# 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.

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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.

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

## Details

References

[1] E. H. Lockwood, A Book of Curves, Cambridge, England: Cambridge University Press, 1961. https://www.cambridge.org/core/books/book-of-curves/F08B52C8FB0563B2F9866DA186FC87F1.

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

## Permanent Citation

Jan Mangaldan

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