Kids—Cabochon area SPOILER: The Wikipedia Epicycloid and Hypocycloid articles give parametric equations *x*(θ) and *y*(θ). Then ½∲(*x dy - y dx*) = ½ ∫ (*x*(*θ*) *dy/dθ* - *y*(*θ*) *dx/dθ*) *dθ*. (Mnemonically, "The (*dθ*)s cancel.") If the big circle has radius* R* = 1 and, for *n* cusps, the small circle has radius *r* = 1/*n*, the area integrals come out (1+1*/n*)(1+2/*n*)π and (1-1*/n*)(1-2/*n*)π . (Note the circular case when *n*➝∞.) So a hypocycloid is like an epicycloid with -*n* cusps, and vice versa! And the area of *n* cabochons is 6π/*n*. For *n* = 4, each cabochon is 3π/8, = the area of the whole astroid! —rwg (But why does a mumblecycloid with *n* = 0 cusps have infinite area?) On Thu, Nov 7, 2019 at 9:58 AM Bill Gosper <billgosper@gmail.com> wrote:

https://upload.wikimedia.org/wikipedia/commons/d/da/Rolling_Hypocycloids.gif This is the decacuspid (n=10) case. What happens when n➝∞? Why do all the curves coincide only at 3 o'clock and 9 o'clock? Why isn't https://en.wikipedia.org/wiki/Epicycloid nearly as complete as https://en.wikipedia.org/wiki/Hypocycloid? Telling us, e.g., the area of Julian's <http://gosper.org/tripenta.gif> trisectrix (astroid) but not the pentasectrix (quadricuspid epicycloid). Kids: I somehow missed (or completely forgot) the cool integral for the area of the curve enclosed by {x(t),y(t)}: ½∲(x dy - y dx). Supposing a unit radius, what's the area of one of those four "cabochons"? —rwg