Sketcher BSplinePoleWeight/de

Beschreibung
Blendet die Darstellung der Kontrollpunktgewichte für die Kontrollpunkte einer B-Spline-Kurve ein oder aus (siehe unten für eine Erklärung der Gewichte).



Anwendung

 * 1) Eine B-Spline-Kurve auswählen.
 * 2) Die Schaltfläche drücken.

Gewicht, kurz erklärt
B-Splines sind im Grunde eine Kombination aus Bézierkurven (sehr schön erklärt in diesem und diesem Video).

Die Bézierkurve wird mit dieser Formel berechnet:

$$\quad \textrm{Bezier}(n,t)=\sum_{i=0}^{n}\underbrace{\binom{n}{i}}_{\text{Polynomausdruck}}\underbrace{\left(1-t\right)^{n-i}t^{i}}_{\text{Polynomausdruck}}\; \underbrace{P_{i}}_{\text{Punktkoordinate}} $$

Dabei ist n der Grad der Kurve. Eine Bézierkurve vom Grad n ist also ein Polygon der Ordnung n. Die Faktoren $$P_{i}$$ sind dabei die Koordinaten der Kontrollpunkte der Bézierkurven. Zur Veranschaulichung siehe diese Seite.

The term weight in FreeCAD is a bit misleading because in literature the factors $$P_{i}$$ 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:

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

whereby $$w_{i}$$ is the weight for the point $$P_{i}$$.

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.

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:



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:



Changing Weights
How to change weights is described in this Wiki page.