Crofton formula

Remarks

Both sides are additive when the curve is cut into pieces, and both are invariant under rigid motions, since $dp \, d\theta$ is the invariant measure of integral geometry. Take a segment of length $\ell$ in the direction $\varphi$, and fix $\theta$. The lines $L(\theta, p)$ meeting it form an interval of $p$ of length $\ell \abs{\cos(\theta - \varphi)}$, and $\int_0^\pi \ell \abs{\cos(\theta - \varphi)} \, d\theta = 2 \ell$. So the formula holds for polygonal curves, and the general case follows by approximating $\gamma$ by inscribed polygons and passing to the limit. Example: a circle of radius $r$ meets $L(\theta, p)$ twice when $\abs{p} < r$, so the right side is $\frac{1}{2} \cdot \pi \cdot 2r \cdot 2 = 2 \pi r$.

Going deeper

For a student

For the boundary of a plane convex body, almost every line that meets the body crosses the boundary twice. So the perimeter equals $\int_0^\pi w(\theta) \, d\theta$, where $w(\theta)$ is the width in the direction $\theta$. This is Cauchy's formula, and it gives Barbier's theorem for curves of constant width. The formula explains Buffon's needle problem. It is the model for Crofton formulas in higher dimensions, which recover intrinsic volumes of convex bodies from intersections with random affine subspaces.