Crofton's Integrals

Consider a convex plane curve K with perimeter L, and the set of points P exterior to K. Further, let t_1 and t_2 be the perpendicular distances from P to K (with corresponding tangent points A_1 and A_2 on K), and let omega=∠A_1PA_2. Then

 int_(P ext. to K)(sinomega)/(t_1t_2)dP=2pi^2

(Crofton 1885; Solomon 1978, p. 28).

If K has a continuous radius of curvature and the radii of curvature at points A_1 and A_2 are rho_1 and rho_2, then

 int_(P ext. to K)(sinomega)/(t_1t_2)rho_1rho_2dP=1/2L^2

(Solomon 1978, p. 28), and furthermore

 int_(P ext. to K)(sinomega)/(t_1t_2)(rho_1+rho_2)dP=2piL

(Santaló 1953; Solomon 1978, p. 28).

See also

Crofton's Formula

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Crofton, M. W. "Probability." Encyclopaedia Britannica, 9th ed., Vol. 19. Philadelphia, PA: J. M. Stoddart, pp. 768-788, 1885.Santaló, L. Introduction to Integral Geometry. Paris: Hermann, 1953.Solomon, H. Geometric Probability. Philadelphia, PA: SIAM, 1978.

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Crofton's Integrals

Cite this as:

Weisstein, Eric W. "Crofton's Integrals." From MathWorld--A Wolfram Web Resource.

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