Point inversion and projection for NURBS curve: control polygon approach

Projecting a test point to a NURBS curve finds the closest point on the curve and point inversion finds the corresponding parameter for this test point. This paper presents an accurate and efficient method to solve both of these problems. We first subdivide the NURBS curves into a set of Bezier curves using knot insertion. For point projection, we extract candidate Bezier subcurves based on the relationship between the test point and the control polygon of the Bezier subcurve. For point inversion, we extract candidate Bezier subcurves based on the strong convex hull property, and then find the approximate candidate points and their corresponding parameter values. Finally, by comparing the distances between the test point and candidate points, we can find the closest point. We improve its accuracy by using the Newton-Raphson method.