A strophoid of a circle with the pole at the center of the circle and the fixed point on the circumference of the circle. Freeth (1878, pp. 130 and 228) described this and various other strophoids (MacTutor Archive).
It has polar equation
(1)
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The area enclosed by the outer boundary of the curve is
(2)
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and the total arc length is
(3)
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(4)
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(OEIS A138498), where , is a complete elliptic integral of the first kind, is a complete elliptic integral of the second kind, and is a complete elliptic integral of the third kind.
If the line through parallel to the y-axis cuts the nephroid at , then angle is , so this curve can be used to construct a regular heptagon.
The curvature and tangential angle are given by
(5)
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(6)
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where is the floor function.