Numerical Techniques for Isotopic Bezier Spline Approximation
A popular technique for Bezier spline approximation is subdivision of the control polygon. This work investigates the crucial topological question of when this piecewise linear (PL) approximation and the underlying spline curve have the same embedding in 3-dimensional space. We compute a neighborhood where the curve and its PL approximant under subdivision have the same embedding.
Furthermore, many perturbations within this neighborhood maintain the initial embedding. The static approximation guarantees are important for graphics and the dynamic assurances are crucial for animation. The use of Newton's method to solve the associated system of equations of degree 2n-1 provides no guarantee of convergence. We present a technique that reduces these equations to quadratics, which can be solved in closed form.