Curves with constant geodesic curvature in the Bolyai-Lobachevskian plane

Transcription

1 Stud. Univ. Babeş-Bolyai Math ), No. 3, Curves with constant geodesic curvature in the Bolyai-Lobachevsian plane Zoltán Gábos and Ágnes Mester Abstract. The aim of this note is to present the curves with constant geodesic curvature of the Bolyai-Lobachevsian hyperbolic plane. By using the Lobachevsian metric the equations of the circle, paracycloid and hipercycloid are given. Furthermore, we determine a new family of curves with constant curvature which was not emphasized before. During the analysis we use Cartesian and polar coordinates. Mathematics Subject Classification 200): 53A35. Keywords: hyperbolic geometry, geodesics, orthogonal trajectory, polar coordinates.. General formulas in Cartesian coordinates We consider the Lobachevsian metric ds 2 = cosh 2 y dx2 + dy 2,.) where is the parameter of the two-dimensional hyperbolic plane. Among the geodesics we can find so-called Euclidean lines too, which can be used as coordinate axes. Therefore we can define a Cartesian coordinate system in the hyperbolic plane. If dx = 0, then ds 2 = dy 2, thus we can use the euclidean method when determining the value of y. The x-value of a point can only be determined by the x-axis, because when dy = 0, then the formula ds 2 = dx 2 can only be used in the case of y = 0. Now let us consider a point P x, y) in the hyperbolic plane. The foot of the perpendicular from P to the x-axis is denoted by P x, 0). Then distance OP corresponds with the x-coordinate of P, while the length of P P equals the y-coordinate of P. As the reflection over the coordinate axes is a symmetry operation, during the analysis we will consider only the first quadrant of the plane. Note that the lines we obtain in the first quadrant have segments in the other quadrants, too. From metric.) it follows that the translation of the origin along the direction of the x-axis is also a symmetry operation.

2 450 Zoltán Gábos and Ágnes Mester If we use s as variable, the functions characterizing the geodesics are of form x = xs), y = ys). The geodesic lines are determined by the following differential equations: d 2 x ds tanh y dx dy = 0,.2) ds ds d 2 y ds 2 sinh y cosh y ) 2 dx = 0..3) ds If we replace variable s with x, equation.2) can be written in the following equivalent form: cosh 2 y dx ds = C,.4) where C is constant. Using.3) and.4) we obtain the d 2 tanh y dx 2 2 tanh y = 0.5) differential equation. If we use variable x, we only need to determine the constants a and a 2 which appear in the equation C 2 tanh y = a cosh x + a 2 sinh x..6) By using.),.4) and.6), for the value of C we get = tanh 2 y d tanh y ) = a 2 + a 2 dx 2..7) In order to determine the geodesic curvature, we use the formula given by Schlesinger: = r g = cosh y { d 2 x ds tanh y dx ds ) dy dy ds ds From.2),.3) and.8) it follows that [ d 2 y ds 2 sinh y cosh y Metric.) can also be obtained by using the metric ) ] 2 dx ds } dx. ds.8) r g = 0..9) ds 2 = dx 2 + dx 2 2 dx 2 0.0) defined in the three-dimensional pseudo-euclidean space, with the help of the following formulas: x = sinh x cosh y, x 2 = sinh y, x 0 = cosh x cosh y..)

3 Curves with constant geodesic curvature General formulas in polar coordinates Using.0) and equalities x = cos ϕ sinh ρ, x 2 = sin ϕ sinh ρ, x 0 = cosh ρ, 2.) we obtain the metric ds 2 = dρ sinh 2 ρ dϕ2, 2.2) where ρ and ϕ represent polar coordinates. If we use s as variable, we obtain the following differential equations which determine the lines of the hyperbolic plane. d 2 ϕ ds coth ρ d 2 ρ ds 2 sinh ρ cosh ρ dϕ dρ = 0, 2.3) ds ds ) 2 dρ = ) ds If we replace variable s with ϕ, equation 2.3) can be written in the following equivalent form: sinh 2 ρ dϕ ds = C ) Also, from 2.4) we get d 2 coth ρ dϕ 2 + coth ρ = ) The geodesics satisfy coth ρ = b sin ϕ + b 2 cos ϕ, 2.7) where b and b 2 are constant values. From.6) and 2.7) it follows that the values x, x 2 and x 0 admit a linear connection. Using 2.2), 2.5) and 2.7), we get for the value of C 2 r g = sinh ρ C 2 2 = 2 [ coth 2 ρ + d coth ρ dϕ ) 2 ] = 2 b 2 + b 2 2 ). 2.8) The geodesic curvature verifies the formula given by Schlesinger: { d 2 ϕ ds coth ρ ) [ dϕ dρ dρ ds ds ds d 2 ρ ds 2 sinh ρ cosh ρ dϕ ds ) 2 ] } dϕ. ds 2.9) Using.) and 2.) it follows that the connection between the Cartesian and polar coordinates is determined by the following equations: sinh x cosh y = cos ϕ sinh ρ, 2.0) sinh y = sin ϕ sinh ρ, 2.) cosh x cosh y = cosh ρ. 2.2)

4 452 Zolta n Ga bos and A gnes Mester Furthermore, from 2.0) and 2.2) we obtain x ρ tanh = cos ϕ tanh. 2.3) 3. The Bolyai-Lobachevsian lines As the geometry in discussion is based on metric.), we differentiate four types of lines. The first family contains lines crossing the origin. The second set consists of lines which cross the x-axis, while the lines of the third family do not cross the x-axis. Then there are the lines which are parallel to the x-axis. a) Based on.6), the lines crossing the origin satisfy y x tanh = a2 sinh. Let us consider a point P x, y) on a line in question, then the tangent vector to the line in P admits dy tan α =. cosh y dx In this case y x tan α = a2 cosh cosh, thus the value of a2 determines the tangent vector in the origin. The angle of intersection between the line and the x-axis is denoted by ϕ, which verifies y x tanh = tan ϕ sinh. 3.) Figure

5 Curves with constant geodesic curvature 453 Obviously, ϕ is the polar angle of point P. If ϕ is set as constant, we get ds2 = dρ, thus the value of ρ determines the distance between the origin and point P measured along the geodesic. The values of x and x2 represented in figure are determined by equation 3.), when y and y. Using equation.7), we obtain 2 C = cos ϕ. b) From.6) it follows that the lines crossing the x-axis and passing through points P0 a, 0) and P 0, b) verify b a x x y cosh coth sinh. 3.2) tanh = tanh The values x and x2 are determined by equation 3.) as y and y figure 2). Figure 2 In the case of polar coordinates we use ρ b a = coth sin ϕ + coth cos ϕ 3.3) obtained from formula 2.7). The value of ϕ is given by equation 3.3) as ρ. Using 2.6) and 2.8), we get for the constants C and C2 the following formulas: s r tanh2 b b a = +, = coth2 + coth2. 3.4) C C2 sinh2 a coth

6 454 Zoltán Gábos and Ágnes Mester c) Figure 3 illustrates that the line passing through the y-axis in point P 0, b) while not crossing the x-axis has a minimum point. Figure 3 The line admits cosh x tanh x m ), 3.5) tanh y = tanh b sinh x where x m denotes the value of x determined by the minimum point. The domain of the line is determined by equation 3.5) as y. In the case of polar coordinates, by using equations 2.9), 2.0), 2.) and 2.2), we get coth ρ = coth b sin ϕ + tanh x m cos ϕ. 3.6) For the values of C and C 2, we obtain formulas = tanh2 b C cosh 2 and = coth 2 b xm C 2 + x m tanh2. 3.7) If x m = 0, the line admits while the values C and C 2 verify tanh y = tanh b cosh x, coth ρ = coth b sin ϕ, 3.8) C = cosh b, C 2 = sinh b. 3.9) d) If a and x m, we obtain the line parallel to the x-axis, passing through P 0, b). The lines of this family satisfy tanh y = tanh b e x, while C =, C 2 = tanh b. ρ coth = coth b sin ϕ + cosϕ, 3.0)

7 4. The orthogonal curves Curves with constant geodesic curvature 455 If in the formulas obtained in the previous section we set a variable as a varying parameter, we obtain families of lines. Each family of lines admits orthogonal lines. Let us denote the original lines by the index. The line passing through point P x, y) admits the following orthogonality condition: cosh 2 y dxdx + dydy = 0. 4.) The lines verify y = y x, p), p = px, y ), 4.2) where the parameter is denoted by p. By deriving this equation with respect to variable x, we obtain dy dx = fx, p). 4.3) Using 4.2) and 4.3), we eliminate the parameter, thus we get dy dx = f[x, px, y )] = F x, y ). 4.4) From 4.) and 4.4) it follows that the orthogonal lines verify cosh 2 y dx + F x, y)dy = ) In the case of polar coordinates we use the formula ρ = ρ ϕ, p). Hence we get dρ = gϕ, p) = Gρ, ϕ ). 4.6) dϕ By applying the orthogonality condition dρdρ + 2 sinh 2 ρ dϕdϕ = 0, 4.7) we get for the orthogonal lines Gρ, ϕ)dρ + 2 sinh 2 ρ dϕ = ) Now let us consider the distance along the line between points P 0, b) and P x, y) represented in figure 4. We denote the length of P P by d. The distance from the origin to P equals ρ. In figure 4 a right-angled triangle is formed, where the hypotenuse is equal to ρ and the other two sides are x and y. Equality 2.2) gives a formula considering these values. Let P 3 be the foot of the perpendicular from P to the line OP. Then two right-angled triangles, namely OP 3 P and P P 3 P are formed. In OP 3 P the length of the hypotenuse OP is denoted by b, while the legs OP 3 and P P 3 are denoted by ρ 0 and c. Therefore we can write cosh b = cosh ρ0 cosh c.

8 456 Zoltán Gábos and Ágnes Mester Figure 4 In triangle P P 3 P the hypotenuse is d, while the value of the legs are ρ ρ 0 and c, thus we get If we eliminate c, we obtain cosh d = cosh ρ ρ 0 cosh d = cosh b cosh c. cosh ρ tanh ρ 0 sinh ρ ). 4.9) In triangle OP 3 P the angle between OP and OP 3 is equal to π 2 ϕ. Hence, from 2.3) we can write tanh ρ 0 = sin ϕ tanh b. 4.0) Applying 4.9) and 4.0), we get cosh d = cosh b cosh ρ sin ϕ sinh b sinh ρ. 4.) If we use Cartesian coordinates, from 4.), 2.2) and 2.) it follows that cosh d = cosh b cosh x cosh y sinh b sinh y. 4.2) In the following sections we determine the orthogonal lines. During the analysis our choice of coordinates may vary depending on the form of calculations. We will prove that the curvature of the orthogonal lines is constant. Moreover, any two orthogonal lines from the same family are parallel. We define parallelism in the following way: let us consider a geodesic and its two orthogonal lines passing through the line in two different points. If the distance between the two points of intersection is constant in the case of any geodesic, we say that the two orthogonal lines are parallel.

9 Curves with constant geodesic curvature The orthogonal curves of the radial lines Radial lines are lines which pass through a common point. We will consider four family of lines. a) The first family consists of lines crossing the origin, which are determined by equation 3.). Here we use p = tan ϕ as parameter. In this case we can write Therefore, by using 4.5) we get F x, y) = coth x sinh y cosh y. cosh y sinh x d cosh x cosh y ) = 0. The expression in bracet is constant. From equation 2.2) it follows that the points of an orthogonal curve are always at the same distance from the origin. Hence we obtain a circle with center O. cosh x cosh y = cosh R, ρ = R, 5.) where R denotes the radius of the circle. If we use polar coordinates, from 5.) and 2.2) it follows that dϕ ds = sinh R. From 2.9) we obtain the formula characterizing the curvature: r g = coth R. 5.2) b) Now we determine the orthogonal curves of lines crossing point P 0 a, 0) Ox. As the translation of the origin along the direction of the x-axis into point P 0 is a symmetry operation, we obtain circles with center P 0, which verify cosh x a cosh y = cosh R. 5.3) The curvature of the orthogonal lines is determined by formula 5.2). If we use polar coordinates, from 5.3), 2.0) and 2.2) it follows that the circles verify cosh a cosh ρ cos ϕ sinh a sinh ρ = cosh R. c) The lines parallel to the x-axis admit the p = tanh b parameter and satisfy equation 3.0). As F = sinh y cosh y,

10 458 Zoltán Gábos and Ágnes Mester from 4.5) it follows that d ln cosh y x ) = 0. The orthogonal curve crossing point P x 0, 0) is called paracycloid, which verifies cosh y = e x x ) We can also determine the equation of the paracycloid by the following way. If a > x 0, the circle with center P 0 a, 0) passing through P x 0, 0) has the radius R = a x 0. If a, from 5.3) we obtain formula 5.4). Thus the paracycloid can be considered a semicircle with infinite radius. d) The orthogonal curves of lines passing through point P 0, b) are lines which cross or do not cross the x-axis. By the use of polar coordinates we obtain Here the parameters are given by coth ρ = coth b sin ϕ + p cos ϕ. 5.5) coth a b and tanh x m. The orthogonal lines admit the following formula: Gρ, ϕ) = ρ cosϕ sinh2 coth b sin ϕ coth ρ Using 4.8), we get coth b cosϕ sin ϕ coth ρ ) dρ + dϕ = = cos ϕ sinh b sinh ρ d cosh b cosh ρ sin ϕ sinh b sinh ρ ) = 0. Hence, by using 4.) it follows that the orthogonal lines are circles with center P 0, b), which verify the following equations: cosh b cosh ρ sin ϕ sinh b sinh ρ = cosh R, 5.6) cosh b cosh x cosh y sinh b sinh y = cosh R. 5.7) We obtain the curvature by considering 2.2) and using the formulas below: Hence we obtain From 5.8) we get sin ϕ = fρ), fρ) = coth b coth ρ cosh R sinh b sinh ρ. 5.8) ). dρ ds = sinh b f 2 sinh R. 5.9) cos ϕdϕ = f 2 dϕ = df dρ dρ.

11 Curves with constant geodesic curvature 459 Thus by using 5.9) we obtain dϕ ds = sinh b sinh R The derivatives of the second ind are as follows: d 2 ρ ds 2 = sinh2 b sinh 2 R f df dρ, d 2 ϕ ds 2 df dρ. 5.0) = sinh2 b f sinh 2 R 2 d2 f dρ 2. Hence, from 2.9) and 5.5) it follows that the curvature is given by formula 5.2). If we consider two circles, the distance between the intersections with a geodesic is equal to the difference of the two radiuses, which is a constant value. This proves that the orthogonal lines determined above are parallel. 6. The orthogonal curves of lines not having common point We will consider two different cases. a) The first family consists of lines being parallel to the y-axis. The orthogonal lines are called hipercycloids having y = b, 6.) where b is a constant value. In the upper half-plane b > 0, while below the x-axis b < 0. The hipercycloids satisfy the orthogonality condition 4.), because in the case of the geodesic satisfying dx = 0 its orthogonal curve verifies dy = 0. From 2.2) and 6.) we obtain Using 6.), 6.2) and.8), we get for the curvature dx ds = cosh b. 6.2) r g = tanh b. b) The lines of the second family do not cross the x-axis and have minimum point on the y-axis. By using 3.8), these lines verify tanh y = p cosh x, p = tanh b, 6.3) where b can be either positive or negative value. In this case F x, y) = tanh x sinh y cosh y. From the 4.5) orthogonality condition we get tanh x sinh y dy + cosh y dx = cosh x d sinh x cosh y ) = 0.

12 460 Zoltán Gábos and Ágnes Mester Thus the orthogonal curve passing through point P x 0, 0) verifies sinh x cosh y = sinh x ) Figure 5 illustrates the geodesics determined by b and b, and their orthogonal curves passing through points P x 0, 0) and P x 0, 0). Figure 5 The tangent field of the orthogonal line admits tan α = dy cosh y dx = coth 2 x 0 y coth2. Thus as y, we obtain tan α =. sinh x0 We use polar coordinates in order to determine the curvature of the lines. From 6.4) and 2.0) we obtain cos ϕ sinh ρ = sinh x ) Also, from 2.2) and 6.5) it follows that dρ ds = tanh 2 x 0 ρ coth2 = gρ). Hence, by using equation 6.5) we obtain dϕ ds = gρ) cot ϕ coth ρ = tanh x 0 The derivatives of the second ind are as follows: d 2 ρ ds 2 = dg 2 2 dρ, cosh ρ sinh 2 ρ.

13 Curves with constant geodesic curvature 46 d 2 x0 ϕ tanh ds 2 = 2 sinh ρ 2 coth 2 ρ ) g. From 2.9) we obtain for the curvature the following formula: r g = tanh x ) The orthogonal curves are parallel to the y-axis. We will prove this by determining the distance between the points P 0 0, b) and P x, y ), illustrated on figure 5. Point P is the intersection point of the geodesic and its orthogonal curve. The coordinates are given by the formulas tanh y = tanh b cosh x, sinh x cosh y = sinh x 0. Hence we obtain From 4.2) we get thus sinh y = sinh b cosh x 0, cosh x y tanh = tanh b. cosh d = cosh b cosh x cosh y sinh b sinh y = cosh x 0, d = x 0. The distance of two orthogonal lines is given by the difference of the values x 0, which proves the parallelism of the orthogonal curves. These orthogonal lines are the duals of the hipercycloids, fact which is illustrated also by the curvatures determined above. This family of lines was not considered in the past. Note that any orthogonal line can be described by formula.). In the case of hipercycloids the value of x 2, while in the case of 6.4) the value of x is constant. Along the circles 5.), 5.3) and 5.7) the following values are constant: cosh a x 0 sinh a x, cosh b x 0 sinh b x 2. References [] Préopa, A., Kiss, E., Staar, G., eds.), Bolyai Emléönyv, Vince Kiadó, Budapest, [2] Bonola, R., A nemeulideszi geometria története, Pávia, 906. [3] Coxeter, H.S.M., Introduction to geometry, Wileg, New Yor, 2 edition, 989. [4] Bolyai, J., Appendix, a tér tudománya, Aadémiai Kiadó, Budapest, 977. [5] Dávid, L., Bolyai geometria az Appendix alapján, Cluj, 944. [6] Schlesinger, L., A tér absolute igaz tudománya, Ábel Kiadó, Cluj-Napoca and Szeged, 202. [7] Lobacsevszij, N.I., Pangeometrie, Leipzig, 902.

14 462 Zoltán Gábos and Ágnes Mester [8] Lobacsevszij, N.I., Geometriai vizsgálato a párhuzamoso elméleténe öréből, Aadémiai Kiadó, Budapest, 95. [9] Sanielevici, S., Buletinul ştiinţific, secţiunea: matematică şi fizică, 95. Zoltán Gábos Babeş-Bolyai University Faculty of Physics, Kogălniceanu Street, Cluj-Napoca, Romania zoltan.gabos@gmail.com Ágnes Mester Babeş-Bolyai University Faculty of Mathematics and Computer Science, Kogălniceanu Street, Cluj-Napoca, Romania mester.agnes@yahoo.com