Page 92 - Geometric Modeling and Algebraic Geometry

90 R. Krasauskas and S. Zube
y =0. This fact may be important for applications in order to avoid singular points
on canal surfaces.

Fig. 5.8. The ellipse, two components of its offset and an evolute of the ellipse.

In other cases different degrees and different number of singular parts can appear.
We will list only the most important information for blending applications: subsets
of double points that contain positive lines.
(0)
For the case P represented by the parabola C : {x − 4ay =0,z = r =0}
2
++
we calculate the following implicit equation of Γ(C) of degree 6:

F = ω − 2 v ω − 18 ax − v 3 vω − a 27 ax − 2 v 3 x ,
3
2
2
2
2
2
where v = y − 2a, ω = x + y + z − r . The double hyperbolic paraboloid H :
2
2
2
2
{x =0, 4 y − 1+4 z − 4 r =0} is obtained by intersecting with the hyperplane
2
2
x =0.
The case H ++ is defined by the curve
C(t)=(a(1 + t )/(2t),b(1 − t )/(2t), 0, 0).
2
2
Then, similarly to the case E ++ , deg Γ(C)=8 and just one double hyperboloid of
1-sheet H : {y =0,x /(a + b )+ z /b − r /b =1} is found.
2
2
2
2
2
2
2
(++)
The case H is defined by the curve
+−
C(t)=(0, 0, 2at/(1 − t ),b(1 + t )/(1 − t )).
2
2
2
Then deg Γ(C)=8 and two double hyperboloids H 1 : {z =0, (x + y )/a −
2
2
2
r /(a − b )=1} and H 2 : {r =0, (x + y )/b − z /(a − b )=1} contain four
2
2
2
2
2
2
2
2
2
families of positive lines. Note that H 2 is a usual hyperboloid of revolution in R .
3
The case H +0 is defined by the hyperbola

C(t)=(2at/(1 − t ), 0,b 1+ t 2 /(1 − t ),b 1+ t 2 /(1 − t ))}.
2
2
2