On Bézier surfaces in three-dimensional Minkowski space

0. Introduction A Bézier surface is defined using mathematical spline functions whereby the resulting surface has a compact analytic description. This enables such surfaces to be easily manipulated, and they also have greater continuity properties. Bézier curves and surfaces are commonly used in computer-aided design [1,2], image processing [3,4], and finite elementmodeling (e.g. [5–7]). Many other authors have studied Bézier curves and surfaces in Euclidean space (e.g. [8–12]). Inmany problems in physics andmechanics, important functionals are often defined on surfaces or on othermultidimensional objects. For example, the space of surfaces with a prescribed border is often studied. An important property studied on surfaces is that of their areas via the various area functionals. Another property relates to the Dirichlet functional, which is defined by the Lagrange functional based on the Laplacian operator. The relationship between both of these functionals is very similar to the relationship between length and energy functionals. The study of surfacesminimizing the area functional with prescribed border, called the Plateau problem, is to date a main topic in Euclidian differential geometry. Such kinds of surface, characterized by the vanishing mean curvature, are called minimal surfaces. Recently, there has been interest in the relevant research communities in studying Bézier surfaces in various spaces as well as the properties of such surfaces subject to functionals. For example, Monterde studied Bézier surfaces ofminimal area in R3 [10,11]. Miao et al. have studied the variational problems of finding Bézier surfaces that minimize the bending energy functional with prescribed border for both triangular and rectangular cases [13]. Xu and Wang have studied the properties of harmonic-type Bézier surfaces over both rectangular and triangular domains [14]. Farin and Hansford proposed a mask derived from the discretization of the Laplacian operator for generating the control net of the resulting Bézier surface [15]. Similarly, Ugail and Monterde studied the generation of Bézier surfaces satisfying the standard Laplace equation as well as the biharmonic equation, which they refer to as harmonic and biharmonic Bézier surfaces, respectively [16]. Later, they ∗ Corresponding author. E-mail addresses: [email protected] (H. Ugail), [email protected] (M.C. Márquez), [email protected] (A. Yılmaz). 0898-1221/$ – see front matter© 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2011.07.065 Author's personal copy 2900 H. Ugail et al. / Computers and Mathematics with Applications 62 (2011) 2899–2912 presented a method for generating Bézier surfaces from the boundary information based on a general fourth-order elliptic partial differential equation, and they point out that both the harmonic and biharmonic Bézier surfaces are related to minimal surfaces, i.e., surfaces that minimize the area among all the surfaces with prescribed boundary data [12]. In a somewhat parallel fashion, Minkowski space is the basis for the study of the physical phenomena described by the theory of relativity which has great geometric and physical meaning. Much work to date, therefore, has been done on timelike and spacelike surfaces inR1 (the three-dimensionalMinkowski space). For example, Treibergs has studied spacelike hypersurfaces with constant mean curvature in Minkowski space [17]. Aledo et al. obtain a Lelievvre-type representation for timelike surfaces with prescribed Gauss map [18]. Abdel-Baky and Abd-Ellah study both spacelike and timelike ruled W-surfaces inR1 which satisfies a nontrivial relation between elements of the set {K , KII ,H,HII}, where (K ,H) and (KII ,HII) are the Gaussian and mean curvatures of the first and the second fundamental forms [19]. Brander et al. constructed the spacelike constant mean curvature surfaces in R1SU(2) with the non-compact real form SU(1, 1) [20]. Lin studied the implications of curvature restrictions on timelike surfaces in R1 that are convex as surfaces in Euclidean 3-space R 3 [21]. Kossowski has studied restrictions on zero mean curvature surfaces in R1 [22]. Georgiev, in his recent work, obtained sufficient conditions for Bézier surfaces to be spacelike [23]. Given the vast studies that have been undertaken both for Bézier curves and surfaces as well as surfaces in Minkowski space, our motivation for this work stems in trying to understand the relationship between the two, i.e. we are interested in understanding the properties of Bézier surfaces in Minkowski space. To do this we first establish the necessary and sufficient conditions for a Bézier surface to exist in Minkowski space. We also study the Plateau problem for Bézier surfaces in Minkowski space by using the Dirichlet functional formulated in the metric of this space. The paper is organised as follows. In Section 1, we first introduce basic notation and recall the conditions of timelike and spacelike surfaces in R1 Minkowski space. Furthermore, we give the definitions and derivative formulas of Bézier curves and surfaces. In Section 2, we define the first fundamental coefficients E, F , and G in terms of coordinates of the control points of the Bézier surface, and then we give the conditions of the timelike case and the spacelike case for Bézier surfaces. In Section 3, we deal with the Plateau–Bézier problem by obtaining conditions on the control points to be extremal of the Dirichlet functional. In Section 4, we give some examples for these cases, and we compare the area functionals for the minimal Bézier surface in R3 and R1. Finally, in Section 5, we conclude the work presented in this paper. 1. Notation and preliminaries Let R1 be the three-dimensional Minkowski space, that is, the three-dimensional real vector space R 3 with the metric ⟨, ⟩ = (dx1) + (dx2) − (dx3), where (x1, x2, x3) denotes the canonical coordinates in R3. An arbitrary vectorv = (v1, v2, v3) in R1 can have one of three Lorentzian casual characters, i.e. it can be spacelike if ⟨v , v ⟩ > 0 or v = 0, timelike if ⟨v , v ⟩ < 0, and null (lightlike) if ⟨v , v ⟩ = 0 and v ≠ 0. For any vectors x = (x1, x2, x3) and y = (y1, y2, y3) in R1, the Lorentz vector product of x and y is defined by x × y = −e1 −e2 e3 x1 x2 x3 y1 y2 y3  =  − x2 x3 y2 y3  , x1 x3 y1 y3  , x1 x2 y1 y2  . LetM be a surface in R1.M is called a timelike surface if the induced metric onM is a Lorentzian metric on each tangent plane. This is equivalent to saying that the unit normal vectorN is spacelike at each point ofM .M is called a spacelike surface if the inducedmetric onM is a positively definite Riemannianmetric on each tangent plane. This is equivalent to saying that the unit normal vector N is timelike at each point ofM . IfM is given with a parameterization, Φ : U ⊂ R2 → R1 (u, v) → Φ(u, v) = (Φ1(u, v), Φ2(u, v), Φ3(u, v)), then the unit normal vector field N onM is given by N = Φu × Φv ‖Φu × Φv‖ , where Φu = ∂Φ/∂u, Φv = ∂Φ/∂v, and × stands for the Lorentzian cross product of R1. The metric ⟨, ⟩ on each tangent plane ofM is determined by the first fundamental form, I = ⟨dΦ, dΦ⟩ = Edu2 + 2Fdudv + Gdv2, with the differentiable coefficients E = ⟨Φu, Φu⟩, F = ⟨Φu, Φv⟩, G = ⟨Φv, Φv⟩. (1) Author's personal copy H. Ugail et al. / Computers and Mathematics with Applications 62 (2011) 2899–2912 2901 SinceM is timelike and spacelike, we have det I = EG − F 2 < 0 and det I = EG − F 2 > 0, respectively. Definition 1. Given n + 1 control points P0, P1, . . . , Pn, the Bézier curve of degree n is defined to be P(t) = n − i=0 PiBi (t), where Bi (t) =  n! (n − i)!i! (1 − t)n−it i, if 0 ≤ i ≤ n 0 otherwise are called the Bernstein polynomials or Bernstein basis functions of degree n [24]. Definition 2. Let Bi (u) and B m j (v) be the Bernstein basis functions of degree n andm, respectively. A Bézier surface with the control points Pij (0 ≤ i ≤ n, 0 ≤ j ≤ m) is the parametric surface defined by P(u, v) = n −