10.2 Intersection problem classification Intersection problems can be classified according to the dimension of the problems and ac- cording to the type of geometric equations involved in defining the various geometric elements (points, curves and surfaces). The solution of intersection problems can also vary according to the number system in which the input is expressed and the solution algorithm is implemented 10.2.1 Classification by dimension /P, P/C, P/S /C, C/S S/S 10.2.2 Classification by type of geometry Parametric Implicit (eg. R=R(t) (eg.f(x,y)=0.,z=0) Rational Procedural Polynomial (algebraic) Figure 10.3: Curve geometry classification 1. Points plicit: R= Ro; R y, Procedural: Intersection of two procedural curves, procedural curve and surface, or three procedure surfaces, eg. offset or blending surfaces Algebraic: f(R)=g(R)=h(r)=0; where f, 9, and h are polynomials A classification of curves is illustrated in Figure 10.3 ● aranetrio R=R(t)A≤t≤B (a) Rational Polynomials(eg: NURBS, rational Bezier) (b)Procedural, eg: offsets, evolutes, ie. the locus of the centers of curvature of Implicit: These require solution of (usually nonlinear)equations10.2 Intersection problem classification Intersection problems can be classified according to the dimension of the problems and ac￾cording to the type of geometric equations involved in defining the various geometric elements (points, curves and surfaces). The solution of intersection problems can also vary according to the number system in which the input is expressed and the solution algorithm is implemented. 10.2.1 Classification by dimension • P/P, P/C, P/S • C/C, C/S • S/S 10.2.2 Classification by type of geometry polynomial Procedural Polynomial (algebraic) (eg. f(x, y) = 0, z = 0) Implicit Rational Parametric (eg. R = R(t) ) Figure 10.3: Curve geometry classification 1. Points • Explicit: R = R0; R = [x, y, z] • Procedural: Intersection of two procedural curves, procedural curve and surface, or three procedure surfaces, eg. offset or blending surfaces. • Algebraic: f(R) = g(R) = h(R) = 0; where f, g, and h are polynomials. 2. Curves A classification of curves is illustrated in Figure 10.3. • Parametric: R = R(t) A ≤ t ≤ B (a) Rational Polynomials (eg: NURBS, rational B´ezier). (b) Procedural, eg: offsets, evolutes, ie. the locus of the centers of curvature of a curve. • Implicit: These require solution of (usually nonlinear) equations 5