Parametric Representation of a Hypersurface Family With a Common Spatial Geodesic
HHypersurface family with a common geodesic
Ergin BAYRAM a and Emin KASAP b Abstract
In this paper, we study the problem of finding a hypersurface family from a given spatial geodesic curve in ℝ . We obtain the parametric representation for a hypersurface family whose members have the same curve as a given geodesic curve. Using the Frenet frame of the given geodesic curve, we present the hypersurface as a linear combination of this frame and analyze the necessary and sufficient condition for that curve to be geodesic. We illustrate this method by presenting some examples. Keywords:
Hypersurface, Frenet frame, geodesic.
MSC: . Introduction
Geodesic is a well-known notion in differential geometry. A geodesic on a surface can be defined in many equivalent ways. Geometrically, the shortest path joining any two points of a surface is a geodesic. Geodesics are curves in surfaces that play a role analogous that of straight lines in the plane. A straight line doesn’t bend to left or right as we travel along it [6]. In recent years, there have been various researches on geodesics. Kumar et al., [20] presented a study on geodesic curves computed directly on NURBS surfaces and discrete geodesics computed on the equivalent tessellated surfaces. Wang et al., [26] studied the problem of constructing a family of surfaces from a given spatial geodesic curve and derived a parametric representation for a surface pencil whose members share the same geodesic curve as an isoparametric curve. A practical method was presented by Sanchez and Dorado, [21] to construct polynomial surfaces from a polynomial geodesic or a family of geodesics, by prescribing tangent ribbons. Sprynski et al., [22] dealt with reconstruction of numerical or real surfaces based on the knowledge of some geodesic curves on the surface. Paluszny, [19] considered patches that contain any given 3D polynomial curve as a pregeodesic (i.e. geodesic up to reparametrization). Given two pairs of regular space curves ( ) u r , ( ) u r and ( ) v r , ( ) v r that define a curvilinear rectangle, Farouki et al., [10] handled the problem of constructing a C surface patch ( ) u, v R for which these four boundary curves correspond to geodesics of the surface. Farouki et al., [11] considered the problem of constructing polynomial or rational tensor-product Bézier patches bounded by given four polynomial or rational Bézier curves defining a curvilinear rectangle, such that they are geodesics of the constructed surface. On the other hand, Wang et al., [26] tackled the problem of finding surfaces passing through a given geodesic. In 2011, given curve was changed to a line of curvature and Li et l., [18] constructed a surface family from a given line of curvature. Bayram et al., [5] gave the necessary and sufficient conditions for a given curve to be an asymptotic on a surface. However, while differential geometry of a parametric surface in ℝ can be found in textbooks such as in Struik [24], Willmore [28], Stoker [23], do Carmo [7], differential geometry of a parametric surface in n ℝ can be found in textbook such as in the contemporary literature on Geometric Modeling [9, 16]. Also, there is little literature on differential geometry of parametric surface family in ℝ [2, 8, 17, 26], but not in ℝ . Besides, there is an ascending interest on fourth dimension [1, 2, 8]. Furthermore, various visualization techniques about objects in Euclidean n-space ( ) n 4 ≥ are presented [3, 4, 14]. The fundamental step to visualize a 4D object is projecting first in to the 3-space and then into the plane. In many real world applications the problem of visualizing three-dimensional data, commonly referred to as scalar fields arouses. The graph of a function ( ) x, y, z : U f ⊂ → ℝ ℝ , U is open, is a special type of parametric hypersurface with the parametrization ( ) ( ) x, y, z, x, y, z f in 4-space. There is an existing method for rendering such a 3-surface based on known methods for visualizing functions of two variables [13]. In this paper, we consider the four dimensional analogue of the problem of constructing a parametric representation of a surface family from a given spatial geodesic in Wang et al. [26], who derived the necessary and sufficient conditions on the marching-scale functions for which the curve C is an isogeodesic, i.e., both a geodesic and a parameter curve, on a given surface. We express the hypersurface pencil parametrically with the help of the Frenet frame { } , , , T N B B of the given curve. We find the necessary and sufficient constraints on the marching-scale functions, namely, coefficients of Frenet vectors, so that both the geodesic and parametric requirements met. Finally, as an application of our method one example for each type of marching-scale functions is given.
2. Preliminaries
Let us first introduce some notations and definitions. Bold letters such as a , R will be used for vectors and vector functions. We assume that they are smooth enough so that all the (partial) derivatives given in the paper are meaningful. Let : I ⊂ → ℝ ℝ αααα be an arc-length curve. If { }
T, N, B , B is the moving Frenet frame along αααα , then the Frenet formulas are given by (1) , , ,,
T NN T BB N BB B ′ = ′ = − + ′ = − + ′ = − k k kk kk where and
T, N, B B denote the tangent, principal normal, first binormal and second binormal vector fields, respectively, ( ) i i 1, 2,3 = k the i-th curvature functions of the curve αααα [14]. From elementary differential geometry we have (2) ( ) ( )( ) ( ) ( )( ) ( ) , ,. T N ′ = ′′ = ′′= s ss k s sk s s αααααααα αααα
By using Frenet formulas one can obtain the followings (3) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
21 1 1 2 13 21 1 1 1 1 2 1 2 1 2 1 1 2 3 2 ,3 2 .
T N BT N B B ′′′ ′= − + + ′ ′′ ′ ′= − + − + − + + + s k s k s k k sk k s k k k k s k k k k s k k k s αααααααα iv The unit vectors and
B B are given by (4) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )
21 2 ,, BB B T N ′ ′′ ′′′ ⊗ ⊗= ′ ′′ ′′′⊗ ⊗ = ⊗ ⊗ s s ss s s ss s s s α α αα α αα α αα α αα α αα α αα α αα α α where ⊗ is the vector product of vectors in ℝ . Since the vectors , , , T N B B are orthonormal, the second curvature k and the third curvature k can be obtained from (3) as 5) ( ) ( ) ( )( )( ) ( ) ( ) ( )( ) ( )
12 123 1 2 , , BB ′′′ = = ii s sk s k ss sk s k s k s αααααααα iv where ‘ i ’ denotes the standard inner product. Let { } , , , e e e e be the standard basis for four-dimensional Euclidean space ℝ . The vector product of the vectors u , v , w u e v e w e = = = = = = ∑ ∑ ∑ is defined by u u u uv v v vw w w w e e e eu v w ⊗ ⊗ = [15, 27]. If u , v and w are linearly independent then u v w ⊗ ⊗ is orthogonal to each of these vectors.
3. Hypersurface family with a common geodesic
A curve ( ) r s on a hypersurface ( ) P P = ⊂ ℝ s,t,q is called an isoparametric curve if it is a parameter curve, that is, there exists a pair of parameters t and q such that ( ) ( ) r P = s s,t ,q . Given a parametric curve ( ) r s , it is called an isogeodesic of a hypersurface P if it is both a geodesic and an isoparametric curve on P . Let ( ) : , r r = ≤ ≤ C s L s L , be a C curve, where s is the arc-length. To have a well-defined principal normal, assume that ( ) r ′′ ≠ ≤ ≤ s L s L . Let ( ) ( ) ( ) ( ) , , , T N B B s s s s be the tangent, principal normal, first binormal, second binormal, respectively; and let ( ) ( ) ( ) , and k s k s k s be the first, second and the third curvature, respectively. Since ( ) ( ) ( ) ( ) { } , , ,
T N B B s s s s is an orthogonal coordinate frame on ( ) r s the parametric hypersurface ( ) [ ] [ ] [ ] : x x P → ℝ s,t,q L ,L T ,T Q ,Q passing through ( ) r s can be defined as follows: 6) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )( )( )
121 2 1 2 1 2 , , ,, , ,
TNP r u v w x BB = + ≤ ≤ ≤ ≤ ≤ ≤ sss,t,q s s,t,q s,t,q s,t,q s,t,q ssL s L T t T Q q Q where ( ) ( ) ( ) ( ) , , and u v w x s,t,q s,t,q s,t,q s,t,q are all C functions. These functions are called the marching scale functions. We try to find out the necessary and sufficient conditions for a hypersurface ( )
P P = s,t,q having the curve C as an isogeodesic. First to satisfy the isoparametricity condition there should exist [ ] [ ] and ∈ ∈ t T ,T q Q ,Q such that ( ) ( ) = P r s,t ,q s , ≤ ≤ L s L , that is, (7) ( ) ( ) ( ) ( ) [ ] [ ] u v w x = = = ≡ ∈ ∈ ≤ ≤ s,t ,q s,t ,q s,t ,q s,t ,qt T ,T q Q ,Q L s L
Secondly, the curve C is a geodesic on the hypersurface ( ) P s,t,q if and only if the principal normal ( ) N s of the curve and the normal ( ) ˆ n s,t ,q of the hypersurface ( ) P s,t,q are linearly dependent, that is, parallel along the curve C, [25] . The normal ( ) ˆ n s,t ,q of the hypersurface can be obtained by calculating the vector product of the partial derivatives and using the Frenet formula as follows ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) P u v Tvu w Nwv x Bxw B ∂ ∂= + − ∂ ∂ ∂+ + − ∂ ∂+ + − ∂ ∂+ + ∂
11 22 33 s,t,q s,t,q s,t,q κ s ss s s,t,qs,t,q κ s s,t,q κ s sss,t,qs,t,q κ s s,t,q κ s sss,t,qs,t,q κ s ss ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) , P u vT Nw xB B ∂ ∂ ∂= +∂ ∂ ∂∂ ∂+ +∂ ∂ s,t,q s,t,q s,t,qs st t ts,t,q s,t,qs st t nd ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) . P u vT Nw xB B ∂ ∂ ∂= +∂ ∂ ∂∂ ∂+ +∂ ∂ s,t,q s,t,q s,t,qs sq q qs,t,q s,t,qs sq q
Remark:
Because, ( ) ( ) ( ) ( ) [ ] [ ] u v w x = = = ≡ ∈ ∈ ≤ ≤ s,t ,q s,t ,q s,t ,q s,t ,qt T ,T q Q ,Q L s L along the curve C, by the definition of partial differentiation we have ( ) ( ) ( ) ( ) [ ] [ ] u v w x= ∂ ∂ ∂ ∂= = ≡ ∂ ∂ ∂ ∂ ∈ ∈ ≤ ≤ s,t ,q s,t ,q s,t ,q s,t ,qs s s st T ,T q Q ,Q L s L By using (7) we have ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ˆ ,
P P Pn T N B B ∂ ∂ ∂= ⊗ ⊗∂ ∂ ∂= − + − s,t ,q s,t ,q s,t ,qs,t ,q s t qs,t ,q s s,t ,q s s,t ,q s s,t ,q s φ φ φ φφ φ φ φφ φ φ φφ φ φ φ where ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) v w xv w xv w x ∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂= =∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂ s,t ,q s,t ,q s,t ,qs s ss,t ,q s,t ,q s,t ,qs,t ,q t t ts,t ,q s,t ,q s,t ,qq q q φφφφ ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) u w xu w xu w xu w xu w xw x s,t ,q s,t ,q s,t ,qs s ss,t ,q s,t ,q s,t ,qs,t ,q t t ts,t ,q s,t ,q s,t ,qq q qs,t ,q s,t ,q s,t ,qt t ts,t ,q s,t ,q s,t ,qq q qs,t ,q s,t ,t ∂ ∂ ∂+ ∂ ∂ ∂∂ ∂ ∂= ∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂= ∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂= ∂φφφφ ( ) ( ) ( ) , w x q s,t ,q s,t ,qq q t ∂ ∂−∂ ∂ ∂ ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) u v xu v xu v xu v xu v xv x s,t ,q s,t ,q s,t ,qs s ss,t ,q s,t ,q s,t ,qs,t ,q t t ts,t ,q s,t ,q s,t ,qq q qs,t ,q s,t ,q s,t ,qt t ts,t ,q s,t ,q s,t ,qq q qs,t ,q s,t ,t ∂ ∂ ∂+ ∂ ∂ ∂∂ ∂ ∂= ∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂= ∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂= ∂φφφφ ( ) ( ) ( ) , v x q s,t ,q s,t ,qq q t ∂ ∂−∂ ∂ ∂ ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( )
1, , 1 0 0 u v wu v wu v wu v wu v wv w s,t ,q s,t ,q s,t ,qs s ss,t ,q s,t ,q s,t ,qs t q t t ts,t ,q s,t ,q s,t ,qq q qs,t ,q s,t ,q s,t ,qt t ts,t ,q s,t ,q s,t ,qq q qs,t ,q s,t ,t ∂ ∂ ∂+ ∂ ∂ ∂∂ ∂ ∂= ∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂= ∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂= ∂φφφφ ( ) ( ) ( ) . v w q s,t ,q s,t ,qq q t ∂ ∂−∂ ∂ ∂ o, ( ) ˆ n s,t ,q || ( ) N s if and only if (8) ( ) ( ) ( ) [ ] [ ]
0, 0,, , . = ≡ ≠∈ ∈ ≤ ≤ s,t ,q s,t ,q s,t ,qt T ,T q Q ,Q L s L φ φ φφ φ φφ φ φφ φ φ
Thus, any hypersurface defined by (6) has the curve C as an isogeodesic if and only if (9) ( ) ( ) ( ) ( )( ) ( ) ( ) [ ] [ ] u v w x = = = ≡ = ≡ ≠ ∈ ∈ ≤ ≤ s,t ,q s,t ,q s,t ,q s,t ,qs,t ,q s,t ,q s,t ,qt T ,T q Q ,Q L s L φ φ φφ φ φφ φ φφ φ φ is satisfied. We call the set of hypersurfaces defined by (6) and satisfying (9) an isogeodesic hypersurface family. To develop the method further and for simplification purposes, we analyze some types of marching-scale functions.
Let marching-scale functions be ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) u l Uv m Vw n Wx p X s,t,q = s t,q ,s,t,q = s t,q ,s,t,q = s t,q ,s,t,q = s t,q , , , , ≤ ≤ ≤ ≤ ≤ ≤
L s L T t T Q q Q where ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , , , , , , l m n p U V W X ∈ s s s s t,q t,q t,q t,q C and ( ) ( ) l m ≠ ≠ s s , ( ) ( ) [ ] n p ≠ ≠ ∀ ∈ s s s L ,L . By using (9) the necessary and sufficient condition for which the curve C is an isogeodesic on the hypersurface ( ) P s,t,q can be given as (10) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) = 0,= 0,= 0,0, U V W XV X V XV W V WW X W X ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ≠∂ ∂ ∂ ∂ t ,q = t ,q = t ,q = t ,qt ,q t ,q t ,q t ,q-t q q tt ,q t ,q t ,q t ,q-t q q tt ,q t ,q t ,q t ,q-t q q t [ ] [ ] , ∈ ∈ t T ,T q Q ,Q . ith a closer investigation of (10), we should have ( ) V ∂ =∂ t ,qt and ( ) V ∂ =∂ t ,qq . So, (10) can be simplified to (11) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) U V W XV VW X W X = = = = ∂ ∂ = = ∂ ∂ ∂ ∂ ∂ ∂ − ≠∂ ∂ ∂ ∂ t ,q t ,q t ,q t ,qt ,q t ,qt qt ,q t ,q t ,q t ,qt q q t [ ] [ ] , , . ∈ ∈ t T ,T q Q Q
Let marching-scale functions be ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) , , , ,, , , ,, , , ,, , , , u l Uv m Vw n Wx p X = = = = s t q s t qs t q s t qs t q s t qs t q s t q , , ≤ ≤ ≤ ≤ ≤ ≤
L s L T t T Q q Q , where ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , , , , , , l m n p U V W X ∈ s s s s t,q t,q t,q t,q C . Also let us choose ( ) ( ) ( ) ( ) V UV U = = = = d q d qq qdq dq . By using (9), the curve C is an isogeodesic on the hypersurface ( ) P s,t,q if and only if the followings are satisfied (12) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , 0,, ,, , 0, n W p Xn X W pW p n X = ≡ ∂ ∂ − ≠ ∂ ∂ s t q s t qs t d q d q s tq s t s t qt dq dq t [ ] [ ] , , , , . ∈ ∈ ≤ ≤ t T T q Q Q L s L Let marching-scale functions be ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) , , , ,, , , ,, , , ,, , , , u l Uv m Vw n Wx p X = = = = s t q s q ts t q s q ts t q s q ts t q s q t , , ≤ ≤ ≤ ≤ ≤ ≤
L s L T t T Q q Q , where ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , , , , , , , , , , l m n p U V W X ∈ s q s q s q s q t t t t C . Also let ( ) ( ) ( ) ( ) V UV U = = = = d t d tt tdt dt . By using (9) we derive the necessary and sufficient condition for which the curve C is an isogeodesic on the hypersurface ( ) P s,t,q as (13) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , 0,, ,, , 0, n W p XW p n Xn X W p = ≡ ∂ ∂ − ≠ ∂ ∂ s q t s q td t s q s q d ts t t t s qdt q q dt [ ] [ ] , , , , . ∈ ∈ ≤ ≤ t T T q Q Q L s L
4. Examples Example 1.
Let ( ) ( ) ( ) r = ≤ ≤ π s s s s s s be a curve parametrized by arc-length. For this curve, ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )
21 2
T rNBB B T N ′= = − = − − ′ ′′ ′′′⊗ ⊗= = − ′ ′′ ′′′⊗ ⊗ = ⊗ ⊗ = − − − s s s ss s sr s r s r ss r s r s r ss s s
Let us choose the marching-scale functions of type I, where ( ) ( ) ( ) ( ) l m n p = = = ≡ s s s s and ( ) ( )( ) ( ) ( ) ( ) , , , 0, , , , , U V W X = − − ≡ = − = − t q t t q q t q t q t t t q q q [ ] [ ] ∈ ∈ ≤ ≤ π t q s o, we have ( ) ( ) ( )( )( )( ) , , ,, , 0,, , ,, , . uvwx = − −≡= −= − s t q t t q qs t qs t q t ts t q q q
The hypersurface ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( )( ) ( ) ( ) , , , , , , , , , ,1 1 3cos sin sin ,2 2 21 1 3sin cos cos ,2 2 21 1 3 6 ,2 2 6 32 2 6 32 2 6 3
P r u T v N w B x B s t q s s t q s s t q s s t q s s t q ss t t q q s t t ss t t q q s t t ss t t q q t t q qs t t q q t t q q = + + + += − − − − − + − − + −+ − − − − + − + − − − − − − [ ] [ ] ≤ ≤ π ≤ ≤ ≤ ≤ ∈ ∈ s t q t q is a member of the isogeodesic hypersurface family, since it satisfies (11). By changing the parameters and t q we can adjust the position of the curve ( ) r s on the hypersurface. Let us choose = = t q . Now the curve ( ) r s is again an isogeodesic on the hypersurface ( ) , , P s t q and the equation of the hypersurface is ( ) ( ) ( ) ( )( ) ( ) ( ) P = − − + + − + + − − − + + − − − − s t q s t q ss t q ss t q t qs t q t q The projection of a hypersurface into 3-space generally yields a three-dimensional volume. If we fix each of the three parameters, one at a time, we obtain three distinct families of 2-spaces in 4-space. The projections of these 2-surfaces into 3-space are surfaces in 3-space. Thus, they can be displayed by 3D rendering methods. So, if we (parallel) project the hypersurface ( ) , , P s t q into the w 0 = subspace and fix = q we obtain the surface ( ) ( )( ) ( ) P + = − − + + − + − − − + ≤ ≤ π ≤ ≤ s t s t ss t ss t t s t w in 3-space illustrated in Fig. 1. Fig. 1 . Projection of a member of the hypersurface family with marching-scale functions of type I and its isogeodesic.
Example 2.
Given the curve parameterized by arc-length ( ) ( ) ( ) r = ≤ ≤ π s s s s s it is easy to show that ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )
21 2
T rN r r rB r r rB B T N ′= = − = − −′ ′′ ′′′⊗ ⊗= = −′ ′′ ′′′⊗ ⊗ = ⊗ ⊗ = − − s s s ss s ss s ss s s ss s s
Let us choose the marching-scale functions of type II, where ( ) ( ) ( )( ) , 1, , 1 n p = + + = + − s t s t s t s t t , and ( ) ( ) ( ) ( )
0, , 1.
U V W X = ≡ = − = q q q q q q
So, we get ( )( )( ) ( )( )( ) ( ) ( ) , , 0,, , 0,, , 1 ,, , 1 . uvwx ≡≡= + + −= + − s t qs t qs t q s t q qs t q s t t From (12) the hypersurface ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( )( ) , , , , , , , , , ,1 3sin 1 cos ,2 21 3cos 1 sin ,2 21 ,3 1s 12 2
P r u T v N w B x B = + + + += + + + − − + + −− + − − + + − s t q s s t q s s t q s s t q s s t q ss s t q q ss s t q q ss t ts t q q ≤ ≤ π ≤ ≤ ≤ ≤ s t q is a member of the hypersurface family having the curve ( ) r s as an isogeodesic. Setting = = t q yields the hypersurface ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) P s t q s s t q ss s t q ss ts s t q = + + + − + + − + − − + + By (parallel) projecting the hypersurface ( ) P s,t,q into the w 0 = subspace and fixing = q we get the surface ( ) ( ) ( )( ) ( ) ( )( ) P s t s s t ss s t ss t = + + + − + + − + − w where, ≤ ≤ π ≤ ≤ s t in 3-space demonstrated in Fig. 2. Fig. 2.
Projection of a member of the hypersurface family with marching-scale functions of type II and its isogeodesic. xample 3.
Let ( ) ( ) ( ) r = ≤ ≤ π s s s s s be an arc-length curve. One can easily show that ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )
21 2
T rN r r rB r r rB B T N ′= = − = − −′ ′′ ′′′⊗ ⊗= = −′ ′′ ′′′⊗ ⊗ = ⊗ ⊗ = − − s s s ss s ss s ss s s ss s s for this curve. If we choose the marching-scale functions of type III, where ( ) ( ) ( ) ( ) , sin , , n p = − = s q s q q s q sq and ( ) ( ) ( ) ( )
0, 1,
U V W X = ≡ = = − t t t t t t then ( )( )( ) ( ) ( ) ( ) ( )
02 0 , , 0,, , 0,, , sin ,, , . uvwx ≡≡= −= − s t qs t qs t q s q qs t q sq t t
Thus, from (13) if we take ≠ q then the curve ( ) r s is an isogeodesic on the hypersurface ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , , , , , , , , ,1 3sin cos sin ,2 21 3cos sin sin ,2 2,3 1 sin ,2 2 P r u T v N w B x B s t q s s t q s s t q s s t q s s t q ss s s q qs s s q qsq t ts s q q = + + + += + − − −− − − − (cid:3)(cid:4)(cid:5)(cid:4)(cid:6) (cid:3)(cid:4)(cid:5)(cid:4)(cid:6) where π ≤ ≤ π ≤ ≤ ≤ ≤ s t q y taking = = t q we have the following hypersurface: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) P s t q s s s qs s s qsq t s q = + − − −− − − − Hence, if we (parallel) project the hypersurface ( ) P s,t,q into the z 0 = subspace we get the surface ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) P s q s s s qs s s qs s q = + − − −− − z where π ≤ ≤ π ≤ ≤ s q in 3-space shown in Fig. 3. Fig. 3.
Projection of a member of the hypersurface family with marching-scale functions of type III and its isogeodesic. . Conclusion
We have introduced a method for finding a hypersurface family passing through the same given geodesic as an isoparametric curve. The members of the hypersurface family are obtained by choosing suitable marching-scale functions. For a better analysis of the method we investigate three types of marching-scale functions. Also, by giving an example for each type the method is verified. Furthermore, with the help of the projecting methods a member of the family is visualized in 3-space with its isogeodesic. However, there is more work waiting to study. For 3-space, one possible alternative is to consider the realm of implicit surfaces ( ) , , , 0 F = x y z t and try to find out the constraints for a given curve ( ) r s is an isogeodesic on ( ) , , , 0 F = x y z t . Also, the analogue of the problem dealt in this paper may be considered for 2-surfaces in 4-space or another types of marching-scale functions may be investigated.
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