It is often necessary to handle randomness and geometry in computer vision, for instance to match and fuse together noisy geometric features such as points, lines or 3D frames, or to estimate a geometric transformation from a set of matched features. However, the proper handling of these geometric features is far more difficult than for points, and a number of paradoxes can arise. We try to establish in this article the basic mathematical framework required to avoid them and analyze more specifically three basic problems: \begin{itemize} \item what is a random distribution of features, \item how to define a distance between features, \item and what is the «mean feature» of a number of feature measurements~? \end{itemize} We insist on the importance of an invariance hypothesis for these definitions relative to a group of transformations. We develop general methods to solve these three problems and illustrate them with 3D frame features under rigid transformations. The first problem has a direct application in the computation of the prior probability of false match in classical model-based object recognition algorithms, and we present experimental results of the two others for a data fusion problem: the statistical analysis of anatomical features (extremal points) automatically extracted on 24 three dimensional images of the head of a single patient. These experiments successfully confirm the importance of the rigorous requirements presented in this article.