use the menu {{MenuCommand|Sketch → Sketcher B-spline tools → [[File:Sketcher_BSplineDecreaseKnotMultiplicity.svg|24px]] 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''<sup>0</sup> continuity) and this is not supported. (To create curves with an "edges", you can create two splines and connect them.) | ||
<!--T:8--> | <!--T:8--> | ||
Revision as of 01:46, 1 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 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 C1 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 
Show/hide B-spline knot multiplicity):
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.
A multiplicity of 2 will change the spline so that the second order derivatives becomes equal (C2 continuity). Here is the same spline where the multiplicity was increased to 2:
B-spline where both knots have now the multiplicity 2. Note that control points were added and changed to achieve this.
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 in this example, where the splines from the images above were taken and their the 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 1is 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.
- Press the button
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 (C0 continuity) and this is not supported. (To create curves with an "edges", you can create two splines and connect them.)
Sketcher
- The tools: New sketch, Edit sketch, Leave sketch, View sketch, View section, Map sketch to face, Reorient sketch, Validate sketch, Merge sketches, Mirror sketch
- Sketcher geometries: Point, Line by 2 point, Create an arc, Arc, Arc by 3 Point, Create a circle, Circle, Circle by 3 Point, Create a conic, Ellipse by center, Ellipse by 3 points, Arc of ellipse, Arc of hyperbola, Arc of parabola, Create a B-spline, Create B-spline, Create periodic B-pline, Polyline (multiple-point line), Rectangle, Create regular polygon, Triangle, Square, Pentagon, Hexagon, Heptagon, Octagon, Create Regular Polygon, Slot, Fillet, Trimming, Extend, External Geometry, CarbonCopy, Construction Mode
- Sketcher constraints
- Geometric constraints Coincident, Point On Object, Vertical, Horizontal, Parallel, Perpendicular, Tangent, Equal Length, Symmetric, Constrain Block
- Dimensional constraints Lock, Horizontal Distance, Vertical Distance, Distance, Radius, Internal Angle, Snell's Law, Internal Alignment, Toggle reference/driving constraint,
- Sketcher tools Select solver DOFs, Close Shape, Connect Edges, Select Constraints, Select Origin, Select Vertical Axis, Select Horizontal Axis, Select Redundant Constraints, Select Conflicting Constraints, Select Elements Associated with constraints, Show/Hide internal geometry, Symmetry, Clone, Copy, Move, Rectangular Array, 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, Convert Geometry to B-spline, Increase degree, Increase knot multiplicity, Decrease knot multiplicity
- Sketcher virtual space Switch Virtual Space
User documentation
- Getting started
- Installation: Download, Windows, Linux, Mac, Additional components, Docker, AppImage, Ubuntu Snap
- Basics: About FreeCAD, Interface, Mouse navigation, Selection methods, Object name, Preferences, Workbenches, Document structure, Properties; Help FreeCAD, Donate
- Help: Tutorials, Video tutorials
- Workbenches: Std Base; Arch, Draft, FEM, Image, Inspection, Mesh, OpenSCAD, Part, PartDesign, Path, Points, Raytracing, Reverse Engineering, Sketcher, Spreadsheet, Start, Surface, TechDraw, Test Framework, Web
- Deprecated or unmaintained workbenches: Complete, Drawing, Robot
- Hubs: User hub, Power users hub, Developer hub
