In this paper, we investigate how to compute the minimum distance between a point and a parametric surface, and then to return the nearest point (foot point) on the surface as well as its corresponding parameter, which is also called the point projection problem of a parametric surface. The geometric strategy algorithm (hereafter GSA) presented consists of two parts as follows. The normal curvature to a given parametric surface is used to find the corresponding foot point firstly, and then the Taylor’s expansion of the parametric surface is employed to compute parameter increments and to get the iteration formula to calculate the orthogonal projection point of test point to the parametric surface. Our geometric strategy algorithm is essentially dependent on the geometric property of the normal curvature, and performs better than existing methods in two ways. Firstly, GSA converges faster than existing methods, such as the method to turn the problem into a root-finding of nonlinear system, subdividing methods, clipping methods, geometric methods (tangent vector and geometric curvature) and hybrid second-order method, etc. Specially, it converges faster than the classical Newton’s iterative method. Secondly, GSA is independent of the initial iterative value, which we prove in Theorem 1. Many numerical examples confirm GSA’s robustness and efficiency.

A Geometric Strategy Algorithm for Orthogonal Projection onto a Parametric Surface

Repeated use of point projection to find the projection of a curve on a surface is rather inefficient as the iteration procedures in point projection is typically slow. A novel curve projection scheme is proposed for computing the orthogonal projection of a progenitor curve onto a parametric surface. Under this scheme, the projection curve is parameterized using the parameter of the progenitor curve. Differential geometric characteristics of the projection curve are analysed. A marching method with error adjustment is used to calculate the projection curve. Several examples are presented and comparisons are made to demonstrate the effectiveness of the proposed scheme.

A second-order algorithm for curve orthogonal projection onto parametric surface

(cid:0) For the purpose of the curves reuse and redesign, an array replication method of curves on manifold triangulation surfaces is proposed. In our method, the curves on mesh surfaces are denoted by geodesic B-splines, which can transform the operation for curves to the operation for control points of curves. By introducing the discrete exponential mapping theory, the control points of the source curves can be mapped into its tangent space, and their normal coordinates can be obtained. Taking advantage of the invariable property of the normal co-ordinates, the corresponding relation between the pre-and post-array control points are established, with which the regular multiple replication of curves can be realized. The normal coordinates can retain the geodesic distance and relative position of the control points, thus ensuring the shape retention of the curves in reuse and array. The position from the shape of the curve array processing is seperated, making sure that the curve generated only related to the local area and ignore the overall size of the surface. What’s more, it is not only easy to ensure the shape retention of the curve, but also reduce the calculation. Experimental results show that, our method is robust, effective, and can meet the requirements of interactive design of curves on mesh surfaces.

Pattern of Spline Curves on Meshes

In this paper, we present a new method for the smooth interpolation of the orientations of a rigid body motion. The method is based on the geometrical Hermite interpolation in a hypersphere. However, the non-Euclidean structure of a sphere brings a great challenge to the interpolation problem. For this consideration and the requirements for practical application, we construct the spherical analogue of classical rational Bezier curves, called generalized rational Bezier curves. The new spherical curves are obtained using the generalized rational de Casteljau algorithm, which is a generalization of the classical rational de Casteljau algorithm to a hypersphere. Then, $$G^2$$G2 Hermite interpolation problem in hypersphere is solved analytically using the generalized rational Bezier curve of degree 5. The new method offers residual free parameters including shape parameters and weights, which guarantee the existence of the interpolant to arbitrary motion data and offer great flexibility for the shape design of the motion. Numerical examples show that our method is far better behaved according to the energy functional which is regarded as a measure of the motion shape.

Generalized rational Bézier curves for the rigid body motion design

Regarding the point projection and inversion problem, a classical algorithm for orthogonal projection onto curves and surfaces has been presented by Hu and Wallner (2005). The objective of this paper is to give a convergence analysis of the projection algorithm. On the point projection problem, we give a formal proof that it is second order convergent and independent of the initial value to project a point onto a planar parameter curve. Meantime, for the point inversion problem, we then give a formal proof that it is third order convergent and independent of the initial value.

https://www.mdpi.com/2073-8994/9/10/210/pdf

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

This paper addresses new methods to solve point inversion problem for parametric surface. As a by-product a new method for ray-surface intersection is also developed. After further analysis, we reduce finding the corresponding parameters of a given point on a surface to the following steps: (1) construct a line segment with the given point as its one end and an arbitrarily selected point on the surface as its other end point, (2) project the line segment onto the surface orthogonally or along a vector or through a central point, or compute the intersection curve segment of a specially created plane and the surface, (3) trace the parameters along the projected curve or intersection curve with the linear convergence or the second-order convergence. As a matter of fact, we formulate some related systems of first-order or second-order ordinary differential equations met by the corresponding projection curve segment of the line segment or by the intersection curve segment. Using the parameters of the selected point as initial values, we trace the desired parameter on surface along the line segment from its one end to another or along the intersection curve segment. In this method, there is no need to consider the sensitivity to the choice of starting points, iteration convergence and so on, which several existing methods must face. The method is simpler than existing methods for it merely concerns first-order information of the surface, if we only ask for linear convergence, and has better error control mechanism, if we seek for second-order convergence. Implementation examples are also given to demonstrate its validity.

Computation of point inversion and ray-surface intersection through tracing along the base surface

This paper presents new methods for G 1 and G 2 continuous interpolation of an arbitrary sequence of points on an implicit or parametric surface with prescribed tangent direction and both tangent direction and curvature vector, respectively, at every point. We design a G 1 or G 2 continuous curve in three-dimensional space, construct a so-called directrix vector field using the space curve and then project a special straight line segment onto the given surface along the directrix vector field. With the techniques in classical differential geometry, we derive a system of differential equations for the projection curve. The desired interpolation curve is just the projection curve, which can be obtained by numerically solving the initial-value problems for a system of first-order ordinary differential equations in the parametric domain associated to the surface representation for the parametric case or in three-dimensional space for the implicit case. Several shape parameters are introduced into the resulting curve, which can be used in subsequent interactive modification such that the shape of the resulting curve meets our demand. The presented method is independent of the geometry and parameterization of the base surface, and numerical experiments demonstrate that it is effective and potentially useful in patterns design on surface.

Constructing up to G 2 continuous curve on freeform surface

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Computation of point inversion and ray-surface intersection through tracing along the base surface

Constructing up to G 2 continuous curve on freeform surface