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+---- Thread: Crofton formula (wikipedia) (/showthread.php?tid=434)
Crofton formula (wikipedia) - mklabgr - 06-16-2026
Summary
The Crofton formula is a fundamental result in integral geometry that connects the length of a curve with the average number of times the curve is intersected by randomly chosen lines. Named after the mathematician Morgan Crofton, the formula shows that instead of measuring a curve directly, its length can be calculated by integrating the number of intersections over all possible lines in the plane.
It provides a bridge between geometry and probability and has important applications in areas such as geometric measure theory, computer vision, tomography, and the study of convex bodies. The formula also leads to elegant proofs of classical results like the isoperimetric inequality, Barbier’s theorem, and relationships between surface area and projections in higher dimensions.
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