The astroid has the Cartesian equation:
where r is the radius of the fixed outer circle, and r/4 is the radius of the rolling circle. Its area is 3πr 2/8, or 3/2 times that of the rolling circle, and its length is 6r.
The astroid is a sextic curve and also a special form of a Lamé curve. It has a remarkable relationship with the quadrifolium (a special case of the rose curve): the radial, pedal, and orthoptic of the astroid are the quadrifolium, while the catacaustic (see caustic curve) of the quadrifolium is the astroid. The astroid is also the catacaustic of the deltoid and the evolute of the ellipse.
Related category• PLANE CURVES
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