Projecting points onto planar parametric curves by local biarc approximation

• Present a point projection and an inversion algorithm for planar parametric curves based on local biarc approximation.
• Possessing higher convergence rate than traditional geometric iteration algorithms.
• Present a framework that adapts to any single-point approximation algorithm.
• Can easily be extended to point projection and inversion on 3D parametric curves and surfaces.

This paper proposes a geometric iteration algorithm for computing point projection and inversion on planar parametric curves based on local biarc approximation. The iteration begins with initial estimation of the projection of the prescribed test point. For each iteration, we construct a biarc that locally approximates a segment on the original curve starting from the current projective point. Then we compute the projective point for the next iteration, as well as the parameter corresponding to it, by projecting the test point onto this biarc. The iterative process terminates when the projective point satisfies the required precision. Examples demonstrate that our algorithm converges faster and is less dependent on the choice of the initial value compared to the traditional geometric iteration algorithms based on single-point approximation.

A comparison of the approximation precision of the first order algorithm, the second order algorithm and our algorithm: P is the test point, the black curve is the original curve, Q0 is the initial point, orange point Q6, yellow point Q5, blue point Q2 and red point Q4 are projective points obtained by the Newton–Raphson method, the first order algorithm, the second order algorithm and our algorithm after the first iteration, respectively. Q3 is the exact closest point. (a) The whole view of the projection; (b) the zoom view of (a).Figure optionsDownload high-quality image (136 K)Download as PowerPoint slide