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12 Approximate Implicitization of Space Curves and of Surfaces of Revolution 225
rz–plane xz–plane
Region of interest: 2 2
1.8 1.8
1.6 1.6
1.4 1.4
1.2 1.2
1 1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Global view:
6 6
4 4

2 2
0 0
–2 –2

–4 –4
–4 –2 0 2 4 6 –4 –2 0 2 4 6
Fig. 12.6. The elimination of r may produce additional branches and singular points. Top row:
Region of interest [0, 2] , Bottom row: global view. Left: Approximate implicitization Z(f)
2
of the profile curve in the rz–plane. Right: Intersection of the approximate implicitization
Z(g) with the xz–plane. The original profile curve is shown in grey.

Example 14. We consider the discretized profile curve shown in Fig. 12.8, left, and
apply the method of Section 12.2 to it. The function F is a bi–quadratic tensor–
product spline function whose domain is the union of the cells shown in the figure.
This leads to an approximate implicit representation of the profile curve (Fig. 12.8,
center) and of the surface (right) of degree 4(×4) × 2. In the surface case, the spline
function is defined with respect to ring–shaped cells, obtained by rotating the cells
shown in the left figure.

12.5 Conclusion

Several techniques for approximate implicitization of space curves and surfaces of
revolution have been presented. These techniques are based on algorithms for (exact
or approximate) implicitization of planar curves. In the case of space curves, a repre-
sentation of two approximately orthogonal surfaces can be obtained, which provides
several advantages, such as a geometrically robust definition of the curve and the