Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 39 (1998), No. 2, 291-305.

Galilei Laguerre Geometry and Rational Circular Offset Surfaces

Johannes Wallner  Horng-Yang Chen  Helmut Pottmann

Institut für Geometrie, Technische Universität Wien Wiedner Hauptstrasse 8-10/113, A-1040 Wien e-mail: hannes@geometrie.tuwien.ac.at

Abstract. In this paper the Laguerre geometry of three-dimensional Galilei space, that is, the geometry of oriented planes and cycles of an affine Cayley-Klein space with absolute line and two conjugate complex absolute points on it, is studied. Several different models are presented: A cyclographic model and an isotropic model together with their duals and an interpretation in terms of geometrical optics of Galilei space. We also have a closer look on Laguerre transformations. Galilei geometry arises naturally in the context of rational surfaces which possess rational circular offset surfaces, and it can also be used for modeling rational circular offset surfaces, e. g., with Galilei cyclides.

1. Introduction

There are two reasons why Laguerre geometry of Galilei space is interesting. First the geometry of planes in Galilei space and the different models of isotropic Möbius geometry connected with it deserve interest.

Second, within the search for rational surfaces with rational offsets, which has its origin in computer aided geometric design applications, we have to treat the case of a circular offset surface, which, for instance, is traced out by a point of the axis of a cylindrical milling tool when shaping a given surface. The geometry of planes of Galilei space here occurs naturally.

On the one hand, we rely on Röschel [13] and Brauner [2]. On the other hand, we follow the work done by Pottmann and Peternell [8, 9, 10, 11].

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 1998 Heldermann Verlag

2. Fundamentals of Galilei Laguerre Geometry

2.1. Galilei Space

We repeat some definitions and elementary properties of Galilei space [13].

Definition. Galilei space is three-dimensional real projective space P_{3}(ℝ) together with the geometry induced by the conformal Galilei group H_{8} and its subgroup, the Galilei motion group B_{6}. H_{8} consists of the transformations which, in homogeneous coordinates, are given by

x′_{0} = x_{0},
x′_{1} = rx_{0} + ρ cos(ϕ)x_{1} + ρ sin(ϕ)x_{2} + ux_{3},
x′_{2} = sx_{0} − ρ sin(ϕ)x_{1} + ρ cos(ϕ)x_{2} + vx_{3},
x′_{3} = tx_{0}  + wx_{3},

(1)

and B_{6} is the subgroup defined by ρ = w = 1.

In the terminology of Cayley-Klein geometries [5], Galilei space is the real part of threedimensional complex projective space P_{3}(ℂ), equipped with the absolute figure

ω ⊃ f ⊃ {I_{1}, I_{2}},  (2)

where ω is the absolute plane x_{0} = 0, f is the horizontal absolute line x_{0} = x_{3} = 0, and I_{1}, I_{2} are the horizontal absolute points (0, 1, ±i, 0)ℂ.

Galilei motions leave the two Galilei distances invariant:

d_{1}^{2}((x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2})) = (z_{2} − z_{1})^{2},  (3)
d_{2}^{2}((x_{1}, y_{1}, z), (x_{2}, y_{2}, z)) = (x_{1} − x_{2})^{2} + (y_{1} − y_{2})^{2}.  (4)

2.2. Galilei Laguerre Space

Galilei Laguerre geometry is the geometry of oriented non-horizontal planes in Galilei space.

Definition. An oriented hyperplane ε of P_{n}(ℝ) is a set

ε = (e_{0}, …, e_{n})ℝ^{+}, e_{i} ∈ ℝ, ε ≠ (0, …, 0).

The hyperplane corresponding to the oriented hyperplane (e_{0}, …, e_{n})ℝ^{+} has the equation

e_{0}x_{0} + … e_{n}x_{n} = 0.  (5)

A point p = (x_{0}, …, x_{n})ℝ ∈ P_{n}(ℝ) is incident with the oriented hyperplane (e_{0}, …, e_{n})ℝ^{+}, if it fulfills (5). If n = 3, an oriented (hyper-)plane is horizontal, if the corresponding plane contains the line x_{0} = x_{3} = 0. Then three-dimensional Galilei Laguerre space G is the set of non-horizontal oriented planes:

G = {(e_{0}, e_{1}, e_{2}, e_{3})ℝ^{+}|(e_{1}, e_{2}) ≠ (0, 0)}.

Obviously for every ε ∈ G there is exactly one coordinate vector (e_{0}, e_{1}, e_{2}, e_{3}) which describes ε and which satisfies e_{1}^{2} + e_{2}^{2} = 1.

Definition. The cycle c = (m_{x}, m_{y}, m_{z}, r), m_{x}, m_{y}, m_{z}, r ∈ ℝ, is the set

c = {(e_{0}, …, e_{3})ℝ^{+} ∈ G|e_{1}^{2} + e_{2}^{2} = 1, e_{0} + e_{1}m_{x} + e_{2}m_{y} + e_{3}m_{z} + r = 0}.  (6)

The carrier of the cycle is the circle of radius |r|, which is contained in the horizontal plane x_{3}/x_{0} = m_{z} and has center (m_{x}, m_{y}, m_{z}).

If r ≠ 0, the planes of a cycle c are tangent to the carrier of c. If r = 0, all ε ∈ c are incident with (1, m_{x}, m_{y}, m_{z})ℝ.

2.3. The Cyclographic Model

Definition. The cyclographic map ζ maps the oriented planes of G to hyperplanes of P_{4}(ℝ):

ζ : G → P_{4}(ℝ)^{∗}, (e_{0}, e_{1}, e_{2}, e_{3})ℝ^{+} ↦ (e_{0}, e_{1}, e_{2}, e_{3}, \sqrt{e_{1}^{2} + e_{2}^{2}} )ℝ.  (7)

By (6), the following is clear:

Lemma 2.1. ζ maps the cycle c = (m_{x}, m_{y}, m_{z}, r) to the set ζ(c) of those hyperplanes of ζ(G), which pass through the point (1, m_{x}, m_{y}, m_{z}, r)ℝ.

We define a quadric

Ω : x_{0} = x_{3} = x_{1}^{2} + x_{2}^{2} − x_{4}^{2} = 0.  (8)

Then for all cycles c = (m_{x}, m_{y}, m_{z}, r), we consider the quadratic cone Γ(c) with vertex (1, m_{x}, m_{y}, m_{z}, r)ℝ and base quadric Ω. It can be parametrized by

\begin{aligned}Γ(c) = & \{λ(1, m_{x} + r cos t, m_{y} + r sin t, m_{z}, 0) + μ(1, m_{x}, m_{y}, m_{z}, r), \\ & t, λ, μ ∈ ℝ, λμ ≠ 0\}. \end{aligned}  (9)

We embed ℝ^{3} in P_{4}(ℝ) by letting (x_{1}, x_{2}, x_{3}) ↦ (1, x_{1}, x_{2}, x_{3}, 0)ℝ. Then, by closer inspection of the definitions, the following holds:

Lemma 2.2. ζ(G) consists of those hyperplanes H ∈ P_{4}(ℝ)^{∗}, which fulfil |H ∩ Ω| = 1. For all cycles c, the carrier of c coincides with Γ(c) ∩ ℝ^{3}.

2.4. The Blaschke Cylinder

Definition. Let δ be the duality defined by

δ : \{ \begin{matrix}P_{4}(ℝ)^{∗} → P_{4}(ℝ), \\ (e_{0}, e_{1}, e_{2}, e_{3}, e_{4})ℝ ↦ (e_{4}, e_{1}, e_{2}, e_{3}, e_{0})ℝ. \end{matrix}  (10)

The set δ ∘ ζ(G) is called Blaschke cylinder, which is justified by the following

Lemma 2.3.

1. δ ∘ ζ(G) is contained in the affine part ℝ^{4} of P_{4}(ℝ) and coincides with the cylinder Δ = S^{1} × ℝ^{2}, whose projective extension has the equation

Δ : x_{1}^{2} + x_{2}^{2} = x_{0}^{2}.  (11)

2. We define H to be the set of hyperplanes of P_{4}(ℝ) which do not contain the point (0, 0, 0, 0, 1)ℝ. Then for all cycles c there is an H(c) ∈ H such that ζ(c) = Δ ∩ H(c) and vice versa.

Proof. The first part is clear by (7). For the second part, we note that by Lemma 2.1 and (10), the points of δ ∘ ζ(c) are contained in the hyperplane (r, m_{x}, m_{y}, m_{z}, 1)ℝ. □

2.5. The Isotropic Model

It is well-known that a quadric can be represented as conformal closure (in some well-defined sense dependent on the quadric) of an affine space. In order to perform this task for the Blaschke cylinder Δ, we apply a stereographic projection to it.

Consider the point w = (0, 1, 0, 0) ∈ Δ and its generator subspace

W = T_{w}(Δ) ∩ Δ : e_{1} = 0, e_{2} = 1.  (12)

Definition. We embed ℝ^{3} in P_{4}(ℝ) by letting (x, y, z) ↦ (1, x, 0, y, z)ℝ. Then σ is the projection of Δ \ W with center w to ℝ^{3}.

The stereographic projection σ is in coordinates described as follows:

σ : \{ \begin{matrix}Δ \ W → ℝ^{3}, \\ (1, e_{1}, e_{2}, e_{3}, e_{0})ℝ ↦ \frac{1}{1 − e_{2}} (e_{1}, e_{3}, e_{0}). \end{matrix}  (13)

If c is the cycle (m_{1}, m_{2}, m_{3}, r), then σ ∘ δ ∘ ζ(c) is the set Ψ of points, which, in coordinates, is given by

Ψ : z = f_{c}(x, y) = −\frac{r + m_{2}}{2} x^{2} − m_{1}x − m_{3}y − \frac{r − m_{2}}{2} .  (14)

If r ≠ −m_{2}, then Ψ is a parabolic cylinder, whose projective closure has the vertex (0, 0, −m_{3}, 1)ℝ. If r + m_{2} = 0, then Ψ is a plane. Equation (14) shows that the σ ∘ δ ∘ ζimages of cycles are precisely the Möbius spheres of twofold isotropic space in the sense of [2].

Oriented planes (e_{0}, 0, 1, e_{3})ℝ^{+} do not have an image under σ ∘ δ ∘ ζ. This leads to the following

Definition. The disjoint union I_{3} = ℝ^{3} ∪ W is called isotropic model of Galilei Laguerre geometry.

Then I_{3} is the conformal closure of affine ℝ^{3} in the sense of twofold isotropic Möbius geometry [2]. We extend σ by:

\overline{σ}(x) := \{ \begin{matrix}σ(x), if x ∉ W, \\ x, if x ∈ W, \end{matrix}  (15)

and define

Λ : G → I_{3}, e ↦ \overline{σ} ∘ δ ∘ ζ(e).  (16)

Then for all cycles c, the set Λ(c) ⊂ I_{3} consists of the graph surface {x, y, f_{c}(x, y)} together with δ ∘ ζ(c) ∩ W.

The inverse map Λ^{−1} maps points of I_{3} to oriented planes of G, and is, in the affine part ℝ^{3} of I_{3} described by:

Λ^{−1} : (x, y, z) ↦ \frac{1}{1 + x^{2}} (2z, 2x, x^{2} − 1, 2y)ℝ^{+}.  (17)

2.6. The Dual Isotropic Model and Geometrical Optics

A plane h = (h_{0}, h_{1}, h_{2}, h_{3})ℝ^{+} ∈ G, h_{1}^{2} + h_{2}^{2} = 1, intersects the horizontal plane x_{3}/x_{0} = c in the oriented line (h_{0} + ch_{3}, h_{1}, h_{2})ℝ^{+}. We use this for the following

Definition. Let the oriented lines a = (a_{0}, a_{1}, a_{2})ℝ^{+}, b = (b_{0}, b_{1}, b_{2})ℝ^{+} of ℝ^{2} and the line c = (c_{0}, c_{1}, c_{2})ℝ be given. Assume that a_{1}^{2} + a_{2}^{2} = b_{1}^{2} + b_{2}^{2} = 1, and denote the non-oriented lines corresponding to a, b by a′, b′. Then c is called bisector of a and b, if c is a euclidean bisector of a′ and b′, and there is a λ ∈ ℝ such that (c_{1}, c_{2}) = λ(a_{1} + b_{1}, a_{2} + b_{2}).

The non-horizontal plane ε_{1} ∈ P_{3}(ℝ)^{∗} is the bisector of ε_{2}, ε_{3} ∈ G if and only if for all horizontal planes η ⊂ ℝ^{3}, the intersection line η∩ε_{1} is the bisector of the oriented intersection lines η ∩ ε_{2} and η ∩ ε_{3}.

The connection between this definition and the next one will become apparent later. The dual isotropic model of Galilei Laguerre space is the conformal closure of a subset of dual three-dimensional projective space. The oriented planes of G whose δ ∘ ζ-image is not in W (cf. (12)), are again represented by planes.

We define a mapping β by

\{ \begin{matrix}β : G \ (δ ∘ ζ)^{−1}(W) → P_{3}(ℝ)^{∗}, \\ (e_{0}, e_{1}, e_{2}, e_{3})ℝ^{+} ↦ (e_{0}, e_{1}, e_{2} − \sqrt{e_{1}^{2} + e_{2}^{2}}, e_{3})ℝ. \end{matrix}  (18)

and define

Definition. The dual isotropic model I_{3}^{∗} of Galilei Laguerre geometry is the disjoint union I_{3}^{∗} = β(G \ (δ ∘ ζ)^{−1}(W)) ∪ W. The mapping Λ^{∗} is defined by

Λ^{∗} : G → I_{3}^{∗} : \{ \begin{matrix}g ↦ β(g) if & δ ∘ ζ(g) ∉ W \\ δ ∘ ζ(g) if & δ ∘ ζ(g) ∈ W \end{matrix}  (19)

The name dual isotropic model is explained by the following

Lemma 2.4. Let π be the duality given by

π : P_{3}(ℝ) → P_{3}(ℝ)^{∗}, (x_{0}, x_{1}, x_{2}, x_{3})ℝ ↦ (x_{3}, x_{1}, −x_{0}, x_{2})ℝ.  (20)

Then for all g ∈ G such that Λ(g) ∈ ℝ^{3} ⊂ I_{3}, π ∘ Λ(g) = Λ^{∗}(g).

Proof. Clear from (7), (13) and (20). □

Lemma 2.5. The mapping Λ^{∗} is one-to-one. Let g_{0} = (0, 1, 0, 0)ℝ^{+}. For all g ∈ G with δ ∘ ζ(g) ∉ W, the plane Λ^{∗}(g) is the bisector of g and g_{0}.

Proof. Because of Lemma 2.4, Λ^{∗} is one-to-one. The statement about the bisector property is verified by an elementary calculation. □

We want to show a connection between Λ^{∗} and geometrical optics. For this purpose, we consider an immersed surface f : U ⊂ ℝ^{2} → ℝ^{3}, such that (i) for the normal vector field n_{f} = f_{x} × f_{y} = (n_{1}, n_{2}, n_{3}) always n_{1} ≠ 0 holds, and (ii) the (euclidean) Gaussian curvature never vanishes. Then we have the following

Lemma 2.6. If f has the above properties, the family of Λ^{∗}-images of the oriented tangent planes

(x, y) ↦ f^{∗}(x, y) = Λ^{∗}((−⟨n_{f}, f⟩, n_{1}, n_{2}, n_{3})ℝ^{+})  (21)

envelopes a surface which is then called the dual isotropic image of f.

Proof. From n_{1} ≠ 0 it follows that the δ ∘ ζ-image of the oriented tangent plane is not in W, so its Λ^{∗}-image is in the projective part of I_{3}^{∗}. The Gaussian curvature of f is not zero, so the differential of the (euclidean) spherical mapping (x, y) ↦ n_{f}(x, y)/∥n_{f}(x, y)∥ is nonsingular. This implies that also the differential of the mapping, which maps (x, y) to the unit normal vector of f^{∗}(x, y), is nonsingular, and the envelope surface exists. □

We consider a bundle of light rays, which is (Galilei) reflected at the surface f. A surface which intersects the reflected rays (Galilei) orthogonally, is called anticaustic surface with respect to the given light rays. It remains to define ‘reflection’ and ‘orthogonal’ in Galilei geometry. This can simply be done by stating that reflection at a vertical plane is defined by euclidean reflection, and all other reflections are images of this one under Galilei motions. Orthogonality between a horizontal line and a plane is defined in exactly the same way. Then we have the following:

Theorem 2.7. Let f be an immersed surface f : U ⊂ ℝ^{2} → ℝ^{3} satisfying the conditions of Lemma 2.6. Then f is an anticaustic surface of the dual isotropic image of f with respect to the bundle of parallel light rays emanating from (0, 1, 0, 0)ℝ.

Proof. The light rays are horizontal, and therefore stay horizontal after the reflection. An elementary calculation shows that the Λ^{∗}-images of the oriented tangent planes along a horizontal level curve of the surface f touch the dual isotropic image of f along a horizontal curve. Therefore, and because of the definition of the bisector plane together with Lemma 2.5, it is sufficient to restrict attention to the situation in a horizontal plane, which is well-known (see e.g. [10]). □

3. Galilei Laguerre Transformations

Definition. A Galilei Laguerre transformation is a bijection G → G which induces a bijection in the set of cycles. The group of Galilei Laguerre transformations will be denoted by L.

Theorem 3.1. For all Galilei Laguerre transformations f ∈ L there is a projective automorphism κ of P_{4} with κ(Δ) = Δ, such that the δ ∘ ζ-image of f coincides with κ|Δ.

Proof. The δ ∘ ζ-image g of an f ∈ L is a bijection Δ → Δ. By Lemma 2.3, g induces a bijection in H, also denoted by g. We are going to show that g is an automorphism of a circle geometry in the sense of [3], p. 992.

In the terminology of [3], a 2-dimensional plane ε ⊄ Δ intersects Δ in a proper circle k = ε ∩ Δ, if the affine span [k] equals ε. We denote the absolute plane x_{0} = 0 by ω, the line x_{0} = x_{1} = x_{2} = 0 by f, and the point (0, 0, 0, 0, 1)ℝ by u. The proper circles of Δ are therefore those conics (ε ∩ f = ∅) and pairs of parallel lines (ε ∩ f = {p}), which are not entirely contained in a generator subspace. Two proper circles k_{1}, k_{2} touch at a if either a ∈ k_{1} = k_{2} or [k_{1}] ∩ [k_{2}] has dimension 1 and is tangent to Δ in a. A conic k_{1} and a pair k_{2} of parallel lines never touch, because [k_{1}] ∩ [k_{2}] then would contain a point of f ∈ [k_{1}].

If three points p_{1}, p_{2}, p_{3} ∈ Δ span a 2-dimensional generator subspace U, there is no H ∈ H such that p_{i} ∈ H. Therefore there is no H′ ∈ H such that g(p_{i}) ∈ H′ and g thus maps U into another generator subspace. g^{−1} enjoys the same properties as g, thus the set X ⊂ Δ is contained in a generator subspace if and only if g(X) is.

For every irreducible conic k ⊂ Δ there exist H_{1}, H_{2} ∈ H, such that H_{1} ∩H_{2} ∩Δ = k, and k is not contained in a generator subspace. Therefore neither is g(k) = g(H_{1}) ∩ g(H_{2}) ∩ Δ, and g(k) is a proper circle.

If k consists of two parallel lines l_{1}, l_{2}, such that [k] ∩ f ≠ u, there exist H_{1}, H_{2} ∈ H such that k = H_{1} ∩ H_{2} ∩ Δ. There is no generator subspace which contains k, and there are generator subspaces U_{i} ⊃ l_{i}, i = 1, 2. Thus g(k) is not contained in a generator subspace, but g(l_{i}) is. Because of g(l_{i}) ⊂ g(H_{1}) ∩ g(H_{2}), we have g(l_{1})∥g(l_{2}) and g(k) is a proper circle with u ∉ [g(k)].

For all lines l ∋ u, also g(l) ∋ u, because for any two points p, q ∈ l there is no H ∈ H such that p, q ∈ H, and therefore there is no H ∈ H such that g(p), g(q) ∈ H, and vice versa. Thus also those proper circles k which pass through u are mapped to proper circles g(k) ∋ u.

We are going to show that g is compatible with the touching relation: Assume that the proper circles k_{1} ≠ k_{2} are conics. Let V = [k_{1}] ∩ [k_{2}] and let U_{a} be the generator subspace of a ∈ Δ. Then k_{1} and k_{2} touch at a, if and only if dim V = 1 and V ∩Δ ⊂ U_{a}. The intersection V ∩ f is empty because neither [k_{1}] nor [k_{2}] intersect f. This implies that (i) there is exactly one H ∈ H with k_{i} ⊂ H and (ii) within this H, whose intersection H ∩Δ is a non-degenerate elliptic cylinder Z ⊃ k_{i}, the set k_{1} ∩ k_{2} has no points outside the Z-generator line H ∩ U_{a}.

The converse is also true: if k_{1} ≠ k_{2} with a ∈ k_{1} ∩ k_{2}, and there is an H ∈ H containing the k_{i}, then dim V = 1; and if k_{1} ∩ k_{2} ⊂ H ∩ U_{a}, then V is tangent to Δ, which means that the k_{i} touch in a.

Conditions (i) and (ii) are invariant with respect to g, and therefore so is the touching relation, when restricted to conics. Pairs of lines, however, touch if and only if one of their lines coincide, so g leaves the touching relation invariant. Thus g is an automorphism of circle geometries in the sense of [3], and we can use Schröder’s theorem ([3], p. 992 or [14]) to deduce the existence of a projective automorphism κ of P_{4} with κ(Δ) = Δ and κ|Δ = g.

□

In [2], H. Brauner considered Möbius geometry of twofold isotropic space. It turns out that isotropic Möbius geometry and Galilei Laguerre geometry are isomorphic. The result in [2], which describes all isotropic Möbius transformations as collineations automorphic for some quadric, which is of the same projective type as the Blaschke cylinder, has been obtained by assuming continuity. This is the reason why we re-proved it on basis of [14].

We are able to write down the δ ∘ ζ-image of Galilei Laguerre transformations:

Theorem 3.2. In homogeneous coordinates, the projective automorphisms κ of the Blaschke cylinder are given by a matrix

A = ( \begin{matrix}A_{0} & 0 \\ A_{1} & A_{2} \end{matrix} ) ,  (22)

where A_{0} is the matrix of a projective automorphism of the unit circle x_{0}^{2} = x_{1}^{2} + x_{2}^{2}, and A_{2} ∈ GL(2, ℝ). Thus L is a 13-dimensional transformation group.

Proof. For all x_{3}, x_{4}, κ((0, 0, 0, x_{3}, x_{4})ℝ) = (0, 0, 0, x′_{3}, x′_{4})ℝ holds, therefore the upper right corner in (22) is zero.

Embed P_{2}(ℝ) into P_{4}(ℝ) by letting (x_{0}, x_{1}, x_{2})ℝ ↦ (x_{0}, x_{1}, x_{2}, 0, 0)ℝ and let π : P_{2}(ℝ) → P_{2}(ℝ) be defined by κ followed by the central projection with center x_{0} = x_{1} = x_{2} = 0 onto P_{2}(ℝ). Then π|S^{1} is a projective automorphism of the unit circle. A is invertible if and only both A_{0} and A_{2} are.

Obviously these conditions are also sufficient for κ being an automorphism of Δ and therefore dim L = 13. □

4. Rational Circular Offset Surfaces

In analogy to the Pythagorean-Normal surfaces of [8, 11] we now will consider rational surfaces with rational circular offsets [9]. A circular offset surface is formed by the offset curves at distance d to the level curves of the original surface in horizontal planes. If we define the Galilei normals to a surface as the horizontal lines perpendicular to level curves, the definition of circular offset surfaces is in accordance with the usual definition of an offset surface as a surface whose distance to the given surface, measured along the common surface normals, is constant.

Definition. A two-dimensional rational parametrization p is a tuple (p_{0}(x, y), …, p_{n}(x, y)) of rational functions such that not all of them are zero. p(x, y) then is defined for all (x, y) except for those which are contained in an algebraic subset of dimension at most one. We say that p is defined for almost all (x, y) and the mentioned algebraic set is negligible. The (possibly degenerate) rational surface Φ(p) of P_{n}(ℝ) defined by p is the smallest projective algebraic variety which contains all points (p_{0}(x, y), …, p_{n}(x, y))ℝ. If Φ(p) is not entirely contained in the absolute plane x_{0} = 0, the set ℝ^{n} ∩ Φ(p) will also be called rational surface (of ℝ^{n}).

If p parametrizes a rational surface of ℝ^{n}, for almost all parameter values (x, y), the point p(x, y)ℝ is defined and actually contained in ℝ^{n} ⊂ P_{n}(ℝ).

Definition. If p(x, y) ≠ 0, the hyperplane h = (h_{0}, …, h_{3})ℝ ∈ P_{3}(ℝ)^{∗} is tangent to p = (p_{0}, …, p_{3}) at p(x, y)ℝ, if ⟨h, p(x, y)⟩ = ⟨h, \frac{∂}{∂x}p(x, y)⟩ = ⟨h, \frac{∂}{∂y}p(x, y)⟩ = 0. If p ∉ ω, the vector λ(h_{1}, h_{2}, 0), λ ∈ ℝ, is then called Galilei normal vector of p in p(x, y)ℝ.

Definition. A rational surface Φ(p) of ℝ^{3} is called a rational circular offset surface, if there are rational functions n_{1}(x, y), n_{2}(x, y) and r(x, y), r nonzero, such that

1. for all (x, y) such that n_{i}(x, y) is defined and p(x, y) ≠ 0, the vector (n_{1}(x, y), n_{2}(x, y), 0) is Galilei normal to p in p(x, y)ℝ and

2. the equation n_{1}^{2} + n_{2}^{2} = r^{2} holds in ℝ(x, y).

The name rational circular offset surface is justified by the following

Lemma 4.1. If p is a parametrization of a rational circular offset surface, then indeed for all d ∈ ℝ there exist rational parametrizations p_{d} of surfaces, which have the property that the distance to Φ(p), measured along the common Galilei normal, equals d.

Proof. Without loss of generality we can achieve n_{1}^{2} + n_{2}^{2} = 1 by changing n_{1} and n_{2} to n_{1}/r and n_{2}/r, respectively. Then an easy calculation shows that the surface given by

p_{d} = (1, \frac{p_{1}}{p_{0}} + dn_{1}, \frac{p_{2}}{p_{0}} + dn_{2}, \frac{p_{3}}{p_{0}} ),

has the following properties: (i) for almost all (x, y), the point p_{d}(x, y) is defined; (ii) if both are defined, the Galilei distance between p_{d}(x, y)ℝ and p(x, y)ℝ equals d. □

We are now going to define the surface Φ^{∗}(p) dual to a rational surface Φ(p). Because of possible degeneracies, we have to distinguish several cases. We define

p_{x}(x, y) = \frac{∂}{∂x} (\frac{p_{1}}{p_{0}} , \frac{p_{2}}{p_{0}} , \frac{p_{3}}{p_{0}} ) and p_{y}(x, y) = \frac{∂}{∂y} (\frac{p_{1}}{p_{0}} , \frac{p_{2}}{p_{0}} , \frac{p_{3}}{p_{0}} ) .  (23)

1. If dim Φ(p) = 2, for almost all (x, y) the vectors p_{x}(x, y) and p_{y}(x, y) are linearly independent and n_{p}(x, y) = p_{x}(x, y) × p_{y}(x, y) is nonzero.

2. If dim Φ(p) = 1, at least one of p_{x} and p_{y}, say p_{x}, is nonzero. There exists a vector n_{0} ∈ ℝ^{3} which is independent of p_{x}(x, y) for almost all (x, y). Thus for almost all (x, y) the vector n_{p}(x, y) = p_{x}(x, y) × n_{0} is nonzero.

3. If dim Φ(p) = 0, let n_{p}(x, y) = (x, y, 1).

Definition. If p and n_{p} are as defined above, the rational surface Φ(p)^{∗} of P_{3}(ℝ)^{∗} parametrized by

p^{∗} = (−n_{1}p_{1} − n_{2}p_{2} − n_{3}p_{3}, n_{1}p_{0}, n_{2}p_{0}, n_{3}p_{0})  (24)

is called the surface dual to Φ(p).

We leave the verification that p^{∗} is actually well-defined to the reader. We have the following

Lemma 4.2. When defined as in (24), p^{∗}(x, y) is tangent to p in p(x, y). Both the iterated dual p^{∗∗} and p define the same surface. If p defines a rational circular offset surface with Galilei normal vector field m_{1}, m_{2}, there is a rational function m_{3} such that p^{∗} and (−m_{1}p_{1} − m_{2}p_{2} − m_{3}p_{3}, m_{1}p_{0}, m_{2}p_{0}, m_{3}p_{0}) define the same surface Φ(p^{∗}).

Proof. In regular points p(x, y) we can use differential geometry to verify the lemma, and the set of parameter values corresponding to singular points is negligible.

Not both n_{1}, n_{2} are zero, so assume that n_{1} is nonzero. Both (n_{1}, n_{2}, 0) and (m_{1}, m_{2}, 0) are Galilei normal to p and therefore (m_{1}, m_{2}, 0) = (m_{1}/n_{1})(n_{1}, n_{2}, 0) holds in ℝ(x, y). Then define m_{3} = m_{1}n_{3}/n_{1}. □

We are going to show that Galilei Laguerre transformations do not destroy the property of being a rational circular offset surface. But first we have to define what dual surface means in terms of Galilei Laguerre space.

Definition. Let p^{∗} = (h_{0}, h_{1}, h_{2}, h_{3}) be a rational parametrization of a surface in P_{3}(ℝ)^{∗} which does not entirely consist of horizontal planes. Then the set of those p^{∗}(x, y)ℝ^{+}, which are not horizontal, is called dual surface to p considered as a subset of G.

Then the following theorems hold:

Lemma 4.3. The transfer δ ∘ζ from Galilei Laguerre space to the Blaschke cylinder Δ maps the duals of rational circular offset surfaces to rational surfaces which are contained in Δ, and vice versa.

Proof. Without loss of generality we can assume a parametrization h of the dual surface which satisfies h_{1}^{2} + h_{2}^{2} = 1. Then δ ∘ ζ(h(x, y)ℝ^{+}) ∈ Δ. Conversely, for a rational parametrization q = (q_{0}, q_{1}, q_{2}, q_{3}, q_{4}) of a surface contained in Δ, q_{0} is nonzero, so without loss of generality q_{0} = 1. This implies q_{1}^{2} + q_{2}^{2} = 1 and p^{∗} = (δ ∘ ζ)^{−1}(q) defines a surface parametrized by p^{∗∗}, which is a rational circular offset surface. □

Theorem 4.4. Galilei Laguerre transformations map the duals of rational circular offset surfaces to the duals of rational circular offset surfaces.

Proof. The δ ∘ ζ-image of a Galilei Laguerre transformation is a projective automorphism of Δ and therefore preserves rationality. The theorem follows immediately from Lemma 4.3. □

The stereographic projection σ defines a bijective map between rational surfaces in Δ \ W and rational surfaces in the isotropic model I_{3} \ W, which gives us a simple construction of rational circular offset surfaces.

Theorem 4.5. The geometric transformation Λ^{−1}, which describes the change from the isotropic model of Galilei Laguerre space to the standard model, maps rational surfaces to the duals of rational circular offset surfaces, with the single exception of one degenerate rational surface consisting of a horizontal point at infinity.

Proof. The stereographic projection preserves rationality and the only rational surface which could lie entirely in W is one which corresponds to a bundle with vertex situated on the absolute horizontal line. The converse is obvious. □

5. Special Rational Circular Offset Surfaces

5.1. Normal Forms of Quadratic Graphs

We consider the graph Ψ = {(x, y, f(x, y)} of the quadratic function

f : ℝ^{2} → ℝ, (x_{1}, x_{2}) ↦ ∑_{i+j≤2} a_{ij}x^{i}y^{j}  (25)

as a subset of ℝ^{3} ⊂ I_{3} and ask for its Λ-preimage in G. It will turn out that Λ^{−1}(Ψ) consists of the set of tangent planes of a cubic surface with special properties (see next section). In order to simplify the discussion, we are looking for geometric transformations τ which do not destroy the Galilei geometric properties of Λ^{−1}Ψ, and transform Ψ with the Λ-image of τ. The result is

Lemma 5.1. We are able to transform the graph surface to the graph of one the following normal forms:

1. z = ax^{2} + by^{2}, a, b ∈ ℝ, b ≠ 0  (26)
2. z = cxy, c ∈ ℝ \ 0.  (27)
3. z = d, d ∈ ℝ.  (28)

Proof. The transformation is achieved by letting ϕ = v = 0 in (1) and adjusting the other parameters appropriately. □

5.2. Parabolic Galilei Cyclides

We consider the isotropic model I_{3} of Galilei Laguerre Geometry and the normal form (26) of quadratic graphs. Λ^{−1} transforms this graph into a rational parametrization of a surface of class 4, whose corresponding point surface Φ has the homogeneous equation

Φ : x_{3}^{2}x_{2} + 2ax_{3}^{2}x_{0} + 2b(x_{1}^{2} + x_{2}^{2})x_{0} + 4abx_{2}x_{0}^{2} = 0.  (29)

If necessary, we also consider the complex extension Φ_{ℂ} ⊂ P_{3}(ℂ), which is defined by the same equation. These surfaces are called parabolic Galilei cyclides. For cyclides in Galilei space, see also [7].

Equation (29) obviously defines an algebraic surface of degree three. Its intersection with the plane ω : x_{0} = 0 at infinity is

Φ ∩ ω : x_{3}^{2}x_{2} = x_{0} = 0,  (30)

so Φ touches ω in all points of the horizontal line x_{3} = x_{0} = 0. The line l : x = 0, y = −2a is easily seen to be contained in Φ.

Definition. An algebraic surface is called a Blutel’s surface [4], if almost all of it is covered by a one parameter family of conic sections, such that for all of these conics the tangent planes in the points of the conic are tangent to a quadratic cone.

Lemma 5.2. The surface Φ carries a one-parameter family of horizontal circles and a oneparameter family of parabolas in vertical planes all of which contain the line l. Φ is a Blutel’s surface with respect to both families of conics.

Proof. Horizontal planes intersect the affine part of Φ in circles or points. The intersections of Φ with planes ε ⊃ l consist, by the surface’s degree, of l and a conic. These conics touch ω in their horizontal points at infinity. They are, therefore, parabolas.

We want to show that the planes tangent to Φ in the points of a horizontal circle form a (Galilei) cone of revolution. For this purpose we notice that for all v ∈ ℝ, along the plane y = const = v, the quadratic graph (26) touches a parabolic cylinder with ideal vertex situated on the line x_{0} = x_{1} = 0, the Λ^{−1}-image of which is a cycle. The carrier of this cycle therefore is contained in Φ. The tangent planes along this circle are, by Λ^{−1}, those with coordinates (2(au^{2} + bv^{2}), 2u, u^{2} − 1, 2v)ℝ (v ∈ ℝ). The line x = 0, y = −2a intersects them in a point p independent of u. This shows that along its horizontal circles, Φ is touched by cones whose vertices p are situated on l.

Analogously it is easily verified that Φ is touched by horizontal parabolic cylinders along the parabolas mentioned above. □

Lemma 5.3. The complex extension Φ_{ℂ} either possesses two conic singularities or one biplanar singularity situated on l. Additionally, Φ_{ℂ} has conic singularities in the absolute points J_{1,2} = (0, i, ±1, 0)ℝ.

Proof. The first part is proved by looking at the intersection points of l and one (and then, all) of the parabolas mentioned above. If one (and then, all) of the parabolas touch l in a point p, {p} is the intersection of Φ with the horizontal plane ε ∋ p. The intersection Φ_{ℂ} ∩ ε, however, consists of [pJ_{1}] ∪ [pJ_{2}], which defines the two tangent planes of p.

The second part is clear from the fact that the midpoints of the horizontal circles contained in Φ are situated on a parabola, which then serves as a base curve for the tangent cones of both J_{1} and J_{2}. □

Figure 1 shows an example of a parabolic Galilei cyclide. Those special Galilei transformations, which have been used to transform a general quadratic graph to its normal form, are affine transformations, whose restrictions to the horizontal planes are translations. Thus all Λ^{−1}-images of quadratic graphs are parabolic Galilei cyclides carrying circles in horizontal planes, whose centers are situated on a parabola with horizontal axis in the plane x = 0.

The general parabolic cyclide then is the image of Φ under a Galilei transformation. Not all of them are Λ^{−1}-images of quadratic graphs. For all of them, however, there is a special choice of the center of the stereographic projection σ, such that they become Λ^{−1}-images.

The Λ-preimages of the quadratic graphs of the remaining normal forms of (26) are described by the following

Lemma 5.4. The normal form (27) is the Λ-image of the surface dual to the cubic conoidal ruled surface with equation

(x^{2} − c^{2})y − 2xz = 0.  (31)

The normal form (28) corresponds to the cycle (0, d, 0, −d).

Proof. The result is verified by elementary calculations. The second part also follows directly from (14). □

Figure 1: Parabolic Galilei Cyclide

5.3. Galilei Cyclides

The most general Galilei cyclides are, in their affine part, generated by circles in horizontal planes z = c, whose centers are situated on a parabola with horizontal axis and contained in a non-horizontal plane, and whose radius is a quadratic polynomial function of c. They either are Galilei cylinders/cones of revolution or are the images of the parabolic cyclide Φ under Galilei transformations and the quadratic transformation

ψ : (x, y, z) ↦ (x, y + βz + γz^{2}, z).

Those Galilei cyclides will be called proper ones. Let p = (m_{x} + r cos ϕ, m_{y} + r sin ϕ, m_{z}) and v = (n_{x}−r cos ϕ, n_{y}−r sin ϕ, n_{z}). Then the tangent vector (p; v) is mapped by dψ to (ψ(p); w) with w = v + (0, n_{z}(β + 2m_{z}γ), 0). This tells us that a surface touching a Galilei cone of revolution along a horizontal circle does not lose this property when being transformed by ψ.

Furthermore, the second component of w depends linearly on m_{z}. This tells us that if the vertices of all those cones are situated on a line l, they also do so after the ψ-transform. This implies:

Theorem 5.5. A proper Galilei cyclide is a twofold Blutel’s surface carrying a horizontal family of circles. The vertices of the cones tangent to the surface along these circles are situated on a straight line.

6. Modeling Rational Circular Offset Surfaces

6.1. Galilei Canal Surfaces

In analogy to [10], where a canal surface has been discretised and approximated by a sequence of Dupin cyclides, we can ask for a discretisation and approximation of Galilei canal surfaces by Galilei cyclides:

Definition. Surfaces of the form

f : I × ℝ → ℝ^{3}, (u, ϕ) ↦ (m_{x}(u) + r(u) cos(ϕ), m_{y}(u) + r(u) sin(ϕ), u).

are called Galilei canal surfaces.

A Galilei cyclide touches such a surface along a horizontal circle u = const, if and only if both surfaces have both the circle and the tangent cone along this circle in common.

Definition. If f is defined as above, its medial axis is the curve

c_{f} : I → ℝ^{3}, u ↦ (m_{x}(u), m_{y}(u), r(u)).

Clearly, two Galilei canal surfaces f and g touch each other along the circle z = u, if and only if their medial axes touch each other in the point c_{f}(u) = c_{g}(u).

If the curve of centers (m_{x}(u), m_{z}(u), u) is of the form a + ub + u^{2}c with a, b, c ∈ ℝ^{3} such that c is horizontal and b is not, c_{f} is a parabola or a straight line, and f becomes a parametrization of a Galilei cyclide. Conversely, every parabola of ℝ^{3} corresponds to a proper Galilei cyclide, and every straight line to a Galilei cylinder/cone of revolution. Thus the problem of approximating a Galilei canal surface is equivalent to the problem of approximating a curve of ℝ^{3} by a quadratic B-spline, which has been studied extensively. A picture is shown in figure 2.

Figure 2: Piecewise Galilei Cyclide Surface

6.2. A Powell-Sabin Method

Let T be a triangulation of a polygonal subset U ⊆ ℝ^{2} and let for each vertex v_{i} a value z_{i} and a linear functional d_{i} ∈ ℝ^{2∗} be given. Then the algorithm of Powell and Sabin [6] gives a piecewise quadratic C^{1}(U) function f with the property f(v_{i}) = z_{i} and df(v_{i}) = d_{i} for all i. Moreover, the points where f is not C^{2} are those of the edges of a six-fold refinement of T . This can be used for interpolating given values and derivatives by a rational circular offset surface, simply by interpolating the appropriate values and derivatives in the isotropic model. For classical Laguerre geometry, this has been done in [8].

Acknowledgements

This work has been supported in part by grant No. P11357-MAT of the Austrian Science Foundation.

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Received March 18, 1997