Page 83 - Geometric Modeling and Algebraic Geometry

5 Canal Surfaces Defined by Quadratic Families of Spheres 81
Let C be a natural cone, and let L be the corresponding hyperbolic line in R .
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All the spheres in R tangent to C correspond to an isotropic hypersurface Γ(L) in
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R . Let v be a directional vector of L, and let p be a point on L. The equation of
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Γ(L) can be calculated easily (see [2])
Γ(L): x − p, x − p v, v − x − p, v =0. (5.2)
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5.3 Conics in R 4
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Our goal is to study quadratic canal surfaces . Since they are defined as envelopes
of quadratic families of spheres, they are encoded by conics in R . The well-known
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examples are Dupin cyclides (see [5, 9]). The corresponding conics C are character-
ized as PE circles, i.e. infinite points of C are lying on Ω (may be a pair of complex
conjugated points or a double point). For example, all conics C contained in Γ(a),
a ∈ R are PE circles. Therefore, Env(C) is a Dupin cyclide [5, 8].
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Fig. 5.2. Conics of type σ = (++) in canonical position.

Fig. 5.3. Conics of type σ =(+0) in canonical position.

Let us classify PE types of conics in R by:
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