I know how to express a circle as a quadratic rational Bezier curve. Now I need to do it for a cubic one. I'm not sure how to choose the weights. Also I haven't found any online resources so I'm thankful for any help!
2026-03-26 16:26:35.1774542395
Semicircle as cubic rational bezier curve
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If you already have a rational quadratic representation of the semi-circle, you can just use standard degree elevation formulae (in 4D) to convert it to a cubic.
One wrinkle is that in your rational quadratic, the weight of the middle control point must be zero. This is a little odd, but it doesn’t matter — the degree elevation process will still work.
The result you get for a unit semi-circle has the following data:
Control points: $(1,0),(1,2),(-1,2),(-1,0)$.
Weights: $1, \tfrac13, \tfrac13, 1$.
This curve is not entirely satisfactory: its parameter distribution is fairly uneven, and the convex hull of the control points is much larger than the curve itself.