Crofton formula [SEP] named after
Crofton formula
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.
Statement.
Suppose formula_1 is a rectifiable plane curve. Given an oriented line "ℓ", let formula_2("ℓ") be the number of points at which formula_1 and "ℓ" intersect. We can parametrize the general line "ℓ" by the direction formula_4 in which it points
and its signed distance formula_5 from the origin. The Crofton formula expresses the arc length of the curve formula_1 in terms of an integral over the space of all oriented lines:
The differential form
is invariant under rigid motions, so it is a natural integration measure for speaking of an "average" number of intersections. The right-hand side in the Crofton formula is sometimes called the Favard length.
Proof sketch.
Both sides of the Crofton formula are additive over concatenation of curves, so it suffices to