The purpose of this talk is to give an introduction to rotational integral geometry and to exemplify a number of its core results and their applications. Integral geometry, introduced by Blaschke in the 1930s, is the theory of invariant measures on geometric spaces (often Grassmannians) and its application to determine geometric probabilities.

We will start by recalling the kinematic Crofton formula, which allows one to retrieve certain geometric characteristics (such as volume, surface area and other intrinsic volumes) of a compact convex set $K$ in $\mathbb R^n$ from intersections with invariantly integrated $k$-dimensional affine subspaces, where $k=0,\ldots,n-1$ is fixed.

Motivated by applications from biology, we suggest a number of variants of Crofton's formula, where the intersecting affine spaces are constrained, for instance to contain the origin (and hence are just linear subspaces) or even are all required to contain a fixed lower-dimensional axis. Corresponding rotational Crofton formulae will be established and explained.

This talk is based on collaboration with E. Dare and on the Springer Monograph in Mathematics Rotational Integral Geometry and its Applications (2025) by Eva B. Vedel Jensen and myself.