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The Lie Model for Euclidean Geometry

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The Lie Model for Euclidean Geometry MM Research Preprints,95–111 No. 19, Dec. 2000. Beijing 95 The Lie Model for Euclidean Geometry Hongbo Li Academy of Mathematics and Systems Science Chinese Academy of Sciences, Beijing 100080 Abstract. In this paper we investigate the Lie model of Lie sphere geometry using Clif- ford algebra. We employ it to Euclidean geometric problems involving oriented contact to simplify algebraic description and computation. 1. Introduction According to Cecil (1992), Lie (1872) introduced his sphere geometry in his dissertation to study contact transformations. The subject was actively pursued through the early part of the twentieth century, culminating with the publication of the third volume of Blaschke’s Vorlesungen ¨uber Differentialgeometrie (1929), which is devoted entirely to Lie sphere geom- etry and its subgeometries, particularly in dimensions two and three. After this, the subject fell out of favor until 1981, when Pinkall used it as the principal tool to classify Dupin hy- persurfaces in R4 in his dissertation. Since then, it has been employed by several differential geometers to study Dupin, isoparametric and taut submanifolds (eg. Cecil and Chern, 1987). It has also been used by Wu (1984/1994) in automated geometry theorem proving. Despite its important role played in differential geometry, Lie sphere geometry has limited applicability in classical geometry. This is because a Lie sphere transformation has classical geometric interpretation only when it is a M¨obius transformation or the orientation-reversing transformation. In other words, a general Lie sphere transformation does NOT have classical geometric interpretation. For classical geometry, Lie sphere geometry can contribute to simplifying description and computation of tangencies of spheres and hyperplanes. Because of this, we would like to “attach” Lie sphere geometry to the homogeneous model of Euclidean geometry in (Li, Hestenes and Rockwood, 2000a) as a supplementary tool. This goal is achieved in this paper. The tool, called the Lie model, is essentially a coordinate-free reformulation of Lie sphere geometry for the purpose of applying it to Euclidean geometry. It is used to solve geometric problems involving oriented contact of spheres and hyperplanes, and can help obtaining sim- plifications. Unfortunately, this may be as much as it can contribute to classical geometry. The model can also be extended to spherical and hyperbolic geometries via their homoge- neous models in (Li, Hestenes and Rockwood, 2000b, c). This paper is arranged as follows: in section ?? we introduce the Lie model using the language of Clifford algebra (Hestenes and Sobczyk, 1984); in section ?? we investigate further basic properties of this model; in section ?? we provide examples to illustrate how to apply it to Euclidean geometry. 96 H. Li 2. The Lie model The Lie model will be established upon the homogeneous model. So first let us review the homogeneous model for Euclidean geometry (Li, Hestenes and Rockwood, 2000a; Li, 1998). 2.1. The homogeneous model n n Let {e1,..., en} be an orthonormal basis of R . A point c of R corresponds to the vector from the origin to the point, denoted by c as well. The origin corresponds to the zero vector. We embed Rn into a Minkowski space of n + 2 dimensions as a subspace. Denote the n+1,1 n+1,1 Minkowski space by R . Let {e−2, e−1, e1,..., en} be an an orthonormal basis of R , 2 2 where −e−2 = e−1 = 1. The 2-space spanned by e−2, e−1 is Minkowski, and has two null 1-subspaces. Let e, e0 be null vectors in the two 1-subspaces respectively. Rescale them to make e · e0 = −1. Now we map Rn in a one-to-one manner into the null cone of Rn+1,1 as follows: c2 c 7→ c´ = e + c + e, for c ∈ Rn. (2.1) 0 2 The range of the mapping is {x ∈ Rn+1,1|x2 = 0, x · e = −1}. (2.2) By this mapping, a point c of Rn can be represented by the null vector c´. In particular, n the origin of R can be represented by e0. Vector e corresponds to the unique point at infinity for the compactification of Rn. This representation is is called the homogeneous model for Euclidean geometry. The homogeneous model can also be described as follows. Any null vector x of Rn+1,1 represents a point or the point at infinity of Rn. It represents the point at infinity if and only if x · e = 0. Two null vectors represent the same point or the point at infinity if and only if they differ by a nonzero scale. The following is a fundamental property of the homogeneous model: Theorem 2.1. Let Br−1,1 be a Minkowski r-blade in Gn+1,1, 2 ≤ r ≤ n + 1. Then Br−1,1 represents an (r − 2)-dimensional sphere or plane in the sense that a point represented by a null vector a is on it if and only if a ∧ Br−1,1 = 0. It represents a plane if and only if e ∧ Br−1,1 = 0. The representation is unique up to a nonzero scale. The (r − 2)-dimensional sphere passing through r affinely independent points a1,..., ar n of R can be represented by a´1 ∧· · ·∧a´r; the (r −2)-dimensional plane passing through r −1 n affinely independent points a1, ..., ar−1 of R can be represented by e ∧ a´1 ∧ · · · ∧ a´r−1. When r = n + 1, the dual form of the above theorem is Theorem 2.2. Let s be a vector of Rn+1,1 satisfying s2 > 0. Then it represents a sphere or hyperplane in the sense that a point represented by a null vector a is on it if and only if a · s = 0. It represents a hyperplane if and only if e · s = 0. The representation is unique up to a nonzero scale. The following are standard representations in the homogeneous model. Lie sphere geometry 97 ρ2 1. A sphere with center c and radius ρ > 0 is represented by c´ − e. 2 2. A hyperplane with unit normal n and distance δ > 0 away from the origin in the direction of n is represented by n + δe. 3. A hyperspace with unit normal n is represented by either of ±n. 4. A sphere with center c and passing through point a is represented by a´ · (e ∧ c´). 5. A hyperplane with normal n and passing through point a is represented by a´ · (e ∧ n). The following are formulas and explanations for some inner products in Rn+1,1: • For two points c´1 and c´2, (c´ − c´ )2 |c − c |2 c´ · c´ = − 1 2 = − 1 2 . (2.3) 1 2 2 2 • For point c´ and hyperplane n + δe, c´ · (n + δe) = c · n − δ. (2.4) It is positive, zero or negative if the vector from the hyperplane to the point is along n, zero or along −n respectively. Its absolute value equals the distance between the point and the hyperplane. ρ2 • For point c´ and sphere c´ − e, 1 2 2 ρ2 ρ2 − |c − c |2 c´ · (c´ − e) = 1 2 . (2.5) 1 2 2 2 It is positive, zero or negative if the point is inside, on or outside the sphere respectively. Its absolute value equals half the distance between the point and the sphere. • For two hyperplanes n1 + δ1e and n2 + δ2e, (n1 + δ1e) · (n2 + δ2e) = n1 · n2. (2.6) ρ2 • For hyperplane n + δe and sphere c´ − e, 2 ρ2 (n + δe) · (c´ − e) = (n + δe) · c´. (2.7) 2 It is positive, zero or negative if the vector from the hyperplane to the center c is along n, zero or along −n respectively. Its absolute value equals the distance between the center and the hyperplane. 98 H. Li ρ2 ρ2 • For two spheres c´ − 1 e and c´ − 2 e, 1 2 2 2 ρ2 ρ2 ρ2 + ρ2 − |c − c |2 (c´ − 1 e) · (c´ − 2 e) = 1 2 1 2 . (2.8) 1 2 2 2 2 It is zero if the two spheres are perpendicular to each other. When the two spheres intersect, (??) equals cosine the angle of intersection multiplied by ρ1ρ2. The following are geometric interpretations of the outer product of s1, s2, which are vectors of nonnegative square in Rn+1,1: 1. If s1 ∧ s2 = 0, s1, s2 represent the same the geometric object. If s1 ∧ s2 6= 0 but 2 (s1 ∧ s2) = 0, then • if s1 represents a point or the point at infinity, it must be on the sphere or hyperplane represented by s2; • if s1, s2 represent two spheres or a sphere and a hyperplane, they must be tangent to each other; • if s1, s2 represent two hyperplanes, they must be parallel to each other. In all these cases we say the geometric objects represented by s1 and s2 are in contact. Let s1 ∧ s2 6= 0. The blade represents the pencil of spheres and hyperplanes that contact both s1 and s2, together with the point of contact, in the sense that a point, the point at infinity, a sphere or a hyperplane represented by a vector s of nonnegative square is in contact with both s1, s2 if and only if s ∧ s1 ∧ s2 = 0. The pencil is called a contact pencil. 2 2. If (s1 ∧ s2) < 0, then s1, s2 must represent two intersecting spheres or hyperplanes. The blade s1 ∧ s2 represents the pencil ofaa spheres and hyperplanes that pass through the intersection of s1 and s2, together with the points of intersection. The pencil is called a concurrent pencil.
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