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 direction{ of T, n, }B respectively. The motion is local which means that W, V, U depend 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 determinant 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. ds √1 + r2τ2 Therefore, T = ωT rτω n. − Also, since dT/d s n = , dT/d s then dT dT ds = ds ds ds 2 2 2 2 = ωω ′ + rκτω T + ω κ (rτω )′ω n rτ ω B. − − ! ! Taking

dT 2 2 = ωω + rκτω 2 + ω2κ (rτω ) ω + ( rτ2ω2)2. ds q ′ − ′ ! !

Hence ωω + rκτω 2 ω2κ (rτω ) ω rτ2ω2 n = ′ T + − ′ n B, Ω Ω − Ω where 2 2 Ω = ωω + rκτω 2 + ω2κ (rτω ) ω + ( rτ2ω2)2. q ′ − ′ Finaly, since ! ! B = T n, × then r2τ3ω3 rτ2ω3 ω3κ (rτω ) ω2 + rτω 2ω + r2κτ 2ω3 B = T + n + − ′ ′ B, Ω Ω Ω ! ! which completes the proof. Theorem 3.1. Let α(s) be a parallel curve to a unit speed curve α(s). Then the metric g of curve α(s) is given by g = g 1 + r2τ2 . (3.7) ! 47 Maryam T. Aldossary and Mashniah A. Gazwani

Proof. As we know that ds 1 = , ds √1 + r2τ2 Then, we have ds = ds 1 + r2τ2. p Therefore, g = g 1 + r2τ2 , which ends the proof. ! 3.2. Parallel Surfaces. Definition 3.2. Let M be an orientable surface and let N be a unit normal vector field of M. A surface M is said to be parallel to M if there is a normal geodesic congruence between M and M such that the distance between corresponding points is constant, i.e. for each X M we have ∈ M : X(u, v) = X(u, v) + rN(u, v), (3.8) where, r = 0 is a real constant. We can say that M and M are parallel surfaces at distance 6 r. The relation between the Gaussian and mean curvatures K, H and K, H of M and M, respectively, are given by [9] K K = , (3.9) µ H rK H = − , (3.10) µ where µ = 1 2rH + r2K = 0. − 6 Also if κ1, κ2 and κ1, κ2 denote the principal curvatures of M and M, respectively, then we have [9] κ1 κ2 κ1 = , κ2 = . 1 + rκ1 1 + rκ2 By direct calculations we find the frame Xi, N associated to the surface M expressed in terms of the original frame X , N to M { i } X aij 0 X i = j , (3.11) N 0 1 N where, aij = δi rg jk L . j − ki Lemma 3.2. Let M be a parallel surface to M in R3. Then

i) The metric glk on M is related to metric g mj on M by lj km glk = a a gmj , (3.12)

and the determinant g of metric gij on M is given by

g = det (gij ) = gij G(i, j), (3.13) where g gij = G(i, j).

48 Motion of Parallel Curves and Surfaces in Euclidean 3-Space R3

ii) The curvature tensor Lij on M is related to the curvature tensor L kj on M by

ik Lij = a Lkj , (3.14)

and the determinant L of The curvature tensor Lij on M are given by

L = det (Lij ) = Lij L(i, j), (3.15)

ij where L L = L(i, j). Proof. Since

glk = Xl, Xk , using (3.11) we get lj km glk = a a gmj . By deﬁnition of parallel surfaces we conclude that N = N, since ik ik Xij = a,j Xk + a Xkj ,

Lij = Xij , N , by using X , N = 0, then we conclude that h i i ik Lij = a Lkj , which completes the proof.

4. M OTION OF PARALLEL CURVES AND SURFACES IN R3 In this section, we study the motion of parallel curves and surfaces in R3, such as the evolution equations of the frame associated, curvatures, metric, components of the ﬁrst and second fundamental forms for the general parallel curves and surface. We depend on integrability conditions in all our calculations.

4.1. Motion of parallel curves.

Theorem 4.1. The evolution equations for Frenet-Serret frame T, n, B associated to the parallel curve α(s) are given by ∂T ∂ω ∂U = + ( rωτ )( τV + κW) T ∂t ∂t ∂s − ∂U ∂(rωτ ) + ω( τV + κW) n ∂s − − ∂t ∂V 1 ∂ ∂V τ ∂U + ω( + τU) (rωτ ) ( + τU) + ( τV + κW) B. ∂s − κ ∂s ∂s κ ∂s − !

49 Maryam T. Aldossary and Mashniah A. Gazwani

∂n ∂ ωω + rκτω 2 ω2κ (rτω ) ω ∂U = ′ − ′ ( τV + κW) ∂t ∂t Ω − Ω ∂s − ! ! rτ2ω2 ∂V + ( + τU) T Ω ∂s (ωω + rκτω 2) ∂U ∂ ω2κ (rτω ) ω + ′ ( τV + κW) + − ′ Ω ∂s − ∂t Ω ! rτ2ω2 1 ∂ ∂V τ ∂U + ( + τU) + ( τV + κW) n Ω κ ∂s ∂s κ ∂s − ! ∂ rτ2ω2 + − ∂t Ω ! 2 2 (ωω + rκτω ) ∂V ω κ (rτω )′ω 1 ∂ ∂V + ′ ( + τU) + − ( + τU) Ω ∂s ! Ω κ ∂s ∂s ! τ ∂U + ( τV + κW) B. κ ∂s −

∂B ∂ r2τ3ω3 rτ2ω3 ∂U = τV + κW ∂t ∂t Ω − Ω ∂s − ! ! ! ω3κ (rτω ) ω2 + rτω 2ω + r2κτ 2ω3 ∂V − ′ ′ + τU T − Ω ∂s ! ! r2τ3ω3 ∂U ∂ rτ2ω3 + τV + κW + Ω ∂s − ∂t Ω ! ! ! ω3κ (rτω ) ω2 + rτω 2ω + r2κτ 2ω3 1 ∂ ∂V − ′ ′ ( + τU) − Ω κ ∂s ∂s ! ! τ ∂U + ( τV + κW) n κ ∂s − r2τ3ω3 ∂V rτ2ω3 1 ∂ ∂V + + τU) + ( + τU) Ω ∂s Ω κ ∂s ∂s ! ! ! ! τ ∂U ∂ ω3κ (rτω ) ω2 + r2τω 2ω + rκτ 2ω3 + ( τV + κW) + − ′ ′ B. κ ∂s − ∂t Ω ! Proof. Differentiating (3.2)-(3.4) w.r.t parameter t, respectively, and using (2.6)-(2.8) completes the proof.

Corollary 4.1. The evolution equation of curvature to the parallel curve α(s) is given by

∂ (ωω + rκτω 2)2 + ( ω2κ (rτω ) ω)2 + ( rτ2ω2)2 ∂κ ∂t ′ − ′ = q . 2 2 2 2 2 2 2 ∂t 2 (ωω ′ + rκτω ) + ( ω κ (rτω )′ω) + ( rτ ω ) p − Corollary 4.2. The evolution equation of torsion to the parallel curve α(s) is given by

50 Motion of Parallel Curves and Surfaces in Euclidean 3-Space R3

2 2 3 2 3 3 ∂τ ∂ ωω ′ + rκτω rκτ ω r τ ω = − + ′ ∂t ∂t − ! Ω Ω Ω ! 2 ! 2 3 2 !3 3 ω κ (rτω )′ω rτ ω r κτ ω − ′ + − ! Ω Ω Ω ! ! τ 3 2 2 2 2 3 ω κ (rτω )′ω + rτω ω′ + r κτ ω − Ω − !2 2 3 2 2 2 2 3 3 3 rτ ω ω κ (rτω )′ω + rτω ω′ + r κτ ω rτ ω + Ω − Ω ′ + Ω . ! Theorem 4.2. The evolution equation of metric g of the parallel curve α(s) is given by ∂g ∂W ∂ = g 2 κU (1 + r2τ2) + (1 + r2τ2) . ∂t ∂s − ∂t ! Proof. Differentiating (3.7) w.r.t parameter t, and using (2.11) completes the proof.

4.2. Motion of parallel surfaces.

Theorem 4.3. The evolution equations for the associated frame Xi, N on M are given by

ik ∂a ij k k ij ∂U j ∂ X + a ( jV Lj U) a ( + V Lji ) X i =  ∂t ▽ − ∂ui  k . (4.1) ∂t N ki ∂U j N  g ( + V Lji ) 0   − ∂ui  Proof. From (3.11) we have ij Xi = a Xj. (4.2) Differentiating (4.2) w.r.t parameter t, we get

∂X ∂aij i = X + aij X , ∂t ∂t j j then substituting (2.15) we derive the evolution equations of tangent of Xi we complete ∂X prove i . ∂t Since N = N by deﬁnition this yields to ∂N ∂N = , ∂t ∂t which completes the proof.

Theorem 4.4. The evolution equations for the metric tensor glk on M are given by ∂g ∂ lk = alj akm g + alj akm V + V 2L U . (4.3) ∂t ∂t mj ▽j m ▽m j − mj ! Proof. Differentiating (3.12) w.r.t parameter t, and using (2.16) completes the proof.

51 Maryam T. Aldossary and Mashniah A. Gazwani

Theorem 4.5. The evolution equations for the curvature tensor Lij on M are given by ik ∂Lij ∂a = L ∂t ∂t kj ik m m m m + a j k U Lkm Lj U + Lkm j V + Ljm k V + V j Lkm . (4.4) h▽ ▽ − ▽ ▽ ▽ i Proof. Differentiating (3.14) w.r.t parameter t, and substituting (2.19) we complete the proof. Lemma 4.1. The evolution equations for the Gaussian curvature K on M are given by ∂K K = (Lij ( U + Vk L ) + 2HU )( 1 2rH ) ∂t µ2 ▽j ▽i ▽j ik − + r(2Lij U + gki U + Vk L Lj LkU) . ▽i ▽j ▽i kj − k j Proof. Differentiating (3.9) w.r.t parameter t, and substituting (2.23) and (2.24) we complete the above equation, we get the results. Theorem 4.6. The evolution equations for the mean curvature H on M are given by ∂H 1 (1 r2K) = − 2Lij L U + gij ( U + Vk L ) Lj LkU ∂t µ2 2 ij ▽j ▽i ▽j ik − k j ! r K (1 rH ) Lij ( U + Vk L ) + 2HU . − − ▽j ▽i ▽j ik ! Proof. Differentiating (3.10) w.r.t parameter t, we get ∂H 1 ∂H ∂K ∂H ∂K = µ r (H rK ) 2r + r2 , ∂t µ2 ∂t − ∂t − − − ∂t ∂t 1 ∂H ∂K = µ + 2Hr 2r2K r µ + rH r2K , µ2 − ∂t − − ∂t ! ! substituting (2.23) and (2.24) we complete the proof. Corollary 4.3. The evolution equations for the determinant g of metric tensor on M are given by ∂g ∂a = G(i, j) g + alj akm V + V 2L U (4.5) ∂t ∂t mj ▽j m ▽m j − mj ! Proof. Differentiating (3.13) w.r.t parameter t, we get

∂g ∂ g ∂ gij ∂ gij = = G(i, j) , ∂t ∂ gij ∂t ∂t substituting (4.3) we complete the proof. Corollary 4.4. The evolution equations for the determinant L of curvature tensor on M are given by ∂L ∂aik = L(i, j) L ∂t ∂t kj + aik U L LmU + L Vm + L Vm + Vm L . ▽j ▽k − km j km ▽j km ▽k ▽j im 52 Motion of Parallel Curves and Surfaces in Euclidean 3-Space R3

Proof. Differentiating (3.15) w.r.t parameter t, we get

∂ L ∂L ∂Lij ∂Lij = = L(i, j) , ∂t ∂Lij ∂t ∂t substituting (4.4) we complete the proof.

5. A PPLICATIONS Here we show some examples to show the geometric meaning of the evolution equations for the geometric quantities.

5.1. Application. Let α(s) be a helix curve evolving in R3 by (2.1), where r c κ = , τ = . r2 + c2 r2 + c2 We now graph the evolving a curve by time in different situations to study the evolving model. For graphing requirements we take r = 1, c = 1 and t = 0, 1, 2, 3, 4 . { }

Figure 1. (W = τt2, U = 0, V = 0).

Figure 2. (W = 0, U = κt, V = 0).

53 Maryam T. Aldossary and Mashniah A. Gazwani

Figure 3. (W = 0, U = 0, V = κ(t + 1)t).

Figure 4. (W = τt2, U = κt, V = κ(t + 1)t).

From figure 1, we see that the evolving curve in the tangent direction of the original curve compress the parallel helix and extending its radius at the same time, while the evolution in the normal direction causes rapid increase of the radius of the helix in figure 2. If the helix evolves in the binormal direction we see a faster increment of the radius of the helix in parallel with an obvious shrinking of its length in figure 3. Finally, we sum all affects as we can see in figure 4.

5.2. Application. Let X u1, u2 be a unit sphere evolving in R3 by (2.14). We now graph the evolving sphere by! time in different situations to study the evolving model with r = 3, t = 0, 1, 2, 3, 4 . { }

54 Motion of Parallel Curves and Surfaces in Euclidean 3-Space R3

Figure 5. (φ1 = t2K, φ2 = 0, ψ = 0).

Figure 6. (φ1 = 0, φ2 = t2 H, ψ = 0).

Figure 7. (φ1 = φ2 = 0, ψ = t(t + 1)K).

Figure 8. (φ1 = t2K, φ2 = t2 H, ψ = t(t + 1)K).

Figure 5 illustrates that as the sphere evolves in its X1 direction we see that the sphere gradual turns z = x2 + y2 where z ∞ as t ∞, while in figure 6 we see that −p | | → | | → 55 Maryam T. Aldossary and Mashniah A. Gazwani the sphere shrinks in u2 direction and expand in the u1 direction at the same time. The sphere blows up as it evolves in the N direction figure 7. Finally, figure 8 shows the combination of the three previous figures as it evolves in the directions of X1, X2 and N. Acknowledgments. The authors wish to thank the referees for their many valuable and helpful suggestions in order to improve this manuscript. The authors would like to express his gratitude to prof. Mohammad Hasan Shahid (New Delhi - Jamia Millia Islamia University) and Prof. H. N. Abd-Ellah (Egpt - Assiut University) for editing and con- structive criticism of the manuscript.

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