# Pedal Coordinates

Pedal coordinates are tangential coordinates that describe the position of a point x on a pedal curve γ by two numbers: the distance from the origin r (the pedal point), and the distance from the origin to the tangent of γ at point x. The tangent of the curve depends on the line from the origin to the point.

The pedal equation is defined as [2]
f(r, p) = 0.

## History and Applications of Pedal Coordinates

The name pedal coordinates appears to be due to H.J. Purkiss, a Victorian-era student at the University of Cambridge, who proposed the name because “they are the polar coordinates of the foot of the perpendicular on the tangent” [3]. However, Purkiss didn’t discover the system. According to B. Pourciau [4] Newton uses pedal coordinates in Principia Mathematica Philosophica Naturalis to show the relationship between the inverse square law and a trajectory; the coordinate system is implied, not explicitly stated [5]. Pedal coordinates are more natural than Cartesian or polar coordinates in some settings, like the study of force problems of classical mechanics in the plane [1].

## References

Pedal curve image: Sam Derbyshire at English Wikipedia, CC BY-SA 3.0, via Wikimedia Commons
[1] Blaschke, P. PEDAL COORDINATES, DARK KEPLER AND OTHER FORCE PROBLEMS. Retrieved January 16, 2022 from: http://arxiv-export-lb.library.cornell.edu/pdf/1704.00897
[2] Yates, R. (1974). Curves and Their Properties. The National Council of Teachers of Mathematics. from: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.853.7639&rep=rep1&type=pdf
[3] The Oxford, Cambridge, and Dublin Messenger of Mathematics, Volume 3. Macmillan and Company, 1866.
[4] Pourciau, B. Reading the Master: Newton and the Birth of Celestial Mechanics.
[5] Olivier Bruneau. ICT AND HISTORY OF MATHEMATICS: the case of the pedal curves from
17th-century to 19th-century. 6th European Summer University on the History and Epistemology in Mathematics Education, Jul 2010, Vienna, Austria. pp.363-370. ffhal-01179909f