Page 91 - Geometric Modeling and Algebraic Geometry

5 Canal Surfaces Defined by Quadratic Families of Spheres 89

2
F = ω + 6+2x − 4 y 2 ω + 13 − 28 y +4 x + x +2 y 2 ω 2
2
2
4
2
2
3

+6 x +2 − 4 y 2 −x +1 − 2 y 2 ω
2
2
3
+ 4 − 12 x +2 y 2 − 3 x − 4 y 2 4 y +5 x 2 − 4 x +2 y 2 ,
2
2
2
2
where ω = x + y + z − r .
2
2
2
2
Moreover, F is an irreducible polynomial. Indeed, a plane curve Γ(E) ∩{r =
const}∩{z =0} is defined by an irreducible polynomial of degree 8 as it is an
offset of the ellipse E (see, e.g. [8]). Hence deg Γ(E)=8 as well.
Theorem 11. Let E be an ellipse of type E ++ . Then the hypersurface Γ(E) is a real
3-dimensional variety of degree 8, and its set of finite real double points has four
parts E ∪ H 1 ∪ H 2 ∪ Ω where:
E : x /a + y /b =1, z = r =0,
2
2
2
2
H 1 : x /(a − b ) − z /b + r /b =1, y =0,
2
2
2
2
2
2
2
H 2 : −y /(a − b ) − z /a + r /a =1, x =0,
2
2
2
2
2
2
2
H 1 and H 2 are hyperboloids of 1-sheet and 2-sheet, resp., (the only non-isotropic
lines on Γ(E) are two rulings of the hyperboloid H 1 ), Ω is the absolute quadric.
Proof. It remains to determine double points. We start from the following geometric
description of Γ(E). The hypersurface Γ(E) consists of points in R that define
4
1
spheres tangent to the ellipse. All spheres touching in one point define an isotropic
line in R . If two isotropic lines have a common point then this point is a double
4
1
point on Γ(E). Therefore, a common point of two isotropic lines corresponds to a
sphere that touches the ellipse E in two points. There are two families of circles on
the (x, y)-plane that touch the ellipse E in two points. One family consists of inside
circles with centers on the x-axis, another one of outside circles with centers on the
y-axis. A pencil of spheres that contains this circle goes through each of such circle,
i.e. spheres of the pencil are tangent to the ellipse E in two points. We notice that
these two families of spheres are defined by two equations y =0 and x =0 in
R . Therefore, the equations of the hiperboloids H 1 and H 2 are obtained from the
4
1
equation F =0 as the hyperplane sections.
Remark 12. For the ellipse E in Theorem 11 hypersurface Γ(E) has also other sin-
gularities (not only double points). Consider a projection of Γ(E) to the (x, y)-plane.
Singularities of the ellipse offsets define the evolute curve (see Fig. 5.8), which is an
envelope of normal lines to the ellipse E in the plane. Let a surface K ⊂ Γ(E) be
an envelope of all isotropic lines in Γ(E). Since the isotropic lines are projected to
normals, K is projected to the evolute, and all points of K are singular in Γ(E). The
well-known parametrization of the evolute enables us to parametrize K:
(b + a )sin t 1 − s 2 1+ s 2
3
2
2
K(t, s)=((b − a )cos t, ,h(t) ,h(t) ),
2
3
2
a 2s 2s
where h(t) =(b sin (t)+ a cos (t)) /a . Note that a general point on K be-
2
2
2
2
2
3
2
longs to a segment of an isotropic line on Γ(E) bounded by hyperplanes x =0 and