Page 209 - Geometric Modeling and Algebraic Geometry

212 C. Liang et al.
11.4.3 Silhouette curves of implicit surfaces

The following samples are taken from http://www-sop.inria.fr/galaad/surfaces/.We
intersect the surface with its polar variety in one direction (here the x direction). In
other words, we intersect the surface with the surface defined by one of its first order
derivative (here ∂ x f), to extract its silhouette. The surfaces that we used are called
respectively Tetrahedral, Q3, Q1 and Barth Sextic (see Fig.11.3):
5) f(x)= x +2x y +2x z +y +2y z +z +8xyz−10x −10y −10z +25
2 2
2
4
2
2
2 2
4
2 2
4
g(x)=4x +4xy +4xz +8yz − 20x
2
2
3
time: 510 msec
6) f(x)=5.229914547374508y z +3.597883597883598x y + y + z − x −
2 2
4
2 2
4
4
19.49816368932737xyz +5.229914547374508x − 7.43880040039534y 2
2
g(x)= −3.59788359788359z +7.43880040039534z x −110.45982909yz +
2 2
2
2
7.195767196x y +4y − 19.49816368932737xz − 14.87760080y
3
2
time: 330 msec
7) f(x)= x +y +z −4x −4y z −4y −4z x −4z −4x y +20.7846xyz+1
4
2
2 2
2
4
4
2 2
2
2 2
g(x)=4x − 8x − 8xz − 8xy +20.7846yz
2
3
2
time: 730 msec
8) f(x)=67.77708776x y z − 27.41640789x y − 27.41640789x z
2 4
2 2 2
4 2
+10.47213596x z −27.41640789y z +10.47213596y x +10.47213596y z
4 2
4 2
4 2
2 4
− 4.236067978x − 8.472135956x y − 8.472135956x z +8.472135956x 2
2 2
2 2
4
− 4.236067978y − 8.472135956y z +8.472135956y − 4.236067978z 4
4
2
2 2
+8.472135956z − 4.236067978
2
g(x) = 135.5541755xy z − 109.6656316x y − 54.83281578xz 4
2 2
3 2
+41.88854384x z +20.94427192y x − 16.94427191x − 16.94427191xy 2
3
3 2
4
− 16.94427191xz +16.94427191x
2
time: 4010 msec
(a) Example 5 (b) Example 6 (c) Example 7 (d) Example 8
Fig. 11.3. Topological descriptions of the silhouette curves of algebraic surfaces.