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  • A formula for the area enclosed by a planar Bezier curve of order n

    Here I present a closed-form expression for the area enclosed by an arbitrary order closed Bezier curve in the plane. The formula in itself is of limited interest given that you can trivially approximate any parametrized closed curve by a polygon and compute the polygon's area instead (say with the shoelace formula), but as always in maths, the real treasure is the friends we made along the way! The proof is a bit involved, but all we need is some basic vector calculus, combinatorics and some knowledge about the Beta function.
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