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Motion of Parallel Curves and Surfaces in Euclidean 3-Space R3 Global Journal of Advanced Research on Classical and Modern Geometries ISSN: 2284-5569, Vol.9, (2020), Issue 1, pp.43-56 MOTION OF PARALLEL CURVES AND SURFACES IN EUCLIDEAN 3-SPACE R3 MARYAM T. ALDOSSARY ∗ AND MASHNIAH A. GAZWANI ∗∗ DEPARTMENT OF MATHEMATICS, COLLEGE OF SCIENCE, IMAM ABDULRAHMAN BIN FAISAL UNIVERSITY, P. O. BOX 1982, CITY (DAMMAM), SAUDI ARABIA ∗E-MAIL:[email protected], ∗∗ E-MAIL:[email protected] ABSTRACT . The main goal of this paper is to investigate motion of parallel curves and surfaces in Euclidean 3-space R3. The characteristic properties for such objects are given. The geometric quantities are described. Finally, the evolution equations of the curvatures and the intrinsic geometric formulas are derived. keywords: Curvature, Evolution, Motion, Parallel curves, Parallel surfaces. 1. I NTRODUCTION AND MOTIVATIONS Image processing and the evolution of curves and surfaces has significant applications in computer vision [20]. As a scale space by linear and nonlinear diffusion’s are defined in [19, 21], image enhancement through an isotropic diffusion’s were studied [17, 5, 23], and image segmentation by active contours are classified in [10, 8, 16]. The evolution of curves has been studied extensively in various homogeneous spaces. The relation- ship between integrable equations and the geometric motion of curves in spaces has been known for a long time. In fact, many integrable equations have been shown to de- scribe the evolution invariant associated with certain movements of curves in particular geometric settings. The dynamics of shapes in physics, chemistry and biology are mod- elled in terms of motion of surfaces and interfaces, and some dynamics of shapes are reduced to motion of plane curves. Evolution of surfaces accompanies many physical phenomena: propagation of wave fronts are descried extensively in [18], and motion of interfaces, growth of crystals [9], geometric integral equations expressed in wide range (see [13, 22, 1]). Geometrically, curves and surfaces evolution means deforming a curve or a surface into another curve or a surface in a continuous way, respectively. For more recent treatment of curves and surfaces evolution, see [16-19]. 2010 Mathematics Subject Classification. Primary 14Q05; Secondary 14Q10. Key words and phrases. Curvature, Evolution, Motion, Parallel curves, Parallel surfaces. 43 Maryam T. Aldossary and Mashniah A. Gazwani 2. P RELIMINARY In this section we present the main results related to the motion of curves and surfaces in Euclidean 3-space R3 [16, 23]. 2.1. Motion of curves in R3 [15]. Let α be a regular curve in Euclidean 3-space, where α : [a, b] R R3. ⊂ → Let α(u, t) denote the position vector of a point on the curve at time t by α (u, t) = α(u) + W (u, t) T (u, t) + U (u, t) n (u, t) + V (u, t) B (u, t) , (2.1) α α such that e(ue, 0 ) = (u), where a metric on the curve is given by ∂α ∂α g(u, t) = , , (2.2) ∂u ∂u and the arc length along the curve is equal to u s(u, t) = g(u, t) du. (2.3) Zo q We may use either u, t or s, t as coordinates of a point on the curve. The Frenet- { } { } ∂α 1 ∂α Serret frame T, n, B is defined in the usual way ( i.e., T = = g− 2 ) which { } ∂s ∂u satisfies Frenet-Serret equations (2.13). Motion of a point on the curve can be specified by the form ∂α = W T + U n + V B, (2.4) ∂t where W, V, U are arbitrary functions represent the component of the velocity in di- rection{ of T, n, }B respectively. The motion is local which means that W, V, U de- pend only{ on local} values of κ, τ and their derivatives according to arc{ length s.} Now we will use the compatibility{ conditions} along above, our proofs in this section is ∂ ∂ ∂ ∂ α(u, t) = α(u, t). (2.5) ∂t ∂u ∂u ∂t If α(u, t) evolves in R3 locally according to the equation (2.4) such that α(u, 0 ) = α(u), then the evolution equations of the Frenet-Serret frame T, n, B associated to the curve, are given by { } ∂T ∂U ∂V = τV + κW n + + τU B. (2.6) ∂t ∂s − ∂s ∂n ∂U 1 ∂ ∂V τ ∂U = τV + κW T + + τU + τV + κW B. (2.7) ∂t − ∂s − κ ∂s ∂s κ ∂s − ! ! ∂B ∂V 1 ∂ ∂V τ ∂U = + τU T + τU + τV + κW n. (2.8) ∂t − ∂s − κ ∂s ∂s κ ∂s − ! ! The evolution equation for curvature κ of α(u, t) is given by ∂κ ∂ ∂U ∂V = τV + κW τ + τU . (2.9) ∂t ∂s ∂s − − ∂s 44 Motion of Parallel Curves and Surfaces in Euclidean 3-Space R3 The evolution equation for torsion τ of α(u, t) is given by ∂τ ∂ 1 ∂ ∂V τ ∂U ∂V = + τU + τV + κW + κ + τU . (2.10) ∂t ∂s κ ∂s ∂s κ ∂s − ∂s The evolution equation for metric g on α(u, t) is given by ∂g ∂W = 2g κU . (2.11) ∂t ∂s − The evolution equation for the arc length s along the curve α(u, t) is given by ∂s u ∂W = √g κU du. (2.12) ∂t Z0 ∂s − Proposition 2.1. The skew-symmetry of Frenet-Serret equations of α(s) is given in matrix form by T′ 0 κ 0 T n′ = κ 0 τ n . (2.13) B −0 τ 0 B ′ − 2.2. Motion of Surfaces in R3. Let M be a surface; evolving is space according to i i k i i i i X u , t = X u + ϕ u , t Xk u , t + ψ u , t N u , t , i = 1, 2 (2.14) suche that X ui, 0 = X ui and its motion of is described by ! ! e e ∂X = ViX + UN, ∂t i i where X is a patch of M, V and U are the velocity components in the tangents vectors Xi and normal N direction, respectively. Then the evolution equations of the frame Xi, N associated to the surface are given by [14] { } k k ∂U j ∂ X iV Li U ( i + V Lji ) X i = ▽ − ∂u k , (2.15) N ∂U N ∂t ki ( + j ) g i V Lji 0 − ∂u i where u are its local coordinates. Moreover, gij and Lij are the metric and curvature tensors, respectively. Also, the evolution equations of the metric tensor and its inverse with determinant, from [14] and [7], respectively are given by ∂gij = V + V 2UL , (2.16) ∂t ▽i j ▽j i − ij ∂glk = 2Llk U gil Vk glk Vl, (2.17) ∂t − ▽i − ▽l ∂g k = 2g kV 2HU . (2.18) ∂t ▽ − It can also express the evolution equations of the curvature tensor, its inverse and deter- minant which follows from [14]. ∂Lij = U UL k L + L Vk + L Vk + Vk L . (2.19) ∂t ▽j ▽i − j ik ik ▽j jk ▽i ▽j ik 45 Maryam T. Aldossary and Mashniah A. Gazwani ∂Ljk = U UL k L + L Vk + L Vk (2.20) ∂t − ▽j ▽i − j ik ik ▽j jk ▽i ! + Vk L Ljk Lij .[7] (2.21) ▽j ik ∂L ij k k = L L V j Lik + j i U + 2 k V 2HU . (2.22) ∂t h ▽ ▽ ▽ ▽ − i Finally, the evolution equations for the Gaussian Curvature K and mean Curvature H are given by ∂K ij k = K L j i U + V j Lik + 2HU . (2.23) ∂t h ▽ ▽ ▽ i ∂H 1 ij ij k j k = 2Lij L U + g j i U + V j Lik Lk Lj U . (2.24) ∂t 2 h ▽ ▽ ▽ − i 3. P ARALLEL CURVES AND SURFACES In this section, we explore the geometry of parallel curves, surfaces and curves lying on parallel surfaces. We investigate the properties of the geometry of such objects in R3. We work to derive the associated moving frame to the parallel object and present the frame found in terms of the original frame, associated to the original curve or surface. All local invariants of the parallel object will be derived. 3.1. Parallel curves. Definition 3.1. Let α(s) be a unit speed curve, we construct a parallel curve α(s) that parametrized by arc length s in Euclidean 3-space R3 as follow α(s) = α(s) + rB(s), (3.1) where r = 0 is a real constant, s = s(s) is the arc length of α and B is the binormal vector to the curve α(6 s). We will study the associated geometry of parallel curves. Lemma 3.1. Let α(s) be a parallel curve to a unit speed curve α(s). Then the associated Frenet- Serret frame T, n, B to α in terms of the frame T, n, B of the original curve α as follow { } T = ωT rωτ n (3.2) − ωω + rκτω 2 ω2κ (rτω ) ω rτ2ω2 n = ′ T + − ′ n B (3.3) Ω Ω − Ω r2τ3ω3 rτ2ω3 ω3κ (rτω ) ω2 + rτω 2ω + r2κτ 2ω3 B = T + n + − ′ ′ B, (3.4) Ω Ω Ω ! ! where d 1 = ′ , ω = , ds √1 + r2τ2 and Ω 2 2 2 2 2 2 2 = (ωω ′ + rκτω ) + ( ω κ (rτω )′ω) + ( rτ ω ) . q − 46 Motion of Parallel Curves and Surfaces in Euclidean 3-Space R3 Proof. Differentiating (3.1) by arc length s we get dα dα dB ds T = = + r , (3.5) ds ds ds ds using (2.13) into (3.5) yields to ds T = (T rτn) . (3.6) − ds Since ds T = T rτn , − ds | | assuming ds 1 = = ω.
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