Page 82 - Geometric Modeling and Algebraic Geometry

80 R. Krasauskas and S. Zube

Fig. 5.1. A Dupin cyclide used for blending circular cones.

5.2 Elements of Laguerre geometry

Here we briefly recall the elements of Laguerre Geometry (cf. [5, 8]). Consider the
space of all oriented spheres in R as a 4-dimensional affine space R , where the first
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4
three coordinates (x 1 ,x 2 ,x 3 ) are the center point of a sphere and the last coordinate
x 4 represents the radius of the sphere. The orientation is defined by the sign of x 4 :
the normals are pointing outwards if x 4 > 0. We denote by R the affine space R 4
4
1
equipped with a pseudo-euclidean (PE) metrics defined by the following PE scalar
product of vectors v, v :

v, v = v 1 v + v 2 v + v 3 v − v 4 v . (5.1)

1 2 3 4
Affine transformations of R that preserve this PE scalar product are called PE trans-
4
1
formations. It will be convenient to consider also the projective extension RP of
4
RP with additional coordinate x 0 . From this projective point of view PE transfor-
4
1
mations are exactly those projective transformations of RP that preserve the ab-
4
solute quadric Ω: x 0 =0, x + x + x − x . A geometric meaning of this metric
2
2
2
2
1 2 3 4
is a tangential distance between spheres in R .
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Affine subspaces A ⊂ R can be of three signature types sign A=(+,..., +,σ),
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1
where σ ∈{+, 0, −}. For example, all lines in R with directional vectors v can be
4
1
classified into three types depending on the sign σ = sign v, v : (+)-lines, (0)-lines,
and (−)-lines (also called positive, isotropic, and negative lines, resp.). 2-Planes also
can have three types: (++)-, (+0)-, (+−)-planes.
For any smooth curve α ⊂ R with tangent (+)-lines almost everywhere, define
4
1
Env(α) as an envelope of the corresponding family of spheres in R . We call such
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envelopes also canal surfaces. Circular cylinder or circular cones (call them both
natural cones) are envelopes Env(L) of (+)-lines L and vice versa. Let Γ(a) denote
the hypersurface x − a, x − a =0. Define Γ-hypersurface Γ(α) of a smooth
curve α in R as the envelope of the family of Γ(α(t)), for all t. Then Env(α)=
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Γ(α) ∩{x 4 =0} and any point x ∈ Γ(α) corresponds to a sphere that touches the
canal surface Env(α). Therefore, for any other curve β canal surfaces Env(α) and
Env(β) touch each other along some curve if and only if β(t) ∈ Γ(α) for all t in
some open interval. Here we exclud the trivial case when α and β are tangent in a
common point (hence the canal surfaces touch along a circle).