Crofton Formula
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, the Crofton formula, named after Morgan Crofton (1826 1915), is a classic result of integral geometry relating the length of a curve to the expected number of times a "random" line intersects it.Suppose is a rectifiable plane curve. Given an oriented line l, let n (l) be the number of points at which and l intersect. We can parametrize the general line l by the direction in which it points and its signed distance p from the origin. Both sides of the Cr...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, the Crofton formula, named after Morgan Crofton (1826 1915), is a classic result of integral geometry relating the length of a curve to the expected number of times a "random" line intersects it.Suppose is a rectifiable plane curve. Given an oriented line l, let n (l) be the number of points at which and l intersect. We can parametrize the general line l by the direction in which it points and its signed distance p from the origin. Both sides of the Crofton formula are additive over concatenation of curves, so it suffices to prove the formula for a single line segment. Since the right-hand side does not depend on the positioning of the line segment,
