Page 221 - Geometric Modeling and Algebraic Geometry

224 M. Shalaby and B. J¨uttler
be introduced by the eliminating r, see Example 11. Indeed, this elimination is
equivalent to computing the polynomial g from
& &
g(x, y, z)= f(− x + y ,z) · f( x + y ,z). (12.15)
2
2
2
2
Note that this produces indead a polynomial, since only even powers of the square
root are present! The product (12.15) leads to a symmetrized version of the ap-
proximate implicitization of the profile curve. Consequently, additional branches
from the half–plane r< 0 may cause problems.
Example 11. Approximate implicitization of the profile curve (a cubic B´ ezier curve)
by a cubic polynomial using the method described in Section 12.2 produces an im-
plicit curve without additional branches and singular points, see Fig. 12.6, left. How-
ever, these problems are present after the elimination step (12.14), see Fig. 12.6,
right. The reason for this phenomenon can be seen from the global view (bottom row
in the picture): the elimination produces a symmetrized version of the approximate
implicitization. Note that methods for exact implicitization of the profile curve have
similar problems.
Remark 12. The first problem can be resolved by using Eq. (12.15) instead of
(12.14).

12.4.2 Implicitization via substitution

In order to avoid the problems of the first approach, we propose to implicitize the
profile curve q(v) in the rz-plane by the zero contour of a bivariate function f(r ,z).
2
The bivariate function f(r ,z) can be chosen from the space of all bivariate functions
2
with even power in r. We may use any basis (e.g., tensor–product B–splines) and
express the bivariate function f(r ,z) as
2

F(r ,z)= c i ϕ i (r ,z) (12.16)
2
2
i∈I
with real coefficients c i , where I is a certain index set. The method for approximate
implicitization described in Section 12.2 is applied to this representation. The ap-
proximate implicit representation of the surface of revolution is then obtained by a
substitution,
g(x, y, z)= F(x + y ,z). (12.17)
2
2
The degree of g with respect to x and y is twice the degree of F with respect to r ,
2
while the degrees with respect to z are equal.
Example 13. We apply this approach to the profile curve of Example 11, using a
polynomial F of total degree 3. The implicit equation of the profile curve has degree
(6,3), and the approximate implicit equation of the surface of revolution has degree
(6,3,3). As shown in Fig. 12.7, we may achieve a similar accuracy in the region of
interest by using an approximate implicitization of the profile curve that is symmet-
ric with respect to the axis of revolution. Due to this symmetry, no problems with
unwanted branches and singular points are present.