BISECTOR CURVES OF PLANAR RATIONAL CURVES IN LORENTZIAN PLANE

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1 INTERNATIONAL JOURNAL OF GEOMETRY Vol. (03), No., BISECTOR CURVES OF PLANAR RATIONAL CURVES IN LORENTZIAN PLANE MUSTAFA DEDE, YASIN UNLUTURK AND CUMALI EKICI Abstract. In this paper, the bisector curves of two planar rational curves are studied in Lorentzian plane. The bisector curves are obtained by two different methods. Consequently, some experimental results are demonstrated.. Introduction The bisector for two objects is de ned as the set of points equidistant from the two objects. The construction of bisectors plays an important role in many geometric computations, such as Voronoi diagrams construction, medial axis transformation, shape decomposition, mesh generation, collision-avoidance motion planning, and NC tool path generation, to mention a few. Hofmann and Vermeer [8] formulate the bisector curve in terms of a system of polynomial equations using some auxiliary variables. By eliminating these auxiliary variables, an implicit equation of the bisector curve can be obtained. However, the process of variable elimination is slow and generates an algebraic bisector curve of high degree. Ho mann [7] also suggests using a dimensionality paradigm in which each polynomial equation is geometrically represented as a hypersurface. The bisector curve of two planar rational curves can be computed via the intersection of three hypersurfaces in R 4. The multiple intersection in R 4 is ine cient, compared with that of intersecting two surfaces in R 3. Choi transforms the problem of computing a bisector curve into that of intersecting two developable surfaces in R 3 []. Farouki and Johnstone show that the bisector of a point and a rational curve in the same plane is a rational curve [4]. Given two planar rational curves, Farouki and Johnstone interpret the bisector as the envelope curve of a one-parameter family of rational point/curve bisectors [5]. The curve/curve bisector is non-rational, in general. Therefore, Farouki and Johnstone approximate the bisector curve with a sequence of discrete points [5]. Keywords and phrases Bisector curves, planar curves, rational polynomials, Lorentzian plane (00)Mathematics Subject Classi cation 5M04, 5M5, 5M30 Received In revised form Accepted

2 48 Mustafa Dede, Yasin Unluturk and Cumali Ekici Farouki and Ramamurthy [6] develop a more precise algorithm that approximates the bisector curve with a sequence of curve segments. The approximation error can be reduced within an arbitrary bound by adaptive re- nement. Elber and Kim suggest a new (symbolic) representation scheme for planar bisector curves, which allows an e cient and robust implementation of a bisector curve construction based on B-spline subdivision techniques by using bisector of two planar curves as a planar curve in R 3 [3]. In this paper, we show that the bisector of two planar curves is also a planar curve in Lorentzian plane. We suggest the representation scheme for planar bisector curves in Lorentzian plane by using the method given in [3]. Also, the interesting examples for planar bisector curves are presented in Lorentzian plane.. Preliminaries Let L be a Lorentzian plane with the Lorentzian inner product <; > given by () < x; y >= x y x y ; where x = (x ; x ) and y = (y ; y ) L. A vector x is said to be timelike if < x; x >< 0, spacelike if < x; x >> 0 or x = 0, and lightlike (or null ) if < x; x >= 0 and x 6= 0. The norm of x is de ned by () kxk = p j< x; x >j An arbitrary curve c(t) in L can locally be spacelike, timelike or null if all of its velocity vectors dc(t) dt are spacelike, timelike or null, respectively. More information about both the Lorentzian plane and space can be found in [, 9, 0,, ]. R (3) 3. Bivariate Functions Let C (t) and C (r) be regular parametric C -continuous plane curves in given by parametrization C (t) = (x (t); y (t)) C (r) = (x (r); y (r)) We compute a bivariate polynomial function F (t; r) = 0 which corresponds to the bisector curve of C (t) and C (r). In this paper, the bivariate functions F i (t; r)(i = ; ) are computed by two di erent methods in Lorentzian plane [3]. 3.. Function F (t; r). The tangent vectors of C (s) and C (r) are obtained by (4) (5) T (t) = (x 0 (t); y0 (t)) T (r) = (x 0 (r); y0 (r)) Thus, the unnormalized normal elds of C (t) and C (r) are obtained by N (t) = (y 0 (t); x0 (t)) N (r) = (y 0 (r); x0 (r))

3 Bisector Curves of Planar Rational Curves in Lorentzian Plane 49 On the other hand, the intersection point of the normal lines of C (t) and C (r) is given as follows (6) C (t) + N (t) = C (r) + N (r) for some and Substituting (3) and (5) into (6) gives the following equation (x (t); y (t)) + (y 0 (t); x 0 (t)) = (x (r); y (r)) + (y 0 (r); x 0 (r)) Simple calculation implies that we have the following two equations in two unknowns and (7) y 0 (t) y0 (r) = x (r) x (t) x 0 (t) x0 (r) = y (r) y (t) By using Cramer s rule in (7), we obtain the solutions of and as bivariate rational functions x (r) x (t) y 0 (r) y (r) y (t) x 0 = (t; r) = (r) ; x (8) 0 (t) x0 (r) y 0 (t) x (r) x (t) x 0 = (t; r) = (t) y (r) y (t) x 0 (t) x0 (r) The two bivariate functions (t; r) and (t; r) provide a bivariate parametrization of the intersection point as follows (9) P (t; r) = C (t) + N (t)(t; r) = C (r) + N (r)(t; r) In addition, since P (t; r) must also be at an equal distance from C (t) and C (r); we have kp (t; r) C (t)k = kp (t; r) C (r)k Using (9) and () implies that (0) N (t)(t; r) = N (r)(t; r) By solving an absolute value equation in (0), we have two cases to consider. In order to represent both two cases in (0) we use " = ; then we have () N (t)(t; r) = "N (r)(t; r) Substituting (8) into (), we get x (r) x (t) y 0 (r) () N y (r) y (t) x 0 (r) "N x 0 (t) x0 (r) y 0 (t) x (r) x (t) x 0 (t) y (r) y (t) = 0 x 0 (t) x0 (r) Consequently, using (5) we have explicit equations of bisector curves as follows (3) P (t; r) = [y 0 (t) x0 (t)][(y (r) y (t))y0 (r) (x (r) x (t))x0 (r)] [y 0 (t) x0 (t)][(y (r) y (t))y0 (t) (x (r) x (t))x0 (t)] = 0

4 50 Mustafa Dede, Yasin Unluturk and Cumali Ekici and P (t; r) = [y 0 (t) x0 (t)][(y (r) y (t))y0 (r) (x (r) x (t))x0 (r)] (4) +[y 0 (t) x0 (t)][(y (r) y (t))y0 (t) (x (r) x (t))x0 (t)] = 0 Both of (3) and (4) represent the bisector curves of C (t) and C (r) in Lorentzian plane. 3.. Function F (t; r). In this method, First of all we obtain the bisector points P (t; r) When a point P (t; r) is on the bisector of two curves C (s) and C (t), there exist (at least) two points C (s) and C (t) such that point P is simultaneously contained in the normal lines L (t) and L (r). As a result, the point P satis es the following two linear equations ( L (t) < P C (s); T (s) >= 0 (5) (6) L (r) < P C (t); T (t) >= 0 Direct computation shows that ( < P; T (t) >=< C (t); T (t) > < P; T (r) >=< C (r); T (r) > Substituting (3) and (4) into (6), then using Cramer s rule leads to the bisector point P (t; r) = (x(t; r); y(t; r)) obtained by x (t)x 0 (t) y (t)y 0 (t) y0 (t) x (r)x 0 (7) x(t; r) = (r) y (r)y 0 (r) y0 (r) x0 (t) y0 (t) ; x 0 (r) y0 (r) x0 (t) x (t)x 0 (t) y (t)y 0 (t) x 0 (8) y(t; r) = (r) x (r)x 0 (r) y (r)y 0 (r) x0 (t) y0 (t) x 0 (r) y0 (r) In addition, point P (t; r) must satisfy the equidistance condition given as follows kp (t; r) C (t)k = kp (t; r) C (r)k From (), it is easy to see that jhp C (t); P C (t)ij = jhp C (r); P C (r)ij Since it is an absolute value equation, there are two cases to consider. Now, we distinguish the following two cases Case. If hp C (t); P C (t)i = hp C (r); P C (r)i, then the bisector P (t; r) must satisfy the following condition given as follows (9) < P (t; r); C (t) C (r) > = C (t) C (r) Case If hp C (t); P C (t)i = hp C (r); P C (r)i, then the bisector P (t; r) must satisfy the following condition given as follows (0) < P (t; r); P (t; r) (C (t) + C (r)) > = C (t) + C (r)

5 Bisector Curves of Planar Rational Curves in Lorentzian Plane 5 Consequently, both of (9) and (0) represent the bisector curve in Lorentzian plane. 4. Examples We demonstrate the bisector curves by the following examples. Throughout this chapter, we assume that P (t; r) and P (t; r) are the solution of (9) and (0) plotted in red and blue in gures, respectively. Example. Let us consider the bisector of two lines C (t) and C (r) in Lorentzian plane given by parametrization () C (t) = (t ; t); C (r) = (; r) From (7) and (8), we have () x(t; r) = 3t + r, y(t; r) = r Using (9), () and (), the bisector curve P (t; r) is obtained as follows (3) P (t; r) = 3t 6t r + r + 4 = 0 From (0), () and (), we get the bisector curve P (t; r) given by (4) P (t; r) = 3t t + 6tr + 8 r 4r = 0 Consequently, the equations (3) and (4) as the bisector of two lines in Lorentzian plane illustrated in Figure b. The bisector of two lines in Euclidean plane is also shown in Figure a. Figure a Figure b Example. In this example, Figure a and Figure b show the bisector curve for a curve and a line give by parametrization C (t) = (t ; t); C (r) = (0; r) The bisector curves are obtained by P (t; r) = t + rt + r 3t 4 = 0 and P (t; r) = t 5t 4 + 6t r + 8t 6 6t 4 r + 8t r r = 0

6 5 Mustafa Dede, Yasin Unluturk and Cumali Ekici Figure a Figure b Figure 3a and Figure 3b show bisector of a circle and line in Euclidean plane and Lorentzian plane, respectively. Figure 3a Figure 3b Consequently, The bisector of two parabolas is illustrated in Figure 4a and Figure 4b in Euclidean plane and Lorentzian plane, respectively. Figure 4a Figure 4b

7 Bisector Curves of Planar Rational Curves in Lorentzian Plane Conclusion This paper enlightens the role of Lorentzian geometry in the computation of bisector planar curves. It is also a collection of basic algorithms and linear constructions for that kind of curves. In the scope of this article, we have shown that the bisector between plane curves is a plane curve in Lorentzian plane. In a further contribution, we will study similar results for point-surface, sphere-surface and plane-surface bisectors in Lorentzian geometry. References [] Catoni, F., Boccaletti, D., Cannata, R., Catoni, V., Nichelatti and E., Zampetti, P., The Mathematics of Minkowski Space-Time With an Introduction to Commutative Hypercomplex Numbers, Birkhauser Verlag, Boston, 008. [] Choi, J. J., Local Canonical Cubic Curve Tracing along Surface/Surface Intersections, Ph.D. Thesis, Dept. of Computer Science, POSTECH, Pohang, Korea, 997. [3] Elber, G. and Kim, M. S., Bisector curves of planar rational curves, Computer Aided Design, 30(998), [4] Farouki, R. and Johnstone, J., The bisector of a point and plane parametric curve, Computer Aided Geometric Design, ()(994), 7-5. [5] Farouki, R. and Johnstone, J., Computing point/curve and curve/curve bisectors, In Design and Application of Curves and Surfaces Mathematics of Surfaces V, R. B. Fisher, Ed., Oxford University Press, 994, [6] Farouki, R. and Ramamurthy, R., Speci ed-precision computation of curve/curve bisectors, International Journal of Computational Geometry & Applications, 8(5/6)(998). [7] Ho mann, C., A dimensionality paradigm for surface interrogations, Computer Aided Geometric Design, 7(6)990, [8] Ho mann, C., and Vermeer, P., Eliminating extraneous solutions in curve and surface operation, Int l J. of Computational Geometry and Applications, ()99, [9] Ikawa, T., Euler - Savary s formula on Minkowski geometry, Balkan J. Geom. Appl., 8()(003), 3. [0] Lopez, R., Di erential geometry of curves and surfaces in Lorentz-Minkowski space, Mini-Course taught at the Instituto de Matematica e Estatistica (IME-USP), University of Sao Paulo, Brasil, 008. [] O Neill, B., Semi Riemannian geometry with applications to relativity, Academic Press, Inc., 983. [] Yaglom, I. M., A simple non-euclidean geometry and its physical basis, Springer- Verlag New York, 979. KILIS 7 ARALIK UNIVERSITY KILIS, TURKEY address mustafadede@kilis.edu.tr KIRKLARELI UNIVERSITY 3900, KIRKLARELI, TURKEY address yasinunluturk@kirklareli.edu.tr ESKISEHIR OSMANGAZI UNIVERSITY 6480, ESKISEHIR, TURKEY address cekici@ogu.edu.tr