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NCB  Deposit # 12


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The Hippopede of Proclus

Dr. Adam Coffman

Department of Mathematical Sciences
Indiana University - Purdue University Fort Wayne 

Hippopede illustration


"Hippopede" means the foot of a horse and/or the shackle attached to the foot of a horse.  The shackle is called a "fetter." 

1.  The implicit equation in the  xy-plane is
where a and b are positive constants.  This is a curve with reflectional symmetry about both the horizontal and vertical axes.

2.  Any Hippopede is the intersection of a torus with one of its tangent planes that is parallel to its axis of rotational symmetry, as demonstrated in the animation.

3.  Converting the implicit equation to polar coordinates gives

Polar equation
so the origin at  r =  0 is a solution, and the remainder of the curve is given by

4.  If  0 < ba,  the point at the origin is an isolated node and the balance of the solution set is a simple closed curve, also called an  Elliptic Lemniscate of  Booth.
The  b  <  a  special case is given by the rational parametrization


Each of these curves is the image of an ellipse under an inversion in a circle.  The circle and the ellipse must have the same center.  The curve is non-convex for b  <  a <  2b,  and convex for  a >  2b when the shape is oval, but is not exactly a strictly defined ellipse.

5.  If  ab, the quartic implicit equation factors into two quadratics;  thus, the curve is the union of two circles, centered at  ( - b,0 ) and  ( b, 0 ), each of radius b, and mutually tangent at the origin.


6.  If  0 <  ab,  the curve intersects itself at the origin, and it is also called a Hyperbolic Lemniscate of  Booth.This is given by the rational parametrization


Each of these curves is the image of a hyperbola under an inversion in a circle.  The circle and the hyperbola must have the same center. 

The animation.
Hippopede animation

Now that you have investigated the equations and read of the special properites, we suggest you use the "Reload" or "Refresh" button at the top of your computer.  This will replay the animation.  Visually "slice" the torus to see the plane lemniscate evolve into a pair of circles, and then the oval shape.

planar Hippopede

planar Hippopede

Circle to lemniscate
Proclus (410 - 485 A.D. ) was a Greek mathematician who is best known for his  Commentaries on the First Book of Euclid's Elements.  This monumental work is often our only historical source on ancient Greek mathematics dating from Thales to Euclid. 

Though educated in Alexandria, he spent most of his highly productive life as head of the Academy of Plato in Athens.   In the tradition of his time, Proclus was buried near his teacher and mentor, and not with his family.

The modern student should realize the notation and graphing we use today were unknown to Proclus.  Still, curiosity led to investigation of highly sophisticated curves.

Circle to lemniscate illustration

Deposit # 12 Links

For other animations from Coffman, see  < >
         Coffman used Maple for the calculations and the raytracer Persistence of Vision, or PoV for short, to generate the graphics.

For more on Proclus, a.k.a. Booth curves, see < >.

For the Hippopede of Eudoxus,  see <  >.

Lawrence, J. D.,  A CATALOG OF SPECIAL PLANE CURVES,  Dover Publications, 1972.  See pp.144-146.


Note:  The Hippopede of Proclus should not be confused with the Hippopede of Eudoxus (c. 408 - 355 B.C.), a much earlier investigation.

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