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1886. On the three-cusped hypocycloid—Addendum*

Published online by Cambridge University Press:  15 September 2017

J. Hadamard
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Abstract

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Type
Mathematical Notes
Information
The Mathematical Gazette , Volume 30 , Issue 288 , February 1946 , pp. 48
Copyright
Copyright © Mathematical Association 1946 

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Footnotes

*

I ventured recently to recall to Professor Hadamard the fact that the asymptotes of any pencil of equilateral hyperbolas are also Simson lines of each triangle whose vertices are three base-points of the pencil. This provides an alternative approach to the generation of the hypocycloid treated in his paper in Math. Gazette, 29, 66-7, 1945.

Professor Hadamard thereupon sent me the following very concise proof of the identity of the asymptotes and Simpson lines and of the fact that they envelop a hypocycloid. He has now very kindly given me permission to communicate his letter to the Gazette

The two small footnotes have been inserted by me.

References

† These projections are on L, L', respectively, by definition of the Simpson lines.

‡ MM' and mm' being parallel diameters of the two circles.