On the Three-Cusped Hypocycloid

Extract

I have had a recent opportunity to recall an early article (1884) which I wrote on the three-cusped hypocycloid. My starting point was the property that the asymptotes of any pencil of equilateral hyperbolas envelop such a hypocycloid. I proved this analytically in the aforesaid article ; perhaps there is some interest in finding geometrical reasons for it.

Principles on pencils of conies are well known. According to these principles :

• (1) The polars of any point a with respect to the various conies of the pencil are concurrent at one and the same point a, which we shall call the corresponding point of a.

• (2) If a describes a straight line D, then a. describes a certain conic C.

• (3) This conic C is also the locus of the poles of D with respect to the conies of the pencil, a consequence being:

• (4) If m, a point of C, is the pole of D with respect to one of the conies H of the pencil and a a point of D with the corresponding point α, then the polar line of a with respect to H is mα.