PROCEEDINGS OF SYMPOSIUM ON COMPUTER GEOMETRY SCG’ 2012, VOLUME 21 pp. 0 - 0 ON SINGULARITIES OF EVOLUTES OF CURVES IN THE PSEUDO-EUCLIDEAN PLANE Viktória Bakurová 1 Comenius University, Faculty of Mathematics, Physics and Informatics Mlynská dolina, 842 48 Bratislava, Slovak Republic, e-mail: [email protected] Abstract. The aim of the paper is to classify singular points of arbitrary order of evolutes of curves in the pseudo-Euclidean plane. We point out a close connection of such singularities to higher-order vertices of base curves. Keywords: Pseudo-Euclidean plane, curve, singular point, evolute. 1. Introduction The concept of evolute is well-known in differential geometry of the Euclidean plane. In this paper, we discuss some properties of this object and classify singularities of these curves in the pseudo-Euclidean plane. The pseudo-Euclidean plane E1;1 is the real affine plane A2 furnished with a nonsingu- lar indefinite quadratic form q(x) = hx; xi , where h; i denotes the pseudo-scalar product. This pseudo-scalar product can be expressed as hx; yi = x1y1 − x2y2 in a suitable basis. We say that x is a space-like vector or a time-like vector or a light-like vector if sgn x = 1 or sgn x = −1 or sgn x = 0, respectively, where sgn x = sgn q(x). We shall make use of a perpendicularity operator x !? x which assigns the vector ? x = (sgn x x2; sgn x x1) to a vector x = (x1; x2). According to the type of the tangent vector at a point of a curve, the point is said to be a space-like or a time-like or a light-like point of the curve. We exclude all light-like points from all curves. In the pseudo-Euclidean plane we can consider parametrized curves similarly as in the Euclidean plane. At every regular (and not light-like) point P (t) of a curve we have the oriented Frenet frame consisting of vectors P 0(t) t(t) = ; n(t) =? t(t) jP 0(t))j and the oriented curvature det(P 0(t);P 00(t)) k(t) = 3 : jhP 0(t);P 0(t)ij 2 Moreover, for a curve expressed in a unit speed parametrization it holds t0(s) = k(s)n(s) n0(s) = k(s)t(s): We refer to [1] and [2] for more details about curves in the pseudo-Euclidean plane. 1This project has been supported by the project UK/371/2012. 0 PROCEEDINGS OF SYMPOSIUM ON COMPUTER GEOMETRY SCG’ 2012, VOLUME 21 pp. 0 - 0 2. Evolute of a curve in the pseudo-Euclidean plane As in the Euclidean plane, the evolute of a curve in the pseudo-Euclidean plane is the locus of all its centers of osculating circles (see [4], Chapter 8). From this we have the following definition: Definition 1. The evolute of a curve, with its inflexion and light-like points removed, is defined as the curve given by 1 E(t) = P (t) − n(t): (1) k(t) Recall that a vertex of a parametrized curve P (t) is a point P (t0), in which the first 0 0 derivate of curvature vanishes, i.e. k (t0) = 0. A point P (t0) is an ordinary vertex if k (t0) = 0 00 and k (t0) 6= 0. A vertex of a curve is a vertex of order l if the first l derivates of curvature 0 (l) vanishes, i.e. k (t0) = 0; : : : ; k (t0) = 0. A vertex of a curve is a vertex of order exactly l if it (l+1) is a vertex of order l and in addition it holds k (t0) 6= 0. The next lemma shows us, how the first four derivates of evolute are expressed: Lemma 1. Consider a unit-speed parametrization of the curve. For the first four derivates of evolute of this curve it holds: k0 E(1) = n k2 ! k0 k00 (k0)2 E(2) = t + − 2 n k k2 k3 ! ! k00 (k0)2 k000 k0k00 (k0)3 E(3) = 2 − 3 t + k0 + − 6 + 6 n k k2 k2 k3 k4 k000 k0k00 (k0)3 E(4) = 3 − 14 + 12 + k0k t k k2 k3 ! k0000 (k0)2 k0k000 (k00)2 (k0)2k00 (k0)4 + 3k00 + − 3 − 8 − 6 + 36 − 24 n k2 k k3 k3 k4 k5 Proof. The assertion of the lemma can be easily shown using Frenet formulas in the pseudo- Euclidean plane. (n) To obtain an adequate expression for derivative E ; n > 4, we use the class Pn;1(k) of functions from [3]. We summarize briefly basic facts on that class. Let Pn be the real vector space of all polynomial functions '(x0; : : : ; xn) and let Pn;1 de- notes its vector-subspace consisting of functions '(x0; : : : ; xn) with the property '(x0; 0;:::; 0) = 0 for every x0 2 R. To the curvature k(t) of the curve P (t) defined on an interval I ⊆ R and to any integer n ≥ 0 we assigning the (n + 1)-tuple of derivatives (k(0); : : : ; k(n)) understood as a function n+1 I ! R . To every function ' 2 Pn or ' 2 Pn;1 we get a one-variable function '(k(0); : : : ; k(n)): I ! R; t ! '(k(0)(t); : : : ; k(n)(t)). 1 PROCEEDINGS OF SYMPOSIUM ON COMPUTER GEOMETRY SCG’ 2012, VOLUME 21 pp. 0 - 0 We denote the set of all such functions as Pn(k) or Pn;1(k), respectively. The most important properties of the classes Pn(k) and Pn;1(k) are following, see [3], Proposition 1. For every n; m ≥ 1 it holds: • if σ 2 Pn;1(k); τ 2 Pm(k) then στ 2 Pn;1(k) • if σ 2 Pn;1(k); τ 2 Pm;1(k) then σ + τ; στ 2 Pmax(n;m);1(k) 0 • d(Pn;1(k)) ⊆ Pn+1;1(k) where d : σ ! σ is the derivate operator Note that the last property follows from the chain rule, particularly. The next theorem is a generalization of the previous lemma and it shows the expression for the derivation of arbitrary order of the evolute of a curve. Theorem 2. Let E(t) be an evolute of a curve P (t) expressed in unit speed parametrization. To every n ≥ 4 there exist polynomial functions αn 2 Pn−2;1 and βn 2 Pn−1;1 such that it holds: 1 E(n) = f(n − 1)knk(n−1) + α (k(0); : : : ; k(n−2)) t+kn−1k(n) + β (k(0); : : : ; k(n−1)) ng : n n kn+1 Proof. We use induction over n. For n = 4, the assertion is true in virtue of Lemma 1. Let n (n−1) (0) (n−2) the formula holds for n. Let us denote an = (n − 1)k k + αn(k ; : : : ; k ) and n−1 (n) (0) (n−1) bn = k k + βn(k ; : : : ; k ) and differentiate: 1 0 E(n+1) = (a t + b n) n n kn+1 1 1 0 = (a0 t + a kn + b0 n + b kt) + (a t + b n) n n n n kn+1 n n kn+1 1 (−1)(n + 1)k(1) = (a0 t + a kn + b0 n + b kt) + (a t + b n) n n n n kn+1 n n kn+2 1 = fka0 + k2b − (n + 1)k(1)a t + k2a + kb0 − (n + 1)k(1)b ng n n n n n n kn+2 Compute the following expression: 0 2 (1) n (1) (n−1) n+1 (n) (0) (n−1) kan + k bn − (n + 1)k an = n(n − 1)k k k + (n − 1)k k + kγn(k ; : : : ; k ) n+1 (n) 2 (0) (n−1) + k k + k βn(k ; : : : ; k ) n (1) (n−1) (1) (0) (n−2) − (n − 1)(n + 1)k k k − (n − 1)k αn(k ; : : : ; k ) n+1 (n) (0) (n−1) = nk k + αn+1(k ; : : : ; k ); (0) (n−1) n (1) (n−1) (0) (n−1) where αn+1(k ; : : : ; k ) = n(n − 1)k k k + kγn(k ; : : : ; k ) 2 (0) (n−1) n (1) (n−1) (1) (0) (n−2) +k βn(k ; : : : ; k )+(n−1)(n+1)k k k −(n−1)k αn(k ; : : : ; k ) denotes polynomial function of the class Pn−1;1. Compute now the following expression: 2 0 (1) n+2 (n−1) 2 (0) (n−2) k an + kbn − (n + 1)k bn = (n − 1)k k + k αn(k ; : : : ; k ) n−1 (1) (n) n (n+1) (0) (n) + (n − 1)k k k + k k + δn(k ; : : : ; k ) n−1 (1) (n) (1) (0) (n−1) − (n + 1)k k k − (n + 1)k βn(k ; : : : ; k ) n (n+1) (0) (n) = k k + βn+1(k ; : : : ; k ); 2 PROCEEDINGS OF SYMPOSIUM ON COMPUTER GEOMETRY SCG’ 2012, VOLUME 21 pp. 0 - 0 (0) (n) n+2 (n−1) 2 (0) (n−2) where βn+1(k ; : : : ; k ) = (n − 1)k k + k αn(k ; : : : ; k ) n−1 (1) (n) (0) (n) n−1 (1) (n) (1) (0) (n−1) +(n−1)k k k +δn(k ; : : : ; k )−(n+1)k k k −(n+1)k βn(k ; : : : ; k ) denotes polynomial function of the class Pn;1. Using these expressions we have: 1 E(n+1) = fnkn+1k(n) + α (k(0); : : : ; k(n−1)) t+knk(n+1) + β (k(0); : : : ; k(n)) ng : n+1 n+1 kn+2 The theorem is proved. 3. Classification of singularities of evolutes in the pseudo-Euclidean plane In this section we discuss singularities of evolute of the curve in the pseudo-Euclidean plane.