A Geometric Orthogonal Projection Strategy for Computing the Minimum Distance Between a Point and a Spatial Parametric Curve

Abstract

Abstract:A new orthogonal projection method for computing the minimum distance between a point and a spatial parametric curve is presented. It consists of a geometric iteration which converges faster than the existing Newton's method, and it is insensitive to the choice of initial values. We prove that projecting a point onto a spatial parametric curve under the method is globally second-order convergence.

Hybrid Second Order Method for Orthogonal Projection onto Parametric Curve in n-Dimensional Euclidean Space

“…They adopt the key technology of degree reduction via clipping to yield a strip bounded of two quadratic polynomials. Curvature information is found for computing the minimum distance between a point and a parameter curve or surface in [6,30]. However, it needs to consider the second order derivative and the method [30] is not fit for n-dimensional Euclidean space.…”

“…Curvature information is found for computing the minimum distance between a point and a parameter curve or surface in [6,30]. However, it needs to consider the second order derivative and the method [30] is not fit for n-dimensional Euclidean space. Hu et al [6] have not proved the convergence of their two algorithms.…”

“…The advantage of these methods is that they can find all solutions, while their disadvantage is that they are computationally expensive and may need many subdivision steps.The third classic methods for orthogonal projection onto parametric curve and surface are geometry methods. They are mainly classified into eight different types of geometry methods: tangent method [22,23], torus patch approximating method [24], circular or spherical clipping method [25,26], culling technique [27], root-finding problem with Bézier clipping [28,29], curvature information method [6,30], repeated knot insertion method [31] and hybrid geometry method [32]. Johnson et al [22] use tangent cones to search for regions with satisfaction of distance extrema conditions and then to solve the minimum distance between a point and a curve, but it is not easy to construct tangent cones at any time.…”

Convergence Analysis on a Second Order Algorithm for Orthogonal Projection onto Curves

“…A vital problem is the projection of a point onto a surface or parametric curve in order to get the foot point and calculate the corresponding parameter values, which is widely applied in the field of Geometric Modeling [1][2][3][4][5][6][7], Computer Graphics and Computer Vision [5][6][7], Computer Aided Geometric Design [8][9][10][11][12], geometric design [13][14][15], geometry sculpt [16], scientific research and engineering applications [17] and computer animation [18][19][20].…”

“…To find the orthogonal projection of a point onto surfaces or parametric curves, Limaien and Trochu [1] constructed an auxiliary function and obtain all its zeros. Based on the geometric method [4], Li et al [2] use the geometric iterative method with second order approximation properties. The common features of [2] and [4] are that they both use the curvature information, and they are independent of the initial value, respectively.…”