DERIVING THE EXACT FORMULA FOR PERIMETER OF AN ELLIPSE USING COORDINATE TRANSFORMATION

Keywords: Coordinate transformation, approximation, formula, perimeter of an ellipse, Integral Elliptic

Abstract

The ellipse can be transformed into a circle by dilating the coordinates of the ellipse relative to the x-axis and y-axis. Therefore, this study aimed to derive the formula for the equation of the perimeter of an ellipse by using the transformation of an ellipse to a circle. This transformation was arranged so that the perimeter of the ellipse was equal to the perimeter of the circle. The type of research was in the review of books, articles, and relevant research reports. The results showed that the ellipse can be transformed into a circle while maintaining its perimeter. So, the perimeter of the ellipse was the same as the perimeter of the circle.

References

A. Brannan, D., F. Esplen, M., & J. Gray, J. (2012). Geometry Second Edition (Second). New York: Cambridge University Press. https://www.cambridge.org/id/academic/subjects/mathematics/geometry-and-topology/geometry-2nd-edition

Abbott, P. (2009). On the Perimeter of an Ellipse. The Mathematica Journal, 11(2), 172-185. https://content.wolfram.com/uploads/sites/19/2009/11/Abbott.pdf

Adlaj, S. (2012). An Eloquent Formula for the Perimeter of an Ellipse. Notice of The AMS, 59(8), 1094–1099. https://doi.org/http://dx.doi.org/10.1090/noti879

Almkvist, P. G., & Berndt, B. (1988). Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, π, and the Ladies Diary (1988). American Mathematical Monthly, 95(1), 585–608. https://doi.org/10.1007/978-3-319-32377-0

Alzer, H., & Qiu, S. L. (2004). Monotonicity theorems and inequalities for the complete elliptic integrals. Journal of Computational and Applied Mathematics, 172(2), 289–312. https://doi.org/10.1016/j.cam.2004.02.009

Archimedes. (2010). The Works of Archimedes Edited in Modern Notation with Introductory Chapters (T. L. Heath, ed.). New York: Cambridge University. https://www.cambridge.org/id/academic/subjects/physics/theoretical-physics-and-mathematical-physics/works-archimedes-edited-modern-notation-introductory-chapters?format=PB

B. Thomas, G. (2018). Thomas’ calculus (based on the original work by George B. Thomas, Jr., Massachusetts Institute of Technology, as revised by Joel Hass, University of California, Davis, Christopher Heil, Georgia Institute of Technology, Maurice D. Weir, Naval Postgraduate (Fourteenth). Pearson Education. https://search.library.uitm.edu.my/Record/wils_911908

B. Villarino, M. (2008). Ramanujan ’ s Perimeter of an Ellipse. http://arxiv.org/abs/math/0506384v1

Barnard, R. W., Pearce, K., & Schovanec, L. (2001). Inequalities for the Perimeter of an Ellipse. Journal of Mathematical Analysis and Application, 260(1), 295–306. https://doi.org/10.1006/jmaa.2000.7128

E. J. F. Primrose. (1973). Maximum Area and Perimeter of a Parallelogram in an Ellipse. The Mathematical Gazette, 57(402), 342–343. https://www.cambridge.org/core/journals/mathematical-gazette/article/3351-maximum-area-and-perimeter-of-a-parallelogram-in-an-ellipse/3008A43D1A6CDACBD5BD99BCBA5E7535

E. Pfiefer, R. (1988). Bounds on the Perimeter of an Ellipse via Minkowski Sums. The College Mathematics Journal, 19(4), 348–350. https://doi.org/10.1080/07468342.1988.11973137

Gusić, I. (2015). On the bounds for the perimeter of an ellipse. The Mathematical Gazette, 99(546), 540–541. https://doi.org/10.1017/mag.2015.102

Hilbert, D., & Cohn-Vossen, S. (2021). Geometry and The Imagination (Second). AMS Chelsea Publishing. http://michel.delord.free.fr/geoim.pdf

J. Purcell, E., & Varberg, D. (2016a). Kalkulus dan Geometri Analitis Jilid 1 (Edisi Ke-5). Jakarta: Penerbit Erlangga. https://repo.iainbatusangkar.ac.id/xmlui/handle/123456789/5650

J. Purcell, E., & Varberg, D. (2016b). Kalkulus dan Geometri Analitis Jilid 2 (Edisi Ke-5). Jakarta: Penerbit Erlangga. https://repo.iainbatusangkar.ac.id/xmlui/handle/123456789/5647

Jameson, G. . J. O. (2015). Inequalities for the perimeter of an ellipse. The Mathematical Gazette, 98(542), 227–234. https://doi.org/10.1017/S002555720000125X

Lockhart, P. (2012). Measurement. Massachusetts: The Belknap Press of Harvard University Pres. https://catalog.lib.kyushu-u.ac.jp/ja/recordID/3447461/

Mazer, A. (2010). The Ellipse : A historical and Mathematical Journey. New Jersey: John Wiley & Sons, Inc. https://elibrary.ru/item.asp?id=19467164

Parker, W. V, & Pryor, J. E. (1944). Polygons of Greatest Area Inscribed in an Ellipse. The American Mathematical Monthly, 51(4), 205–209. https://doi.org/10.1080/00029890.1944.11999068

Rohman, H. A., & Jupri, A. (2019). Investigating the Equation and the Area of Ellipse Using Circular Cylinder Section Approach. Universitas Pendidikan Indonesia, 4(1), 210-214. http://science.conference.upi.edu/proceeding/index.php/ICMScE/article/view/245

Published
2022-04-15
How to Cite
Abdul Rohman, H. (2022). DERIVING THE EXACT FORMULA FOR PERIMETER OF AN ELLIPSE USING COORDINATE TRANSFORMATION. Alifmatika: Jurnal Pendidikan Dan Pembelajaran Matematika, 4(1), 1-16. https://doi.org/10.35316/alifmatika.2022.v4i1.1-16
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