7.4 Examples

In this section, we give examples of distance computations for the five cases: a space point and a variable point on a 3D space curve (P/C); a space point and a variable point on a surface (P/S); two variable points located on two 3D space curves (C/C); two variable points, one of which is located on a space curve and the other is located on a surface (C/S); two variable points located on two surfaces (S/S). The curves and surfaces involved are expressed as rational Bézier entities whose property and degree are listed in Table 7.1.

The curve in the P/C example is a high degree rational curve which gives rise to a system with 4 roots as shown in Fig. 7.1. The surface in the P/S example is a high degree rational surface which also generates a system with 4 roots (see Fig. 7.2). In the C/C example, one curve is a parabola and the other is a quarter circle represented as a rational Bézier curve; the resulting system of equations has three roots (see Fig. 7.3). In the C/S example, the curve is a parabola and the surface is a saddle-like rational Bézier surface; there is only one root, which occurs at the intersection point of the curve and the surface as illustrated in Fig. 7.4. In the S/S example, the two surfaces are ruled rational surfaces with one root (see Fig. 7.5).

Figure 7.1: Distances of a point and a high degree rational Bézier curve (adapted from [461])

Figure 7.2: Distances of a point and a high order rational Bézier surface (adapted from [461])

Figure 7.3: Distances of a rational Bézier curve and an integral Bézier curve (adapted from [461])

Figure 7.4: Distances of a rational Bézier curve and a rational Bézier patch (adapted from [461])

Figure 7.5: Distances of two linear-quadratic Bézier patches (adapted from [461])