A Fast Point Projection Approach of NURBS surfaces

In precision metrology, point projection is required when matching mea-
surement data with the reference NURBS template. In this paper we present a fast
approach to calculate the foot-point parameters of the orthogonal projection point.
The entire NURBS surface is rstly divided into 'simple' Bezier patches. The cor-
rect parameter spans can be determined according to the convex-hull property of
NURBS. A 'jumping mechanism' is established to help determine the correct switch-
ing direction between parameter intervals. To improve the reliability of the rene-
ment result, a second-order approximation approach is adopted. It approximates the
local curve on the NURBS surface along the optimal incremental direction using an
arc. The normal curvature of this arc is calculated based on the theory of dieren-
tial geometry. In this way, point projection becomes a single variable optimisation
problem. This method is insensitive to the initial guess and can achieve a quadratic
convergence rate. Finally numerical examples are given to demonstrate the validity
and eciency of this method.