Surface Curve Design by Orthogonal Projection of Space Curves Onto Free-Form Surfaces

Abstract

A novel technique for designing curves on surfaces is presented. The design specifications for this technique derive from other works on curvature continuous surface fairing. Briefly stated, the technique must provide a computationally efficient method for the design of surface curves that is applicable to a very general class of surface formulations. It must also provide means to define a smooth natural map relating two or more surface curves. The resulting technique is formulated as a geometric construction that maps a space curve onto a surface curve. It is designed to be coordinate independent and provides isoparametric maps for multiple surface curves. Generality of the formulation is attained by solving a tensorial differential equation formulated in terms of local differential properties of the surfaces. For an implicit surface, the differential equation is solved in three-space. For a parametric surface the tensorial differential equation is solved in the parametric space associated with the surface representation. This technique has been tested on a broad class of examples including polynomials, splines, transcendental parametric and implicit surface representations.

Hybrid Second-Order Iterative Algorithm for Orthogonal Projection onto a Parametric Surface

“…with a test point p = (4, 5,6). It is known that the uniquely corresponding orthogonal projection point and parametric value of the test point p are (0.86886081685860457, 1.0860760210732557, 0.93459272735858134) and (α, β) = (0.86886081685860457, 1.0860760210732557), respectively.…”

“…Suppose a parametric surface s(u, v) = (u, v, cos(u 2 + v 2 )), u, v ∈ [0, 2] with a test point p = (4, 5,6). It is known that the uniquely corresponding orthogonal projection point and parametric value of the test point p are (1.0719814278710903,1.3399767848388629,-0.98067565161631654) and (α, β) = (1.0719814278710903, 1.3399767848388629), respectively.…”

“…It is a very interesting problem in geometric modeling, computer graphics and computer vision [1]. Both projection and inversion are essential for interactively selecting curves and surfaces [1,2], for the curve fitting problem [1,2], for reconstructing surfaces [3][4][5] and for projecting a space curve onto a surface for surface curve design [6,7]. It is also a key issue in the ICP (iterative closest point) algorithm for shape registration [8].…”

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

“…It is an interesting problem due to its importance in geometric modeling, computer graphics and computer vision [1]. Both projection and inversion are essential for the interactively selecting curves [1,2], the curve fitting problem [1,2], and the reconstructing curves problem [3][4][5]. It is also a key issue in the ICP (iterative closest point) algorithm for shape registration and rendering of solid models with boundary representation and projecting of a spatial curve onto a surface for curve surface design [6].…”

On Helical Projection and Its Application in Screw Modeling

“…Using transformation matrices, any type of 3D projection can be generated by computers. This makes the projection methods not only means of visualization or graphic representation but also a way for modern computer systems to solve design and manufacturing problems [12,13]. The ever-increasing use of computers in industry also entails new data models for efficiently manipulating and communicating product information [14].…”