Euler Savary s Formula On Complex Plane C

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1 Applied Mathematics E-Notes, 606, 65-7 c ISSN Available free at mirror sites of amen/ Euler Savary s Formula On Complex Plane C Mücahit Akbıyık, Salim Yüce Received 7 October 05 Abstract In this article, we consider a base curve, a rolling curve and a roulette on complex plane. We investigate the third one when any two of the base curve, the rolling curve, and a roulette are known.we obtain Euler Savary s formula, which gives the relation between the curvatures of these three curves. Introduction Complex numbers have important role in mathematics. It has caused new numbers such as dual numbers, hyperbolic numbers, quaternions and octonions, etc. Complex numbers are not only used in mathematics but also have essential concrete applications in a variety of scientific and related areas such as physics, chemistry, biology, economics, electrical engineering, statistics, signal processing, control theory, electromagnetism, fluid dynamics, quantum mechanics, cartography, and vibration analysis. On the Euclidean plane E, let us consider two curves: a base curve and rolling curve which are denoted by B and R, respectively. Assume that X is a point which is relative to a rolling curve R. Suppose that the rolling curve R rolls without splitting along the base curve B. Then, the locus of the point X makes a curve which is called roulette and denoted by X. For instance, if B is a line, R is a circle and X is a point on R, then X is cycloid. Euler Savary s formula is a very famous theorem which gives relation between curvatures of the roulette and these base curve and rolling curve. It is used on quite serious fiel of mathematics and engineering. It is worked by Alexander and Maddocks, [3], Buckley and Whitfield, [4], Dooner and Griffi s, [5], Ito and Takahaski, [6], Pennock and Raje, [7], Wang at all, [8]. However, in 956, Müller, [9], obtained Euler Savary s formula for one parameter motion in Euclidean plane E. In 003, T. Ikawa, [], examined Euler Savary s formula in Minkowskian geometry and also showed a new way for a generalization of the Euler Savary s formula in the Euclidean plane in this article. In 00, Masal at al., [0], expressed Euler Savary s formula for one parameter motion in the complex plane C. In this paper, we research a generalization of the Euler Savary s formula in the complex plane C by a different way from [0]. Also, we study on a base curve, a rolling curve and a roulette in the complex plane C. Mathematics Subject Classifications: 53A7. Department of Mathematics, Yildiz Technical University, Esenler, Istanbul 340, Turkey Department of Mathematics, Yildiz Technical University, Esenler, Istanbul 340, Turkey 65

2 66 Euler Savary s Formula on Complex Plane Preliminaries Let α : I R C from an open interval I into C be a planar curve with arc length parameter s. Let the curve α be defined by α s = α s + iα s, []. Then, the unit tangent vector of the curve α at the point α s is defined by T s = α s + iα s. Frenet formulas of the curve α have the following equations T = κn, N = κt, where N is the unit normal vector of the curve α and κ is curvature of α in [, ]. However, we can define a new curve α A s = α s + x s T s + y s N s, which is called associated curve, where x s, y s R. If we calculate the velocity vector of α A s, then we get d α A s dx s dy s = κ s y s + T s + κ s x s + N s. Besides, we can write the associated curve as α A s = x s + iy s, with respect to the frame {α s ; T s, N s}. Also, according to the frame {α s ; T s, N s}, the velocity vector of the associated curve is calculated as where d α A s v s + iv s := κ s y s + = v s + iv s, dx s + i κ s x s + dy s. Moreover, assume that s A is the arc length parameter of α A, then the Frenet frame {T A, N A } of α A hol T A s A = κ A s A N A s A, N A s A = κ A s A T A s A, where κ A is the curvature of α A. Let θ be a slope angle of α and ω a slope angle of α A. Then, we can write κ A s A = dω = dω = κ + dφ A A v s + iv s, where φ = ω θ.

3 M. Akbıyık and S. Yüce 67 3 Euler Savary s Formula In this section, we investigate the Euler Savary s formula which gives the relation between the curvatures of a base curve, a rolling curve, and a roulette. In addition to this, we investigate the third one when any two of the base curve, the rolling curve, and the roulette are known on the complex plane. CASE. Let the base curve and the rolling curve be given. Let B be the base curve with curvature κ B. Assume that X is a point relative to the rolling curve R and the roulette of the locus of this point X is denoted by X. Then, we can consider that X is an the associated curve of B. So the relative coordinate {x, y} of X with respect to the curve B satisfies the following equation v s + iv s = κ s y s + dx s + i κ s x s + dy s, from the equation. Moreover, we know that when the rolling curve rolls without splitting along the base curve B at each point of contact, the relative coordinate {x, y} is also a relative coordinate of X with respect to the curve R for a suitable parameter s R. In this situation, the associated curve is a point X and the following is provided: v s R + iv s R = κ s R y s R + dx s R Then, we have + i κ s R x s R + dy s R = 0. dx s R + i dy s R = + κ R s R y s R iκ R s R x s R. If we substitute these equations into, we get v s R + iv s R = κ R κ B y + i κ B κ R x. However, we can write the associated curve on the polar coordinate with respect to {B s; x, y} as follows: X = re iφs, where r is the distance from the origin point B s to the point X. Then, from the equation, we calculate d X = dr e iφs + ire iφs R dφ s R = κ R r sin φ + i κ R r cos φ. If we solve this equation with respect to r dφ, then we find r dφ = κ R r + Ime iφ. 3

4 68 Euler Savary s Formula on Complex Plane Furthermore, we know that So, we get κ X = from the equations 3 and 4. So, we may give the following theorem: = κ B + dφ κ B κ R x + y κ B + dφ κ B κ R r. 4 rκ X = κ B κ R κ B κ R + Imeiφ r κ B κ R, 5 THEOREM. Assume that a curve R rolls without splitting along a curve B on the complex plane C. Let X be a locus of a point that is relative to R. Let Q be a point on X and R be a point of contact of B and R correspon to Q relative to the rolling relation. By r, φ, we denote a polar coordinate of Q with respect to the origin R and the base line B R. Then, the curvatures κ B, κ R and κ X of the curves B, R, and X, respectively, satisfy rκ X = ± + Imeiφ r κ B κ R. CASE. Assume that the base curve and the roulette are given. Suppose that B s B = u s B + iv s B is a base curve with the arc length parameter s B. Let us draw the normal to the roulette X for a point Q of the curve B and let the point R = x s B + iy s B be the foot of this normal. Then the length of the normal QR is d Q, R = x s B u s B + y s B v s B. 6 However, by considering the equation 6 on the rolling curve R, this equation represents the length of the point X relative to R and a point of R. So, the orthogonal coordinate f s B + ig s B of R is given by the following equations: f s B + ig s B = x s B u s B + i y s B v s B, df dg + i =. B B CASE 3. Now, assume that the rolling curve R and the roulette X are given. Suppose that X s A = x s A + iy s A is the roulette with arc length parameter s A and R s R is given by the polar coordinate r s R with the arc length parameter s R. Because of the normal of X is N s A = y s A +ix s A, a point u s B +iv s B of the point curve B is given by u s B + iv s B = x s A + iy s A ± r s R N

5 M. Akbıyık and S. Yüce 69 or So, we can write u s B + iv s B = x s A r s R y s +i y s A ± r s R x s. 7 or du + i dv = dx A + i dy A ± dr N ± r dn A A A A du + i dv = dx A + i dy A ± dr A A r κ X dx A + i dy A dy A. + i dx A A Then, we find du +i dv = dx A rκx dr dy dy A +i rκx ± dr A A A dx A where κ X is the curvature of X. Since s R is also the arc length of B, we have and A du dv + i =, dr rκx + =. From this differential equation, we can solve s A = s A s R. By substituting this equation into equation 7, we can have the orthogonal coordinate of B. EXAMPLE. Let s find the curvature of the trajectory curve of a point on the circle R and circles B and R at the moment t = 5 when circle R = cos θ+i sin θ roles without splitting along B = cos θ + i sin θ where θ [0, π]. Here t shows the moment when circle R rolls splitting along circle B., Figure : The rolling at the moment t = 0, t = 5, t = 30, t = 45, t = 9, respectively.

6 70 Euler Savary s Formula on Complex Plane SOLUTION. When a circle R roles without splitting along a circle B, a trajectory curve of a point on R is a line, see, Figure. So, we know that the curvature of this trajectory curve is zero. If we want to solve from Euler Savary s formula, since r = and φ = π 4 at the moment t = 5 and κ B = and κ R =, we find So, κ X = 0. κx = + = 0. DISCUSSION. If κ B κ R > 0, then we have r Ime iφ = κ B κ R, r from the equation 5, where r = r κ X is the distance from origin point to the curvature center of X. In addition, if κ B κ R < 0, then we get r Ime iφ = κ B κ R, r from the equation 5, where r = r + κ X is the distance between origin and the curvature center of X. Consequently, the Euler Savary s formula in the complex plane C coincides with the Euler Savary s formula in the Euclidean plane E in [9]. But, in [0], the Euler Savary formula in the complex plane C is found as r i e iφ = κ B κ R. r It is different from Euler Savary s formula in the Euclidean plane. Note that the left hand-side of the equation is complex number and right hand-side of the equation is real number. So, it is provided when φ = π + kπ, k Z. But, sometimes φ may not equal to π + kπ. References [] R. A. Silverman, Complex Analysis with Applications, Dover Publications, Inc., New York, 974. [] A. Pressley, Elementary Differential Geometry, Springer-Verlag London Lim., Second Edition, 009 [3] J. C. Alexander and J. H. Maddocks, On the Maneuvering of Vehicles, SIAM J. Appl. Math., 48988, [4] R. Buckley and E. V. Whitfield, The Euler-Savary Formula, Math. Gaz., 33949,

7 M. Akbıyık and S. Yüce 7 [5] D. B. Dooner and M. W. Griffi s, On the Spatial Euler-Savary Equations for Envelopes, J. Mech. Design, 9007, [6] N. Ito and K. Takahashi, Extension of the Euler-Savary Equation to Hypoid Gears, JSME Int.Journal. Ser C. Mech Systems, 4999, 8 4. [7] G. R. Pennock and N. N. Raje, Curvature theory for the double flier eight-bar linkage, Mech. Theory, 39004, [8] D. L. Wang, J. Liu and D. Xaio, A unified curvature theory in kinematic geometry of mechanism, Science in China Series E., 4998, [9] W. Blaschke, H. R. Müller and E. Kinematik, Verlag Oldenbourg, München, 959. [0] M. Masal, M. Tosun and A. Z. Pirdal, Euler Savary formula for the one parameter motions in the complex plane C, International Journal of Physical Sciences, 500, [] T. Ikawa, Euler-Savary s formula on Minkowski geometry, Balkan Journal of Geometry and Its Applications, 8003, 3 36.