The properties of a hypocycloid were recognized by James White, an.    From Wordnik.com.  [Kinematics of Mechanisms from the Time of Watt] Reference

The result is a series of enormous hypocycloid designs which recorded the hidden patterns created by the ride as it turned.    From Wordnik.com.  [Boing Boing] Reference

Thus in Fig. 39, the diameters of the two pitch circles are to each other as 4 to 5; the hypocycloid has 5 branches, and 4 pins are used.    From Wordnik.com.  [Scientific American Supplement, No. 470, January 3, 1885] Reference

The original hypocycloid is shown in dotted line, the working curve being at a constant normal distance from it equal to the radius of the roller; this forms a sort of frame or yoke, which is hung upon cranks as in Figs. 36 and 38.    From Wordnik.com.  [Scientific American Supplement, No. 470, January 3, 1885] Reference

Upon examination it will be seen, although we are not aware that attention has previously been called to the fact, that this differs from the ordinary forms of "pin gearing" only in this particular, viz., that the elementary tooth of the driver consists of a complete branch, instead of a comparatively small part of the hypocycloid traced by rolling the smaller pitch-circle within the larger.    From Wordnik.com.  [Scientific American Supplement, No. 470, January 3, 1885] Reference

Sj-hypocycloid created December 31, 2008, last edited January 01.    From Wordnik.com.  [ArmchairGM Popular Stuff] Reference

Following the analogy, it would seem that the next step should be to employ two branches with only one pin; but the rectilinear hypocycloid of Fig. 38 is a complete diameter, and the second branch is identical with the first; the straight tooth, then, could theoretically drive the pin half way round, but upon its reaching the center of the outer wheel, the driving action would cease: this renders it necessary to employ two pins and two slots, but it is not essential that the latter should be perpendicular to each other.    From Wordnik.com.  [Scientific American Supplement, No. 470, January 3, 1885] Reference

It is self-evident that the hypocycloid must return into itself at the point of beginning, without crossing: each branch, then, must subtend an aliquot part of the circumference, and can be traced also by another and a smaller describing circle, whose diameter therefore must be an aliquot part of the diameter of the outer pitch-circle; and since this last must be equal to the sum of the diameters of the two describing circles, it follows that the radii of the pitch circles must be to each other in the ratio of two successive integers; and this is also the ratio of the number of pins to that of the epicycloidal branches.    From Wordnik.com.  [Scientific American Supplement, No. 470, January 3, 1885] Reference

Take in succession as radii the chords A 1, A 2, A 3, etc., of the describing circle, and with centres 1, 2, 3, etc., on the base circle, strike arcs either externally or internally, as shown respectively on the right and left; the curve tangent to the external arcs is the epicycloid, that tangent to the internal ones the hypocycloid, forming the face and flank of a tooth for the base circle.    From Wordnik.com.  [Mechanical Drawing Self-Taught] Reference

Should the junction of two of these arcs fall within the breadth of a tooth, as at D, evidently both the face and the flank on one side of that tooth will be different from those on the other side; should the junction coincide with the edge of a tooth, which is very nearly the case at F, then the face on that side will be the epicycloid belonging to one of the arcs, its flank a hypocycloid belonging to the other; and it is possible that either the face or the flank on one side should be generated by the rolling of the describing circle partly on one arc, partly on the one adjacent, which, upon a large scale, and where the best results are aimed at, may make a sensible change in the form of the curve.    From Wordnik.com.  [Mechanical Drawing Self-Taught] Reference