5.2.2 Classification by type of geometry

In this subsection, we classify the various types of geometric specification of points, curves and surfaces that we will use in formulating various intersection problems:

1. Points
   1. Explicit: .
   2. Procedural: Intersection of two procedural curves, a procedural curve and a procedural surface, or three procedural surfaces.
   3. Implicit algebraic: Intersection of three implicit surfaces, or equivalently , where , , are polynomial functions and .
2. Curves
   1. Parametric: , .
      1. (Rational) (piecewise) polynomial: Bézier, rational Bézier, B-spline, NURBS.
      2. Procedural: offsets, evolutes, etc.
   2. Implicit algebraic: A 2-D planar curve is given by , while a 3-D space curve is given by intersection of two implicit algebraic surfaces .
3. Surfaces
   1. Parametric: , .
      1. (Rational) (piecewise) polynomial: Bézier, rational Bézier, B-spline, NURBS.
      2. Procedural: offsets, blends, generalized cylinders, etc.
   2. Implicit algebraic: .