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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

$\displaystyle x$ $\textstyle =$ $\displaystyle x_0-{g'(f'^2+g'^2)\over f'g''-f''g'}$  
$\displaystyle y$ $\textstyle =$ $\displaystyle y_0+{f'(f'^2+g'^2)\over f'g''-f''g'}.$  

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


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.

© 1996-9 Eric W. Weisstein