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I was just reading here about degree elevation in Bezier curves and I noticed that in the diagrams of the progressively higher degree curve, that the control points began to approximate the curve itself. I tried some more curves and they seem to follow this general rule as well, but are there any counter examples to this?

If not, it has me thinking... one way people render bezier curves is by "flattening" them... breaking them up into sections that are either linearly or sometimes quadratically interpolatable within some tolerance. Maybe there is some play elevating a curve quite a number of times and then using the control points as a curve approximation instead? It doesn't seem like it'd be a better idea at first blush, but maybe it'd be easier to implement in GPGPU due to less branching or something...

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Well, if you keep performing degree elevation to a Bezier curve, the control points will eventually converge to the curve itself. So, in some sense you can indeed use the degree elevated control polygon as an approximation to the Bezier curve itself. However, since the convergence is relatively slow, you will have to elevate the curve to a very high degree for the control polygon to resemble the curve's shape.

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