A STUDY OF SOLVING LIMIT PROBLEMS BY USING MONOGENIC SURFACES. 1. Introduction

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1 Analele Universităţii Oradea Fasc. Matematica, Tom XIV (2007), A STUDY OF SOLVING LIMIT PROBLEMS BY USING MONOGENIC SURFACES LIDIA ELENA KOZMA Abstract. In this paper, the possibility of solving limit problems by using monogenic surfaces is studied. The method opens new ways for approximate solving of limit problems. The approximation of the solution is analyzed by using the Bézier surfaces, and a theorem is formulated for the existence of approximate solution. The use of the Hermite surfaces is also remarked. 1. Introduction In my papers [1], [2], [3], I considered the bijective correspondence between the monogenic functions of the following form f(z) = f(x, y) = u(x, y) + iv(x, y), i 2 = 1, z D C, f C 2 (D) and the real surfaces of the form (1.1) (S) r = iy + ju(x, y) + kv(x, y), (1.2) where (i, j, k) are the Euclidian unit vectors and v(x, y) is the harmonic conjugate of function u(x, y), the same as in relation (1.1). Surface (1.2) was defined to be monogenic only if function (1.1) is monogenic. We state the properties of monogenic surface (S) : 2000 Mathematics Subject Classification. 32A38. Key words and phrases. Monogenic functions, Gauss curvature, Bézier surfaces, Hermite surfaces. 73

2 74 LIDIA ELENA KOZMA A. The total Gauss curvature of the monogenic surface is negative. B. The mean curvature of the monogenic surface is null (the surface (S) is minimal). The formulation of (Dirichlet) limit problem was given in [2] and [3] under the following form: Let a surface (S) be determined under the form (1.2), knowing that u = 0 in D, u C = R(x, y), where R(x, y) is a function given on the curve C, even (smooth) on portions. 2. The possibility of determining a monogenic surface by using Bézier surfaces Next, we analyze the possibility of determining surface (S) from (Dirichlet) limit problem, by using the Bézier surface. We state as in [4], a few information concerning Bézier surfaces: If the base B m, respectively B n, is the Bernstein polynomial base given by B m i (x) = C i m x i (1 x) m i, i = 0, m B n j (y) = C j n y j (1 y) n j, j = 0, n, (2.1) then the surface associated to a network of points b ij ; i = 0, m; j = 0, n will be s(x, y) = m n b ij Bi m (x)bj n (y), (x, y) [0, 1], (2.2) i=0 j=0 and it is called Bézier surface of grades (m, n) and the matrix of points (b ij ) is called control network of the Bézier surface. A surface polygonal on portions, meaning the reunion of all squares of tips b ii, b ij+1, b i+1j, b i+1j+1 ; i = 0, m 1; j = 0, n 1, called the control polyhedron of the Bézier surface, is associated to a Bézier surface.

3 A STUDY OF SOLVING LIMIT PROBLEMS 75 We justify the choice of Bézier surfaces for solving the formulated limit problem, because we know points (b ij ) in the Dirichlet problem (previously formulated). As we know the function u(x, y) on the curve C S, we can suppose that the points (b ij ) on the determined surface (S) are known. While determining the Bézier surfaces, the corner points of the control network interfere. While determining the surface (S) from the Dirichlet problem, the control points are placed continuously on the surface. In this situation it seems that we have a great variety of choices for points from the control network. We underline the followings: The abundance of points in the control network does not modify the form of the Bézier surface [4]. This observation concords with the identity principle which we extended [3] for monogenic surfaces. In [3] we proved the following result: If a monogenic surface r f (x, y) coincides with a surface r ϕ (x, y), on a line of points (x n, y n ) converging to point (x 0, y 0 ), then the two surfaces r f and r ϕ coincide in all the points of the domain where the function f(x, y) is monogenic. The conclusion of the above statements is that we can determine a Bézier surface with the network of points (b ij ), chosen from: a) knowing function u(x, y) on the frontier (C) of the domain D; b) determining function v(x, y), harmonic conjugate of function u(x, y), in each point (x, y) C (with approximation of an a- dditional constant). The vectors associated to a Bézier layer. The geometric characterization of the differentiable surface is done with the help of vectors r x, r y, r xy associated to each point of the surface. Let us take the particular case m = n = 3 (bi-cubic Bézier surface). The tangent vectors r x, r y in the network points b 00, b 0n, b m0, b mn n = m = 0, 1, 2, 3 will be:

4 76 LIDIA ELENA KOZMA r x (0, 0) = m(b 10 b 00 ) ; r x (1, 0) = m(b m0 b m 1,0 ) r x (0, 1) = m(b 1n b 0n ) ; r x (1, 1) = m(b mn b m 1,n ) r y (0, 0) = n(b 01 b 00 ) ; r y (0, 1) = n(b 0n b 0,n 1 ) r y (1, 0) = n(b m1 b m0 ) ; r y (1, 1) = n(b mn b m,n 1 ). (2.3) The vector r xy = 2 r, named in CAGD the twist vector, is of x y special interest 2 m 1 r x y = mn i=0 n j=0 11 b ij B m 1 i (x)b n 1 j (y), (2.4) where 11 b ij = b i+1 j+1 b i j+1 b i+1j b ij 2 r Example: When b 10 b 11 = b 00 b 01, the twist vector x y = 0. From the point of view of the monogenic surface (1.2), the solution to the above formulated Dirichlet problem, has the total Gauss curvature negative (condition A) and it is constant for a given movement model. The expression of the total curvature in [1] is: 2 2 f x y K = [ 1 + f 2 2 z ] f. (2.5) z Thus, result two conditions: Condition I, which is required from the Bézier surfaces as solution of the Dirichlet problem: choosing the points of the control network in such a way that K = constant, and 2 f x y = 2 r x y 0 in any point of the surface. Condition II, which we impose on the Bézier surface and is given by condition (B) of the monogenic surface. From the general theory of Bézier surfaces we know that a Bézier surface is the convex hull of its control points. We can choose the minimal

5 A STUDY OF SOLVING LIMIT PROBLEMS 77 surface, from all the Bézier surfaces with certain points of control (which also satisfy condition I) if this thing is possible. Analytically, condition (B) is equivalent to the Cauchy-Riemann conditions. It results that the points of the network will have to fulfill the condition r x r y = 0, or the vectors given by relation (2.3) should be perpendicular in every (b ij ) point of the chosen network. From the previously analyzed date, we express the following boundary problem: ( ) Let us determine a surface (S) of the shape: (S) r = iy + ju(x, y) + kv(x, y) with functions u(x, y); v(x, y) C 2 (D), D R 2, v(x, y) harmonically conjugated with function u(x, y) if we know: { u = 0 in D u P = R(x, y) where R(x, y) is a function given by the network of points P = {b ii, b ij+1, b i+1j, b i+1j+1 } ; i = 0, m; j = 0, n called control polyhedron. The above problem allows an approximate solution (Bézier surface) given by the following theorem: Theorem 2.1. Any solution of the boundary problem ( ) allows a Bézier surface (which follows the shape ) as approximate solution, if the following conditions are met: (i) 11 b ij = b i+1j+1 b ij+1 b i+1j b ij 0; i = 0, m; j = 0, n. (ii) r x r y = 0 for any vectors calculated in points b ij (Relation 2.3) The Bézier surface is of (m, n) level and has the form (2.2) Proof. Indeed, from condition (i) 11 b ij 0, there results the fact that the Bézier surface has the Gauss curvature: K < 0 (condition A).

6 78 LIDIA ELENA KOZMA Relation (ii) r x r y = 0 is equivalent both with the Cauchy-Riemann conditions for functions u(x, y) and v(x, y), but by [1], with condition (B) met by a monogenic surface. The quality of approximation. The Bézier surface thus obtained approximates the surface of the problem ( ) solution. We have in mind two aspects: (1) The quality of the approximation depends on the number of points considered in network (P ) (the control polyhedron). If the network (P ) of points make up a line of points convergent towards a point in the network, then according to the Identity Principle for monogenous surfaces in [3] (re-expressed in this paper), the determined Bézier surface coincides with the solution of the expressed boundary problem. (2) The determination of surface (S) depends both on function u(x, y) and on function v(x, y), the harmonic conjugate of u(x, y). It is known the function u(x, y) on the points b ij (the corner points), and thus we can determine v(x, y) with an approximation of an additive constant. For this reason, instead of the approximation of surface (S 1 ) (S 1 ) r = iy + ju + kv, we can obtain the approximation of surface (S 2 ): (S 2 ) r = iy + ju + k(v + c) where c = constant Surfaces (S 1 ) and (S 2 ) both meet the conditions in the existence theorem expressed above, even these do not coincide. From this point of view we consider the method as approximate. We can also notice the fact that the Bézier surface has an explicit equation z = F (x, y), and the surface required by problem ( ) has a vectorial form (1.2).

7 A STUDY OF SOLVING LIMIT PROBLEMS 79 In what follows, we intend to analyze the solution of the boundary problem ( ) for the determination of surface (S), by using Hermite surfaces. 3. Hermite surfaces and limit problems formulated with the help of monogenic surfaces We analyze the case of bi-cubic Hermite surface. Let us take (S) = ims, a bi-cubic parametric surface by 3 3 s(x, y) = a ij x i y j, (x, y) [0, 1] (3.1) i=0 j=0 Expressing the polynomial functions in Hermite base, leads us to a matrix (h ij ), which geometrically characterizes the surface, s(0, 0) s(0, 1) s y (0, 0) s y (0, 1) s(1, 0) s(1, 1) s y (1, 0) s y (1, 1) s x (0, 0) s x (0, 1) s xy (0, 0) s xy (0, 1) s x (1, 0) s x (1, 1) s xy (1, 0) s xy (1, 1) [ = B 00 B 01 B 10 B 11 ] (3.2) B 00 is a block of points from the surface, and the others are blocks of derivatives r x, r y, 2 r in the chosen corner points (tips) of the surface. x y Using conditions I and II, we will be able to determine the Hermite surface attached to the Dirichlet problem, H s(x, y) = [ H0(x) 3 H1(x) 3 H2(x) 3 H3(x) ] 0(y) 3 3 h ij H1(y) 3 H2(y) 3 (3.3) H3(y) 3 with (h ij ) the matrix h from relation (3.2). In applications to the Dirichlet type limit problems, the matrix (h ij ) will have to be calculated by using conditions I and II. In the applications

8 80 LIDIA ELENA KOZMA which use these models, the condition s xy 0 has to be imposed, unlike in the general model of approach in the CAGD (Computer Aided Geometric Design) graphics. This model is applicable only under the condition of knowing supplementary geometrical data referring to the solution surface (relation 3.2). For this reason, we recommend the use of the Bézier surfaces which only ask for the points on the surface to be known. We note that the Bézier surface does not pass through the b ij network of the given points, it only gets very close to them. Conclusion. The approximation of a surface of form (1.2) by means of a Bézier surface of the form (2.2) will provide an image graphically close to surface (1.2), although the means of transformation of surface (2.2) into (1.2) was not decided upon analytically. The model introduced in this paper may open new ways of graphic visualization of certain approximate solutions for the boundary problems. References [1] L.E. Kozma, A possible visual image of the extended monogeneous function, Bute and PAMM, Balaton Almady, May 2003, pg , Proceeding of PAMM conference). [2] L.E. Kozma, About the Dirichlet and Neumann boundary value problems expressed by means of monogenous quaternions, Carpathian Journal of Mathematics vol. 20 No. 2, 2004, pg [3] L.E. Kozma, About the use of monogeneous surfaces in solving Plaine Boundary Problems, in Proceedings of International Conference on Complex Analysis and Related Topics, Cluj-Napoca, august, 14-19, [4] Emilia Petrişor, Modelare Geometrică Algoritmică, colecţia Universitaria, Editura Tehnică, Bucureşti Received 8 November 2006 North University of Baia Mare, Department of Mathematics and Computer Science, Victoriei 76, Baia Mare, Romania address: kozma lidia@yahoo.com