Sketcher BSplinePoleWeight: Difference between revisions

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(the weight change had been changed for FC 0.19 -> refer to existing description)
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<languages/>
<languages/>
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{{Docnav
{{Docnav
|[[Sketcher_BSplineKnotMultiplicity| Show/Hide B-spline knot multiplicity]]
|[[Sketcher_BSplineKnotMultiplicity|Show/hide B-spline knot multiplicity]]
|[[Sketcher_ArcOverlay|Show/hide circular helper for arcs]]
|[[Sketcher_BSplineApproximate|Convert Geometry to B-spline]]
|[[Sketcher_Workbench|Sketcher]]
|[[Sketcher_Workbench|Sketcher]]
|IconL=Sketcher_BSplineKnotMultiplicity.svg
|IconL=Sketcher_BSplineKnotMultiplicity.svg
|IconR=Sketcher_BSplineApproximate.svg
|IconR=Sketcher_ArcOverlay.svg
|IconC=Workbench_Sketcher.svg
|IconC=Workbench_Sketcher.svg
}}
}}
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{{GuiCommand
{{GuiCommand
|Name=Sketcher BSplinePoleWeight
|Name=Sketcher BSplinePoleWeight
|MenuLocation=Sketch → Sketcher B-spline tools → Show/Hide B-spline control point weight
|MenuLocation=Sketch → Sketcher visual → Show/hide B-spline information layer → Show/hide B-spline control point weight
|Workbenches=[[Sketcher_Workbench|Sketcher]]
|Workbenches=[[Sketcher_Workbench|Sketcher]]
|Version=0.19
|Version=0.19
|SeeAlso=[[Sketcher_CompCreateBSpline|Sketcher Create B-spline]]
|SeeAlso=[[Sketcher_CreateBSpline|Sketcher CreateBSpline]]
}}
}}


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The [[Image:Sketcher_BSplinePoleWeight.svg|24px]] [[Sketcher_BSplinePoleWeight|Sketcher BSplinePoleWeight]] tool shows or hides the [[B-Splines|B-spline]] control point weight in all sketches.
Shows or hides the display of the '''weights''' for the control points of a B-spline curve (see [[#Explanation|below]] for an explanation of weights).


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{{Caption|B-spline with control point weights displayed in brackets}}
{{Caption|The numbers (green, in square brackets) refer to the weights of the control points.}}


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


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# There are several ways to invoke the tool:
# Select a B-spline and apply.
#* Press the {{Button|[[Image:Sketcher_BSplinePoleWeight.svg|16px]] [[Sketcher_BSplinePoleWeight|Show/hide B-spline control point weight]]}} button.
#* Select the {{MenuCommand|Sketch → Sketcher visual → Show/hide B-spline information layer → [[Image:Sketcher_BSplinePoleWeight.svg|16px]] Show/hide B-spline control point weight}} option from the menu.


==Weight Explanation== <!--T:16-->
==Changing weights== <!--T:29-->


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⚫
How to change weights is described on [[B-Splines#Changing_the_Weight|this page]].
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).


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The Bézier curve is calculated using this formula:

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<math>\quad
\textrm{Bezier}(n,t)=\sum_{i=0}^{n}\underbrace{\binom{n}{i}}_{\text{polynomial term}}\underbrace{\left(1-t\right)^{n-i}t^{i}}_{\text{polynomial term}}\; \underbrace{P_{i}}_{\text{point coordinate}}
</math>

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''n'' is hereby the degree of the curve. So a Bézier curve of degree ''n'' is a polygon with order ''n''. The factors <math>P_{i}</math> are hereby in fact the coordinates of the Bézier curves' control points. For a visualization see [https://pomax.github.io/bezierinfo/#control this page].

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The term weight in FreeCAD is a bit misleading because in literature the factors <math>P_{i}</math> are often called weights as well. FreeCAD's weights are something different. The idea of these weights is to modify the spline so that the different control points are "weighted". The idea is that a point with weight 2 should have twice as much influence than a point with weight 1. This is achieved by using this different formula to calculate the spline:

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<math>\quad
\mathrm{Rational\ Bezier}(n,t)=\cfrac{\sum_{i=0}^{n}\binom{n}{i}\left(1-t\right)^{n-i}t^{i}w_{i}P_{i}}{\sum_{i=0}^{n}\binom{n}{i}\left(1-t\right)^{n-i}t^{i}w_{i}\;\;\;\,}
</math>

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whereby <math>w_{i}</math> is the weight for the point <math>P_{i}</math>.

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This is a new class of Bézier curves because despite the points are indeed weighted as desired, the curve is no longer a polynomial but a fractional polynomial. Therefore these curves are called rational Bézier curves and the B-splines is then called rational B-splines.

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The consequence is that you gain more flexibility in defining the spline shape. If all weights are equal, the shape of the spline does not change. So the weights relative to each other is important, not the value alone. For example this spline has exactly the same shape as the one in the first image:

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[[File:sketcher_BSplineWeightDouble.png|468px]]
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{{Caption|Same B-spline as in first image but with different absolute weight values}}

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A weight of zero would be a singularity in the equation to calculate the rational Bézier curves, therefore FreeCAD assures that it cannot become zero. Nevertheless, small values have the same effect as if the control point would almost not exist:

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[[File:sketcher_BSplineWeightZero.png|468px]]
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{{Caption|Same B-spline with an almost zero weight control point}}

==Changing Weights== <!--T:29-->

<!--T:30-->
⚫
How to change weights is described in [[B-Splines#Changing_the_Weight| this Wiki page]].


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{{Docnav
{{Docnav
|[[Sketcher_BSplineKnotMultiplicity|Show/Hide B-spline knot multiplicity]]
|[[Sketcher_BSplineKnotMultiplicity|Show/hide B-spline knot multiplicity]]
|[[Sketcher_ArcOverlay|Show/hide circular helper for arcs]]
|[[Sketcher_BSplineApproximate|Convert Geometry to B-spline]]
|[[Sketcher_Workbench|Sketcher]]
|[[Sketcher_Workbench|Sketcher]]
|IconL=Sketcher_BSplineKnotMultiplicity.svg
|IconL=Sketcher_BSplineKnotMultiplicity.svg
|IconR=Sketcher_BSplineApproximate.svg
|IconR=Sketcher_ArcOverlay.svg
|IconC=Workbench_Sketcher.svg
|IconC=Workbench_Sketcher.svg
}}
}}


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{{Sketcher Tools navi{{#translation:}}}}
{{Sketcher_Tools_navi{{#translation:}}}}
{{Userdocnavi{{#translation:}}}}
{{Userdocnavi{{#translation:}}}}
{{clear}}

Latest revision as of 09:00, 23 April 2024

Other languages:

Sketcher BSplinePoleWeight

Menu location
Sketch → Sketcher visual → Show/hide B-spline information layer → Show/hide B-spline control point weight
Workbenches
Sketcher
Default shortcut
None
Introduced in version
0.19
See also
Sketcher CreateBSpline

Description

The Sketcher BSplinePoleWeight tool shows or hides the B-spline control point weight in all sketches.

The numbers (green, in square brackets) refer to the weights of the control points.

Usage

  1. There are several ways to invoke the tool:
    • Press the Show/hide B-spline control point weight button.
    • Select the Sketch → Sketcher visual → Show/hide B-spline information layer → Show/hide B-spline control point weight option from the menu.

Changing weights

How to change weights is described on this page.


Sketcher
User documentation
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