Bernoulli-Euler Double Generation Theorem

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This Demonstration generates a circle roulette in 3D in two equivalent ways, in accordance with the "double generation" theorem of Euler and Bernoulli. We get a hypotrochoid or epitrochoid, depending on whether the rolling circle is inside or outside the fixed circle.


The Bernoulli–Euler double generation theorem for circle roulettes can be stated as follows. Let the pitch (or fixed) circle have radius of 1. Then:

1. An epicycloid generated by a rolling circle of radius r is equivalent to a hypocycloid (or pericycloid) with a rolling circle of radius .

2. A hypocycloid generated by a rolling circle of radius r is equivalent to a hypocycloid with a rolling circle of radius 1-r.

The theorem can be suitably generalized to epitrochoids and hypotrochoids, in which the point mounted on the rolling circle is not necessarily placed at the circle's rim.


Contributed by: Jan Mangaldan (June 13)
Open content licensed under CC BY-NC-SA



[1] E. H. Lockwood, A Book of Curves, Cambridge, England: Cambridge University Press, 1961.

[2] R. C. Yates, A Handbook on Curves and Their Properties, Washington, DC: National Council of Teachers of Mathematics, 1952.


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