On Affine Geometry of Space Curves

Transcription

1 On Affine Geometry of Space Curves Ali Mahdipour Shirayeh Abstract In this expository paper, we explain equivalence problem of space curves under affine transformations to complete the method of Spivak and find all the invariants. Furthermore, we propound a necessary and sufficient condition for the invariants and study the shapes of space curves with constant curvatures in detail. Then we suggest their applications in physics, computer vision and computer graphics. Key words: affine geometry, curves in Euclidean space, differential invariants. A.M.S Subject Classification: 53A15, 53A04, 53A55. 1 Introduction This study is devoted to explain the author s paper, [7], which can be viewed as a continuation of the work [9], where he began the classification of space curves up to special affine transformations. We determine all of differential invariants and our method is different from the method of Guggenheimer and other existing methods. Also, for the first time, we prove a necessary and sufficient condition for the invariants in order that complete the classification. Classification of curves has a significant place in geometry, physics, mechanics, computer vision and image processing. In geometrical sense, a plane curve with constant curvature, up to special affine transformations may be either an ellipse, a parabola or a hyperbola [9]. In contrast, we classify the shapes of space curves of constant curvatures which has a wide variety of applications in physics, computer vision and image processing. The general form of these shapes are exist in [3], but here we try to discuss them in more details. 2 Maurer-Cartan form Let G GL(n, R) be a matrix Lie group with Lie algebra g and P : G Mat(n n) be a matrix-valued function which embeds G into Mat(n n). Definition 2.1 The 1-form ω B = {P(B)} 1 dp B of G is called the Maurer-Cartan form. It is often written ω = P 1 dp. The Maurer-Cartan form is in fact the unique left invariant g valued 1-form on G such that ω Id : T Id G g is the identity map. The Maurer-Cartan form ω satisfies in Maurer-Cartan equation d ω = ω ω. School of Mathematics, Iran University of Science and Technology, Narmak, Tehran , Iran. E- mail: mahdipour@iust.ac.ir 1

2 Theorem 2.2 (Cartan [4]) Let G be a matrix Lie group with Lie algebra g and Maurer-Cartan form ω. Let(M be a manifold on which there exists a g valued 1-form φ satisfying d φ = φ φ. Then for any point x M there exist a neighborhood U of x and a map f : U G such that f ω = φ. Moreover, Any two such maps f 1, f 2 must satisfy f 1 = L B f 2 for some fixed B G (L B is the left action of B on G). Corollary 2.3 Given maps f 1, f 2 : M G, then f1 ω = f 2 ω, that is, this pull-back is invariant, if and only if f 1 = L B f 2 for some fixed B G. Definition 2.4 An affine transformation of the Euclidean space R 3 is the composition of a translation in R 3 among with an element of the general linear group GL(3, R). It is called special or unimodular, if its matrix part is an element of SL(3, R). 3 Classification of space curves Let c : [a, b] R 3 be a curve in three dimensional space which we call the space curve, be of class C 5 and det(c, c, c ) 0, (3.1) for any point of the domain, that is, we assume that c, c and c are linear independent. Moreover, we assume that det(c, c, c ) > 0. For the curve c, we consider a new curve, namely α c (t) : [a, b] SL(3, R), defined by α c (t) := (c, c, c ) {det(c, c, c )} 1/3 which is well defined on the domain of c into the special linear group SL(3, R). We can study the new curve in respect to special affine transformations, i.e. the action of special affine transformations on first, second and third differentiations of c. Theorem 3.1. Let c and c are two space curves. c and c are the same with respect to special affine transformations, i.e. c = A c when A = τ B for translation τ in R 3 and B SL(3, R) if and only if α c = L B α c where L B is a left translation generated by B. After some computations, we find that α 1 c det(c,c,c ) 3det(c,c,c ) 0 α c is in the following multiple of dt det(c,c,c ) det(c,c,c ) 1 det(c,c,c ) 3det(c,c,c ) det8c,c,c ) det(c,c,c ) 0 1 2det(c,c,c ) 3 det(c,c,c ) We may use of a proper parametrization σ : [a, b] [0, l] such that the parameterized curve γ = c σ 1 satisfies in condition det(γ (s), γ (s), γ (w)) = 0 and so it is sufficient that we. 2

3 assume det(γ (s), γ (s), γ (s)) = 1. Thus we conclude that σ, namely the special affine arc length, is defined as follows t [ 1/6du. σ := det(c (u), c (u), c (u))] a Every curve parameterized by σ up to special affine transformations is introduced with the following invariants χ 1 = det(c, c, c ), χ 2 = det(c, c, c ). (3.2) Theorem 3.2 Every space curve of class C 5 satisfying in condition (3.1) under the action of special (unimodular) affine transformations is determined by its natural equations χ 1 = χ 1 (σ) and χ 2 = χ 2 (σ) of the first and second special affine curvatures (3.2) as functions (invariants) of the special affine arc length. Theorem 3.3 Two space curves c, c : [a, b] R 3 of class C 5 which satisfy in condition (3.1) are special affine equivalent if and only if χ c 1 = χ c 1 and χc 2 = χ c 2. The generalization of the affine classification of curves in an arbitrary finite dimensional space has been discussed in [6]. 4 Geometric interpretations applied to physics and computer vision Let a curve be parameterized with special affine arc length σ and with constant first and second affine curvatures χ 1 and χ 2 fulfilled in relation α c(σ) = α c (σ).(b), for some b sl(3, R) via the right action of the Lie algebra. Maurer-Cartan matrix of SL(3, R) is a base for Lie algebra sl(3, R) and one can write α c(σ) = α c (σ). ( we obtain α c (σ) = exp σ. 0 0 χ χ χ χ ) different forms which we divide these forms in the following cases:. By solving this first order equation, that, for different values of χ 1 and χ 2 it has a Theorem 4.1 Each curve of class C 5 in R 3 satisfied in condition (3.1) with constant affine curvatures χ 1 and χ 2, up to special affine transformations, is the trajectory of a one parameter subgroup of special (unimodular) affine transformations, that is, a curve of cases I-VI. Corollary 4.2 In the physical sense, we may assume that each space curve X : [a, b] R 3 is the trajectory of a particle with a specified mass m in R 3 and in the view of an observer, that is influenced under the effect of a force F. We have conservation laws as (F F ) F and (P F ) F where P = m v is the momentum of the particle. If these invariants of the trajectory are constant, then the shape of the motion is similar to one of the six cases mentioned in the above theorem. 3

4 Figure 1: (a) χ 1 = χ 2 = 0. (b) χ 1 = 0, χ 2 > 0. (c) χ 1 = 0, χ 2 < 0. (d) χ 1 > 0, χ 2 = 0. (e) χ 1 < 0, χ 2 = 0. (f) χ 1, χ 2 > 0. (g) χ 1, χ 2 < 0. (h) χ 1 < 0 < χ 2. (i) χ 1 > 0 > χ 2. Corollary 4.3 In computer vision and computer graphics, we may suppose that each space curve is one of the characteristic curves on a 3-dimensional object, that are feasible minimum segment curves that completely signify the object in the viewpoint of an observer. Also, if by an effect provided by (orientation-preserving) rotations and translations in R 3 we change the position of a picture without any change in characteristic lines, then these curves will be equivalent under special affine transformation. If a characteristic line has constant affine curvatures χ 1 and χ 2, then it will be similar to one of the cases of curves mentioned in Theorem 4.1. References [1] E. Calabi, P.J. Olver, and A. Tannenbaum, Affine geometry, curve flows, and invariant numerical approximations, Adv. in Math. 124 (1996) [2] A. Gray, Modern Differential Geometry of Curves and Surfaces, CRC Presr, Boca Raton, Florida, [3] H. Guggenheimer, Differential Geometry, Dover Publ., New York,

5 [4] T.A. Ivey and J.M. Landsberg, Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems, A.M.S., [5] W. Kühnel, Differential Geometry, Curves Surfaces Manifolds, 2nd edition, translated by B. Hunt, A.M.S., [6] M. Nadjafikhah and A. Mahdipour Sh., Affine classification of n-curves, Balkan J. Geom. Appl., Vol. 13, No. 2 (2008) [7] M. Nadjafikhah and A. Mahdipour-Shirayeh, Geometry of Space Curves up to Affine Transformations, to appear (2009). [8] P.J. Olver, Equivalence, invariants, and symmetry, Cambridge Univ. Press, Cambridge, [9] M. Spivak, A Comprehensive Introduction to Differential Geometry, Vo,. II, Publish or Perish, Wilmington, Delaware,