By John M. Snyder
Deals a brand new and symbolic method of geometric modelling which addresses the matter of the way to construct shape-defining parametric services. Operators comprise mathematics operators, vector and matrix operators, and integration, differentiation, constraint resolution, and international optimization
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Extra info for Generative Modeling for Computer Graphics and CAD. Symbolic Shape Design Using Interval Analysis
Sample text
The generative modeling approach extends this work by generalizing the kind of inclusion functions used and providing for au tomatic generation of the inclusion function. The bounds thus obtained are much tighter and are automatically computed from the shape representation. Although it has received little attention, the use of interval methods in computer graphics and geometric modeling is not new. Mudur and Koparkar [MUDU84] have presented an algorithm for rasterizing parametric surfaces us ing interval arithmetic.
Two forms of solution are useful: finding any point that solves the system, or finding all points that solve it, assuming there is a finite set of solutions. The operator also requires a parametric function specifying a rectilinear domain in which to solve the constraints. For example, the constraint solution operator can be used to find the intersections between two planar curves. Let 7^ (s) and 7 ^ ( t ) be two curves in R^. These curves could be represented using the curve operator on page 39, or using any of the other operators.
10) • finding whether a point is inside or outside a region defined with parametric boundaries, called point-set classification For example, consider a parametric surface 5 ( u , v) whose image over a 2D rectangle D = [iXo>^i] x b o ? v i ] forms the boundary of a compact 3D region. , Ν always points outside the interior of the bounded region). The point-set classifica tion problem involves determining whether a point ρ € is inside or outside the region. Let d{u, v) be the distance between a point on S and p: d{u,v) = \\S{u,v)-pl Let (t¿"^", ν""") be a point in D that minimizes d, and let 5"^" = 5(tx"^",v"'^") Note that N^^ is either parallel or anti-parallel to the direction S^^ - p .