Definitions
from The American Heritage® Dictionary of the English Language, 4th Edition
 n. The plane locus of a point fixed on a circle that rolls on the inside circumference of a fixed circle.
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. The locus of a point on the circumference of a circle that rolls without slipping inside the circumference of another circle.
from the GNU version of the Collaborative International Dictionary of English
 n. A curve traced by a point in the circumference of a circle which rolls on the concave side in the fixed circle. Cf. epicycloid, and trochoid.
from The Century Dictionary and Cyclopedia
 n. In geometry, a curve described by a point on the circumference of a circle which rolls upon the inside of another circle.
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 n. a line generated by a point on a circle that rolls around inside another circle
Etymologies
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Examples

The result is a series of enormous hypocycloid designs which recorded the hidden patterns created by the ride as it turned.

The properties of a hypocycloid were recognized by James White, an

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.

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.

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 pitchcircle within the larger.

Sjhypocycloid created December 31, 2008, last edited January 01,

It is selfevident 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 pitchcircle; 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.

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.
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