Bezier curves are polynomial parametric curves defined using control points in the Bernstein basis. Quadratic and cubic Bezier curves are commonly used, with the latter having no closed form for arc length. Computing arc length parametrization for quadratic Beziers has no closed form solution either. By assuming Schanuel’s conjecture, a deep dive into abstract math reveals that no such formula exists. Lin’s theorem, involving irreducible polynomials and elementary numbers, is leveraged to show the impossibility of achieving the desired result for quadratic Bezier curves. Despite conditional results, Lin’s theorem effectively demonstrates the limitations of closed form expression in this context.
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