Radial Curve

Let $C$ be a curve and let $O$ be a fixed point. Let $P$ be on $C$ and let $Q$ be the Curvature Center at $P$. Let $P_1$ be the point with $P_1 O$ a line segment Parallel and of equal length to $PQ$. Then the curve traced by $P_1$ is the radial curve of $C$. It was studied by Robert Tucker in 1864. The parametric equations of a curve $(f,g)$ with Radial Point $(x_0,y_0)$ are

$$x = x_0-{g'(f'^2+g'^2)\over f'g''-f''g'}$$

$$y = y_0+{f'(f'^2+g'^2)\over f'g''-f''g'}.$$

Curve	Radial Curve
Astroid	Quadrifolium
Catenary	Kampyle of Eudoxus
Cycloid	Circle
Deltoid	Trifolium
Logarithmic Spiral	Logarithmic Spiral
Tractrix	Kappa Curve

References

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 40 and 202, 1972.

Yates, R. C. ``Radial Curves.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 172-174, 1952.