Sketcher BSplinePoleWeight

Show/Hide B-spline knot multiplicity

Convert Geometry to B-spline

Sketcher

Sketcher BSplinePoleWeight
Menu location
Sketch → Sketcher B-spline tools → Show/Hide B-spline control point weight
Workbenches
Sketcher
Default shortcut
None
Introduced in version
0.19
See also
Create B-spline

Description

Shows or hides the display of the weight for the control points of a B-spline curve (see B-spline).

B-spline with control point weights displayed in brackets

File:Sketcher BSplineWeightHide.png

Same B-spline without weights

The weights determine how the control points are taken into account. The term weight is hereby a bit misleading because in literature the coordinates of the control points are often described as weights since they appear as factor in the definition of the Bézier curves:

$\quad {\textrm {Bezier}}(n,t)=\sum _{i=0}^{n}\underbrace {\binom {n}{i}} _{\text{polynomial term}}\underbrace {\left(1-t\right)^{n-i}t^{i}} _{\text{polynomial term}}\underbrace {w_{i}} _{\text{weight}}$

n is hereby the degree of the curve. So a Bézier curve of degree n is a polygon with order n. B-splines are basically a combination of Bézier curves (nicely explained in this and this video).

The spline weight in FreeCAD is however something different. The idea is to modify the spline so that the different control points are "weighted". So a point with weight 2 should have twice as much influence than a point with weight 1. This is achieved by using this formula to calculate the spline:

$\quad {\textrm {Bezier}}(n,t)=\sum _{i=0}^{n}\underbrace {\binom {n}{i}} _{\text{polynomial term}}\underbrace {\left(1-t\right)^{n-i}t^{i}} _{\text{polynomial term}}\underbrace {w_{i}} _{\text{weight}}$

Therefore, 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:

Same B-spline as in first image but with different absolute weight values

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

Same B-spline with a zero weight control point

Usage

1. Select a B-spline and apply.

Show/Hide B-spline knot multiplicity

Convert Geometry to B-spline

Sketcher

Sketcher

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• 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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