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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_1O 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(t),g(t)) with radial point (x_0,y_0) and parameterized by a variable t are given by

x=x_0-(g^'(f^('2)+g^('2)))/(f^'g^('')-f^('')g^')
(1)
y=y_0+(f^'(f^('2)+g^('2)))/(f^'g^('')-f^('')g^').
(2)

Here, derivatives are taken with respect to the parameter t.

curveradial curve
astroidquadrifolium
catenarykampyle of Eudoxus
cycloidcircle
deltoidtrifolium
logarithmic spirallogarithmic spiral
tractrixkappa curve

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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.

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Radial Curve

Cite this as:

Weisstein, Eric W. "Radial Curve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RadialCurve.html

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