[math-fun] astroid: 3 recipes and a question

For a circle of radius r rolling inside the unit circle: 1) r=1/4: trajectory of point on circumference. 2) r=3/4: trajectory of point on circumference. (opposite dir) 3) r=1/2: boundary of area swept by diameter. All draw x^(2/3) + y^(2/3) = 1. What density function rho(r) must a planet have so that an astroid is a brachistochrone? --rwg