kokonaisgeometria

Kokonaisgeometria, meaning "total geometry" or "integral geometry" in Finnish, is a branch of mathematics that studies geometric properties that are invariant under certain transformations. It bridges differential geometry and probability theory, often dealing with the measurement of geometric objects and their intersections within a given space.

The core idea is to find quantities that remain unchanged when the geometric objects or the space itself are subjected to a specific group of transformations, such as translations, rotations, or more general motions. This invariance is crucial for understanding fundamental geometric characteristics that persist regardless of changes in position, orientation, or other transformations.

Integral geometry finds applications in various fields. In physics, it can be used in problems related to particle detection or imaging where the observed signals are integrals over geometric paths. In probability, it helps in calculating the likelihood of geometric events, such as the probability that a randomly placed line intersects a given shape. The study involves techniques from measure theory, differential forms, and harmonic analysis. It seeks to understand how geometric measures, like length, area, or volume, behave under transformations and how to define and compute probabilities related to geometric configurations.