Hippopede: Definition, Examples

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A hippopede is a closed curve with the polar equation [1]

r2 = 4b(absin2θ).

Or, alternatively by the implicit equation

(x2 + y2)2 = cx2 + dy2.

The hoppopede, defined implicitly with c=16(red), 13(green), and 10(blue).

When d > 0, the curve becomes an oval and is called an Oval of Booth. When d < 0, the curves resembles a figure and is sometimes called the lemniscate of Booth. When d = -c, the hippopede is equivalent to the lemniscate of Bernoulli.

In three dimensions, the hippopede is the intersection of a cylinder and a sphere; you can think of it as a figure 8 drawn on a sphere [2]. The cylinder pierces the sphere; from the inside, the cylinder is tangent to the sphere.

3d hippopede
A hippopede created by the intersection of a cylinder and a sphere.

History of the Hippopede

The Greek mathematician Eudoxus of Cnidus (40 to 355 BC) made the earliest known attempt to model planetary motion; he invented a scheme of nested concentric spheres; when a planet attached to both spheres rotated around different axes in opposite directions, it would trace out a hippopede retrograde motion. Eudoxus’s hippopedes were fixed in size and shape, so if you changed the parameters to match a planet’s speed during retrograde, you couldn’t match its arc (or vice versa) [3].

The hippopede is also called a horse fetter, because it looks like the loop of ropes used for restraining a horse’s feet (alternatively, a horse fetter is a training exercise to keep a horse from leaning in one direction). It was Eudoxus who gave the curve the name hippopede; horse fetters were a favorite practice in riding schools in ancient Greece [4]. It’s uncertain though, whether Eudoxus named the shape after the riding practice or the tethering device.


Curves created with Desmos.com.
[1] Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 144-146, 1972.
[2] Neugebauer, O. (2012). A History of Ancient Mathematical Astronomy. Springer Berlin Heidelberg.
[3] To Save the Phenomena.
[4] Dreyer, J. (1906). History of the Planetary Systems from Thales to KEpler..

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