Clipping with Surfaces

Nothing in the above description of the clipping process restricts the polygons to two dimensions or the clipping function to two variables. The process may be used to clip one surface by another. This extends a function in the Wickham-Jones package that clips a surface with a plane. Here are three examples.

Figure 7 shows the Monkey Saddle, , plotted over the square , . The three-fold symmetry of the surface may be emphasized by clipping it to the inside of the cylinder . The clipped surface is shown in Figure 8.

Figure 7. .

Figure 8. Clipped by .

Clipping a plane by a surface gives a section of the surface. An application of this idea is shown in Figures 9 and 10. The Gaussian and mean curvatures of a surface at a point may be defined in terms of the curvature of the curve of intersection of the surface with a plane through the point and containing the surface normal. Figure 9 shows a surface () and a sampling of planes containing the normal to the surface at a fixed point. Clipping the planes by the surface reveals the normal sections. Figure 10 shows the sections with maximal and minimal (signed) curvature. The package contains a utility, Grid, to generate planes to be clipped.

Figure 9. Normal planes.

Figure 10. Normal sections.

In Figure 11 the surface is generated by rotating a limaçon about its line of symmetry. A hole has been punched in the surface to reveal the inner lobe. Figure 12 indicates how the clipping was accomplished. The figure shows a cross section (clipped by a plane) of the rotated limaçon together with the surface used to punch the hole. The punch surface is also a surface of revolution. The profile curve is rotated about its axis of symmetry—yielding —and then moved into position by an appropriate geometric transformation.

Figure 11. Rotated limaçon.

Figure 12. Punch in position.