Piecewise algebraic surface patches
The author presents tools for modelling with free form algebraic surfaces. An algebraic surface may consist of one or more sheets of infinite extent. It will only be concerned with the part of an algebraic surface within an arbitrary tetrahedron. It will refer to such a surface region as an algebraic surface patch (asp). Consider the scalar function defined by the polynomial equation \(w=f(s,t,u,v)\). We adopt the notation \(S=(s,t,u,v)\), so \(f(s,t,u,v)=f(S)\). This function assigns a unique value \(w=f(S)\) to each point S. A contour surface of the function is comprised of all points S for which f(S) is a constant. We define an asp as the contour \(f(S)=0\) clipped by the tetrahedron. A degree n asp can be defined using Bernstein-Bézier-Farin polynomials as follows. We assign a weight \(w_{ijkl}\) to each control point, and the function \(w=f(S)\) is defined \[ (1)\quad f(s,t,u,v)=w_{ijkl}\frac{n!}{i!j!k!l!}s^ it^ ju^ kv^ l,\quad i+j+k+l=n;\quad s+t+u+v=1. \] This scheme facilitates the design of a free form asp. The value of f(S) at any of the 4 tetrahedral vertices is the value of the weight of that vertex. This means that the algebraic surface f(S), can be forced to interpolate a corner vertex by setting the weight of that vertex to zero. If all the weights of all \(n+1\) control points along an edge are zero, the entire edge interpolates the surface \(f(S)=0\). This is also easily verified from (1). The contribution of a particular control point's weight to the function f(S) in (1) can be shown to be maximum at the control point. The value of the function f(S) along any of the 6 tetrahedral edges can be expressed as a univariate Bernstein polynomial whose coefficients are the \(n+1\) control point weights along the edge. A crucial property of this asp formulation is that it inherits most of the tools of Bernstein-Bézier- Farin curves and surfaces: we can subdivide the surface by subdiving the tetrahedron. A promising application for algebraic surface patches is in modelling solids, since the implicit surface equation defines the half-spaces f(S)\(\leq 0\) and f(S)\(\geq 0\). Algebraic surfaces patches could be ray traced very efficiently, since the ray intersection equation is a univariate polynomial whose degree is the degree of the surface.
scientific article; zbMATH DE number 3926203