On Geometry of Isophote Curves in Galilean space
OOn Geometry of Isophote Curves in
Galilean space
Z¨uhal K¨u¸c¨ukarslan Y¨uzba¸sı † and Dae Won Yoon ‡ † Department of MathematicsFırat University23119 Elazig, TurkeyE-mail address: [email protected] ‡ Department of Mathematics Education and RINSGyeongsang National UniversityJinju 52828, Republic of KoreaE-mail address: [email protected]
Abstract
In this paper, we introduce isophote curves on surfaces in Galilean 3-space. Apartfrom the general concept of isophotes, we split our studies into two cases to get the axis d of isophote curves lying on a surface such that d is an isotropic or a non isotropicvector. We also give the method to compute isophote curves of surfaces of revolution.Subsequently, we show the relationship between isophote curves and slant(general) heliceson surfaces of revolution obtained by revolving a curve by Euclidean rotations. Finally,we give an example to compute isophote curves on isotropic surfaces of revolution. The isophote curve method is one of the most efficient methods that can be used to analyzeand visualize surfaces by lines of equal light intensity. Isophote curve whose normal vectorsmake a constant angle with a fixed vector(the axis) is one of the curves to characterizesurfaces such as parameter, geodesics and asymptotic curves or lines of curvature. Moreover,this curve is used in computer graphics and it is also interesting to study for geometry.The isophote curve of a given surface is calculated with two steps: firstly the normalvector field n ( s, t ) of the surface is computed, and secondly the surface point is traced as (cid:104) n ( s, t ) , d (cid:105)(cid:107) n ( s, t ) (cid:107) = cos β, Corresponding author: Dae Won Yoon.2010
AMS Mathematics Subject Classification: a r X i v : . [ m a t h . G M ] J u l n Geometry of Isophote Curves 2where β is a constant angle(0 ≤ β ≤ π ).Isophote curve is called a silhouette curve when the angle β is given as a right angle suchthat (cid:104) n ( s, t ) , d (cid:105)(cid:107) n ( s, t ) (cid:107) = cos π , where d is the fixed vector.From past to present, there have been a lot of researchers about isophote curves and theircharacterizations in [3, 4, 6, 7].In this paper, our aim is to investigate isophote curves on surfaces in Galilean space andfind its axis d such that it is an isotropic and a non isotropic vector by means of the GalileanDarboux frame. According to the axis d , we split our studies into two cases to find the axisof isophote curves lying on a surface in Galilean space. Moreover, we give the method tocompute isophote curves of surfaces of revolution obtained by revolving a curve by Euclideanand isotropic rotations. In accordance with the Erlangen Program, due to F. Klein, each geometry is associated witha group of transformations, and hence there are as many geometries as groups of transfor-mations. Associated with group of transformations that in physics guarantees the invarianceof many mechanical systems, the Galilei group, is the so-called Galilean geometry. Thatis, Galilean geometry is one of the nine Cayley-Klein geometries with projective signature(0 , , + , +). The absolute of the Galilean geometry is an ordered triple { ω, f, I } , where ω isthe ideal (absolute) plane, f the line in ω and I the fixed elliptic involution of f .We introduce homogeneous coordinates in G in such a way that the absolute plane ω is given by x = 0, the absolute line f by x = x = 0 and the elliptic involution by(0 : 0 : x : x ) → (0 : 0 : x : − x ).The group of motions of G is a six-parameter group given (in affine coordinates) by¯ x = a + x, ¯ y = b + cx + y cos ϕ + z sin ϕ, ¯ z = d + ex − y sin ϕ + z cos ϕ. A plane is called Euclidean if it contains f , otherwise it is called isotropic or i.e., planes x = consant are Euclidean, and so is the plane ω . Other planes are isotropic. In other words,an isotropic plane does not involve any isotropic direction.A Galilean scalar product of two vectors x = ( x , y , z ) and y = ( x , y , z ) in the Galilean3-space G is defined as (cid:104) x, y (cid:105) = (cid:40) x x , if x (cid:54) = 0 or x (cid:54) = 0 ,y y + z z , if x = 0 and x = 0and a Galilean norm of x is given by || x || = (cid:40) | x | , if x (cid:54) = 0 , (cid:112) y + z , if x = 0 . n Geometry of Isophote Curves 3A Galilean cross product of x and y on G is defined by x × y = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e e x y z x y z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where e = (0 , ,
0) and e = (0 , , α be an admissible curve of the class C ∞ in G , and parametrized by the invariantparameter s , defined by α ( s ) = ( s, f ( s ) , g ( s )) . Then the Frenet frame fields of α ( s ) are given by T ( s ) = α (cid:48) ( s ) ,N ( s ) = 1 κ ( s ) α (cid:48)(cid:48) ( s ) ,B ( s ) = T ( s ) × N ( s ) , where the curvature κ ( s ) and the torsion τ ( s ) of α ( s ) are written as, respectively, κ ( s ) = (cid:113) f (cid:48)(cid:48) ( s ) + g (cid:48)(cid:48) ( s ) ,τ ( s ) = det ( α (cid:48) ( s ) α (cid:48)(cid:48) ( s ) α (cid:48)(cid:48)(cid:48) ( s )) κ ( s ) . Here
T, N and B are said to be the tangent, principal normal and binormal vectors of α ( s ).On the other hand, the Frenet formula of the curve is given by (cf. [9]) T (cid:48) = κN,N (cid:48) = τ B,B (cid:48) = − τ N. (2.1)Consider a C r -regular surface M , r ≥
1, in G parameterized by X ( u , u ) = ( x ( u , u ) , y ( u , u ) , z ( u , u )) . We denote by x u i , y u i and z u i the partial derivatives of the functions x , y and z with respectto u i ( i = 1 , ds of a surface M in G isgiven by ds = (cid:18) ds ds (cid:19) , where ds = ( g du + g du ) and ds = h du + 2 h du du + h du . Here g i = x u i and h ij = (cid:104) ˜ X u i , ˜ X u j (cid:105) ( i, j = 1 ,
2) means the Euclidean scalar product of the projections ˜ X u i ofvectors X u i onto the yz -plane.n Geometry of Isophote Curves 4The unit normal vector field n of a surface M is defined by n = 1 ω (0 , x u z u − x u z u , x u y u − x u y u ) , where the positive function ω is given by ω = (cid:112) ( x u z u − x u z u ) + ( x u y u − x u y u ) . Let { T, Q, n } be a Galilean Darboux frame of α ( s ) with T as the tangent vector of a curve α ( s ) in G and n be the unit normal to a surface and Q = n × T . Then the Galilean Darbouxframe is expressed as T (cid:48) = k g Q + k n n,Q (cid:48) = τ g n,n (cid:48) = − τ g Q, (2.2)where k g , k n and τ g are the geodesic curvature, normal curvature and geodesic torsion of α ( s )on M , respectively. Also, (2.2) implies κ = k g + k n , τ = − τ g + k (cid:48) g k n − k g k (cid:48) n k g + k n ,k g = k cos φ and k n = − k sin φ, (2.3)where φ is an angle between the surface normal vector n and the binormal vector B of α, ([12]). A curve α ( s ) is a geodesic (an asymptotic curve or a line of curvature) if and only if k g ( k n or τ g ) vanishes, respectively.On the other hand, the usual transformation between the Galilean Frenet frames and theDarboux frames takes the form Q = cos φN + sin φB, (2.4) n = − sin φN + cos φB. Artykbaev was introduced an angle between two vectors in Galilean space as follows:
Definition 2.1. ([1]) Let x = (1 , x , x ) and y = (1 , y , y ) be two unit non-isotropic vectorsin G . Then an angle ϑ between x and y is defined by ϑ = (cid:112) ( y − x ) + ( y − x ) . (2.5) Definition 2.2. ([1]) An angle between a unit non-isotropic vector x = (1 , x , x ) and anisotropic vector y = (0 , y , y ) in G is defined by ϕ = x y + x y (cid:112) y + y . (2.6) Definition 2.3. ([1]) An angle θ between two isotropic vectors x = (0 , x , x ) and y =(0 , y , y ) parallel to the Euclidean plane in G is equal to the Euclidean angle between them.That is, cos θ = x y + x y (cid:112) x + x (cid:112) y + y . (2.7)n Geometry of Isophote Curves 5 The starting point of this section is to get the fixed vector d of an isophote curve via itsGalilean Darboux frame.Let M be an admissible regular surface and α : I ⊂ R → M be an unit speed curveparametrized by α ( s ) = ( s, α ( s ) , α ( s )) as an isophote curve for some s ∈ I .In order to prove the results, we split it into two cases according to the fixed vector d . Case 1. d is an unit isotropic vector.Since n is the unit isotropic normal vector of a surface M , we have (cid:104) n, d (cid:105) = cos θ = constant . (3.1)If we differentiate (cid:104) T, d (cid:105) = 0 with respect to s , using the Galilean Darboux frame (2.2), thenwe obtain k g (cid:104) Q, d (cid:105) + k n (cid:104) n, d (cid:105) = 0 , (3.2)which implies (cid:104) Q, d (cid:105) = − k n k g cos θ. (3.3)Taking account of the derivative of (3.1) we get τ g (cid:104) Q, d (cid:105) = 0 , (3.4)where if (cid:104) Q, d (cid:105) = 0, k n = 0 which means that α should be an asymptotic curve or τ g = 0which means that α should be a line of curvature. Then, for k n = 0 , d can be written as d = cos θn, (3.5)since d is a constant vector, τ g should be equal zero. Also this is the trivial result.For τ g = 0 , d can be written as d = − k n k g cos θQ + cos θn. (3.6)Since (cid:107) d (cid:107) = 1, we get k n k g = ± tan θ. (3.7)In this situation, we conclude that φ = ± θ or φ = π ± θ. From (2.3) and (2.4) in terms of the Galilean Frenet frame, we get d = ( − k n k cos θ − k g k sin θ ) N + ( − k n k sin θ + k g k cos θ ) B. (3.8)If we differentiate (3.6) using (3.7) and τ g = 0, we get d (cid:48) = 0 , that is, d is a constant isotropicvector. From now on, we suppose if α is a unit-speed isophote curve, then α is also a line ofcurvature.n Geometry of Isophote Curves 6 Theorem 3.1.
Let α be a unit-speed isophote curve on a surface M in G with a fixed unitisotropic vector d as the axis of the isophote curve. In that case, we have the following: i ) If α is a geodesic curve, then α is a straight line. ii ) If α is an asymptotic curve on M , then it is a plane curve, and the fixed vector d isspanned by B. Proof. i ) If α is a geodesic curve, then we have k g = 0 and so from (3.2) it follows that k n = 0 , also τ g = 0 . By substituting k g and k n into (2.3), we get κ = 0, that is, α is a straightline. ii ) If α is an asymptotic curve, we have k n = 0 . From (2.3) and (3.8), we obtain that d = k g k cos θB. Also, by substituting τ g = 0 and k n = 0 into (2.4), we get τ = 0 . It means that α is a planecurve. (cid:3) Theorem 3.2.
Let α be a unit-speed isophote curve on a surface M in G with a fixed unitisotropic vector d as the axis of the isophote curve. The axis d is perpendicular to the principalnormal line of α if and only if either α is a straight line, or an asymptotic curve on M withtaking k n k g = tan θ or α is a curve with k n k g = − tan θ. Proof. If α is a unit-speed isophote curve with k n k g = tan θ , then from (3.8), we get (cid:104) N, d (cid:105) = − k g k sin θ = 0 , from this equation, we have k g = 0 or sin θ = 0.If k g = 0 then, from Theorem 3.1, α is a straight line.If sin θ = 0, then k n = 0 , that is, α is an asymptotic curve.If we take k n k g = − tan θ , then we can easily get (cid:104) N, d (cid:105) = 0 . (cid:3) Theorem 3.3.
Let α be a unit-speed isophote curve on a surface M in G with a fixed unitisotropic vector d as the axis of the isophote curve. The axis d is perpendicular to the principalbinormal line of α such that k n k g = tan θ if and only if θ equals π .Proof. If α is a unit-speed isophote curve with k n k g = tan θ , then from (3.8), we get (cid:104) B, d (cid:105) = k g k ( − sin θ + cos θ ) = 0 . Since α is a non-geodesic curve, − sin θ + cos θ = 0 . So, tan θ = 1. We know that 0 ≤ θ ≤ π , then we get θ = π . (cid:3) Theorem 3.4. If α is a silhouette curve on M, and d is a unit isotropic vector such that itis parallel to Q , then the curve α is a plane curve. n Geometry of Isophote Curves 7 Proof.
If a fixed vector d is a unit isotropic vector and is parallel to Q , then we have d = ± Q, (cid:104) T, d (cid:105) = 0 . By differentiating above equations with respect to s , we obtain τ g n = 0 , k g (cid:104) Q, d (cid:105) + k n (cid:104) n, d (cid:105) = 0 . Since α is a silhouette curve with (cid:104) n, d (cid:105) = 0, we get τ g = 0 , k g = 0 , from this, we have τ = 0 . It means that α is a plane curve. (cid:3) Case 2.
Now, our aim is to find a fixed unit non-isotropic vector d as the axis of anisophote curve.Since n is the unit isotropic normal vector of a surface M , we have (cid:104) n, d (cid:105) = ϕ = constant . (3.9)Let α be a unit speed admissible isophote curve. If we differentiate (cid:104) T, d (cid:105) = 1 (3.10)with respect to s, using the Galilean Darboux frame (2.2) then we have k g (cid:104) Q, d (cid:105) + k n (cid:104) n, d (cid:105) = 0 . (3.11)It follows from (2.6) that we find (cid:104) Q, d (cid:105) = − k n k g ϕ. (3.12)Taking account of the derivative of (cid:104) n, d (cid:105) = ϕ and using the Galilean Darboux frame (2.2) τ g (cid:104) Q, d (cid:105) = 0 , (3.13)where if (cid:104) Q, d (cid:105) = 0, then from (3.12) we get k n = 0 which means that α should be anasymptotic curve. Then, for k n = 0 , d can be written as d = T + ϕn. (3.14)Since d is a constant vector, k g = ϕτ g . Thus, we have the following result: Corollary 3.5.
Let α be a unit-speed isophote curve on a surface M in G with a fixed unitnon-isotropic vector d as the axis of the isophote curve. If α is a geodesic curve or a line ofcurvature, then α is a straight line. If τ g = 0, that is, α is a line of curvature, then d can be written as d = T − k n k g ϕQ + ϕn. (3.15)Since d is a constant vector, k g = k n = 0, which implies κ = 0, that is, α is a straight line.n Geometry of Isophote Curves 8 Theorem 3.6.
Let α be a silhouette curve on M and d be a unit non-isotropic vector. i ) If d lies in the plane spanned by T and Q , then α is a plane curve. ii ) If the axis d is spanned by T , then α is a geodesic curve.Proof. i ) Since α is a silhouette curve and d is a unit non-isotropic vector, we get (cid:104) T, d (cid:105) = ± . (3.16)If we differentiate (3.16) with respect to s , then we get k g (cid:104) Q, d (cid:105) = 0 . Since d is lied in the plane spanned by T and Q, we get k g = 0. Also, if we differentiate (cid:104) n, d (cid:105) = 0 with respect to s , we get τ g (cid:104) Q, d (cid:105) = 0 , it follows that τ g = 0 . Also, by substituting τ g = 0 and k g = 0 into (2.3), we get τ = 0 . Thus, α is a plane curve. ii ) If d is spanned by T , then we get d = T. If we differentiate above equation, then d (cid:48) = k g Q , it follows that k g = 0, that is, the curve isa geodesic curve. (cid:3) We investigate an isophote curve among surfaces in Galilean space. Now we give someexamples for this subject. To see this, notice that in G surfaces of revolution are obtainedby revolving a curve by Euclidean or isotropic rotations as follows, respectively, x = x, (4.1) y = y cos t + z sin t,z = − y sin t + z cos t, where t is the Euclidean angle and x = x + ct, (4.2) y = y + xt + c t ,z = z, where t ∈ R and c = constant > x = constant , y + z = r , r ∈ R . n Geometry of Isophote Curves 9The invariant r is the radius of the circle. Euclidean circles intersect the absolute line f inthe fixed points of the elliptic involution ( F , F ).The trajectory of a point under isotropic rotation is an isotropic circle whose normal formis z = constant , y = x c . The invariant c is the radius of the circle. The fixed line of the isotropic rotation is theabsolute line f [11]. For some more studies, see [2, 5].If a curve α ( s ) = ( f ( s ) , , g ( s )) , ( g ( s ) >
0) is rotated by Euclidean rotations, then asurface of revolution is parametrized by S ( s, t ) = ( f ( s ) , g ( s ) sin t, g ( s ) cos t ) . (4.3)If a curve α ( s ) is parametrized by the arc-length, then we take f ( s ) = s. Then, the unitisotropic normal vector field n ( s, t ) of S is defined by n ( s, t ) = S s × S t (cid:107) S s × S t (cid:107) , (4.4)where S s and S t are the partial differentiations with respect to s and t, respectively. Then,the isotropic normal vector is given by n ( s, t ) = (0 , sin t, cos t ) , it becomes in terms of the Frenet frame as follows: n ( s, t ) = − sin tB + cos tN. (4.5) Proposition 4.1.
Let a curve α ( s ) be a general helix with the isotropic axis d . Then, for t = ( k +12 ) π ( k ∈ Z ) , the curve α ( s ) on surfaces of revolution given by (4.3) of revolution isan isophote curve with the axis d .Proof. Substituting t into (4.5), we get n ( s, t ) = ∓ B. If α ( s ) is a general helix with the axis d , then (cid:104) B, d (cid:105) =constant. Therefore, we get (cid:104) n ( s, t ) , d (cid:105) = ∓ (cid:104) B, d (cid:105) = constant.Thus α ( s ) is an isophote curve with the axis d on the surfaces of revolution. (cid:3) Proposition 4.2.
Let a curve α ( s ) be a slant helix with the isotropic axis d . Then, for t = kπ ( k ∈ Z ) , the curve α ( s ) on surfaces of revolution given by (4.3) is an isophote curvewith the axis d . n Geometry of Isophote Curves 10 Proof.
Substituting t into (4.5), we get n ( s, t ) = ∓ N. If α ( s ) is a slant helix with the axis d , then (cid:104) N, d (cid:105) =constant. Therefore, we get (cid:104) n ( s, t ) , d (cid:105) = ∓ (cid:104) N, d (cid:105) = constant.Thus α ( s ) is an isophote curve with the axis d on the surfaces of revolution. (cid:3) If a curve α ( s ) = ( f ( s ) , , g ( s )) , ( g ( s ) >
0) is rotated by isotropic rotations, then a surfaceof revolution is parametrized by S ( s, t ) = ( f ( s ) + ct, st + c t , g ( s )) . (4.6)If a curve α ( s ) is parametrized by the arc-length, then we take f ( s ) = s. Then, the isotropicsurface normal is given by n = 1 (cid:113) ( g (cid:48) ( s ) c ) + s (0 , g (cid:48) ( s ) c, s ) , it becomes in terms of the Frenet frame as follows: n = 1 (cid:113) ( g (cid:48) ( s ) c ) + s (cid:0) − g (cid:48) ( s ) cB + sN (cid:1) . (4.7) Proposition 4.3.
Let an isotropic axis d is given by (0,d y , d z ). i ) If d y = 0 and g ( s ) is a second order function, then the curve α ( s ) on surfaces ofrevolution given by (4.6) is an isophote curve. ii ) If d z = 0 and g ( s ) is a second order function, then the curve α ( s ) on surfaces ofrevolution given by (4.6) is an isophote curve.Proof. i ) If d y = 0 , then we get d = λ N, ( λ ∈ R ).Using this above condition on (4.7), we get (cid:104) n, d (cid:105) = λ s (cid:113) ( g (cid:48) ( s ) c ) + s . From the above equation, we can get g ( s ) = s c + A, A ∈ R. Thus we obtain (cid:104) n, d (cid:105) = λ √ .ii ) If d z = 0 , then we get d = − λ B, ( λ ∈ R ) . Using this above condition on (4.7), weget (cid:104) n, d (cid:105) = λ g (cid:48) ( s ) c (cid:113) ( g (cid:48) ( s ) c ) + s . From the above equation, we can get g ( s ) = s c + A, A ∈ R. Thus we obtain (cid:104) n, d (cid:105) = λ √ . (cid:3) Therefore, the rotating curve is an isotropic circle on surfaces of revolution. We also showthe surfaces (4.6) for g ( s ) = s c + A in Figure 1. Corollary 4.4.
The generating curve α ( s ) = ( f ( s ) , , g ( s )) on surfaces of revolution givenby (4.6) becomes both a general helix and a slant helix with the axis d .n Geometry of Isophote Curves 11 Figure 1: Isotropic surface of revolution for c = 1 and A = 0. References [1] A. Artykbaev, Total angle about the vertex of a cone in Galilean space, Math. Notes, (1988), 379–382[2] M. Dede, C. Ekici and W. Goemans, Surfaces of revolution with vanishing curvature inGalilean 3-space, J. Math. Phys. Anal. Geo., (2) (2018), 141-152.[3] F. Do˘gan and Y. Yaylı, On isophote curves and their characterizations, Turkish J. Math., (5) (2015), 650-664.[4] F. Do˘gan and Y. Yaylı, Isophote curves on spacelike surfaces in Lorentz-Minkowski space E , arXiv preprint arXiv:1203.4388.[5] A. Kazan and H. B. Karadag, Weighted Minimal and Weighted Flat Surfaces of Revo-lution in Galilean 3-Space with Density, Int. J. Anal. Appl., (3) (2018), 414-426.[6] K. J. Kim and I. K. Lee, Computing isophotes of surface of revolution and canal surface,Comput-Aided Des., (2003), 215-223.[7] J. J. Koenderink, A. J. van Doorn, Photometric invariants related to solid shape, J.Modern Opt., (7) (1980), 981-996.[8] E. Molnar, The projective interpretation of the eight 3-dimensional Homogeneous ge-ometries, Beitr. Algebra Geom., (1997), 261–288.[9] B. J. Pavkovic and I. Kamenarovic, The equiform differential geometry of curves in theGalilean space G3, Glas. Mat., (42) (1987), 449-457.[10] O. R¨oschel, Die Geometrie des Galileischen raumes, Habilitationsschrift, Leoben, 1984.n Geometry of Isophote Curves 12[11] Z. M. Sipus, Ruled Weingarten surfaces in Galilean space, Period. Math. Hungar, (2)(2008), 213–225.[12] T. S¸ahin, Intrinsic equations for a generalized relaxed elastic line on an oriented surfacein the Galilean space, Acta Math. Sci.,33