Page 85 - Geometric Modeling and Algebraic Geometry

5 Canal Surfaces Defined by Quadratic Families of Spheres 83
tangents are marked by small circles. Isotropic directions and asymptotes are shown
by thin dashed and grey lines, respectively.
Proof. Without loss of generality we suppose that a 2-plane containing the given
conic passes through the origin and has the basis {e 1 ,e 2 }, {e 1 ,e 3 + e 4 }, {e 3 ,e 4 },
depending on the signature σ = (++), (+0), (+−), respectively. Also let the center
of the conic (or the vertex in parabolic case) be in the origin. Then in all cases there
exist linear PE transformations that have the following matrix form (when restricted
to these 2-planes with the fixed basis):

cos ϕ ∓ sin ϕ 10 cosh θ ± sinh θ
, , ± , ϕ,ρ,θ ∈ R. (5.3)
sin ϕ ± cos ϕ ρ ±1 sinh θ ± cosh θ
It is easy to recognize rotation, shear and boost (or hyperbolic rotation) transforma-
tions possibly composed with reflections.
Case σ = (++). E ++ , P ++ , H ++ are usual Euclidean types of ellipse, parabola
and hyperbola that can be rotated to the canonical positions. Here we distinguish a
[2]
circle case E , since it has two ‘circular points’ (0, 1, ±i, 0, 0) lying on Ω.
++
Case σ = (+0). Only one direction is isotropic (shown as dashed vertical lines
in Fig. 5.3) and all others are positive. Any positive direction can be moved to any
other positive one using a shear transformation. Hence an axis of a parabola P +0 and
an asymptote of a hyperbola H [1] can be moved to the horizontal position. Similarly
+0
asymptotes of a hyperbola H +0 can be transformed to the symmetric position. The
conjugated hyperbola H +0 has a different PE type, since it has isotropic tangents.

An ellipse E +0 has a pair of points with isotropic tangents, and one can move line
connecting these points to the horizontal position.
Case σ =(+−). There are two fixed isotropic directions. Positive and negative
directions are in between. Using boost transformation one can move any positive
(resp. negative) direction to the horizontal (resp. vertical) direction. This allows to
transform all cases to the canonical ones shown in Fig. 5.4. For example, in the
case E +− we choose a vector connecting the origin with a point on the ellipse with
a biggest PE length, and transform it to a horizontal position using an appropriate
boost.

Corollary 2. Let L be a (+)-line, and C is a conic in R . If the cone Env(L) touches
4
the quadratic canal surface nv(C) along a curve which is neither a line nor a circle
[2] (++) (+0)
then C has one of the following types: E ++ , P ++ , H ++ , P , H +0 , H , H .
+− +−
+0
Proof. Without loss of generality we identify L with the x 1 -axis. Then by Eq. (5.2)
Γ(L) has the equation x + x = x . Consider a projection π :(x 1 ,x 2 ,x 3 ,x 4 ) →
2
2
2
2 3 4
(0,x 2 ,x 3 ,x 4 ) to the hyperplane {x 1 =0}⊂ R . The conic C is contained in
4
Γ(L) (since the touching curve is non circular), and its projection π(C) is a conic
in π(Γ(L)) (since the touching curve is not a line). On the other hand π(Γ(L)) =
Γ(L) ∩{x 1 =0}, and all its infinite points are contained in the absolute quadric Ω.
[2] [2] [2]
Hence π(C) has infinite points on Ω, so it is one of three PE circles E , P , H .
++ +−
+0
Note that Ω in the infinite hyperplane x 0 =0 has the same equation as a sphere,