In texture analysis intensities of a diffraction experiment are recorded as experimentally accessible pole figures which are modeled as a "pole figure projection" of an orientation probability density function. The recovery of the orientation density function by means of inversion of the pole figure projection has been a major issue ever since its origin as it permits the approximate numerical determination of possibly anisotropic macroscopic properties. Major publications in texture analysis have usually stressed the uniqueness of the mathematics involved in its inversion. problem. In the mathematical discipline of integral geometry the Radon transform is a favorite subject of concern. It associates mean values with respect to lower dimensional manifolds to a function defined on some multidimensional manifold. In particular, it developed methods to recover functions defined on Euclidean spaces, hyperbola and spheres, from their Radon transforms. Except for applications in medical technologies mathematicians are often not aware of other routine applications in engineering. In our exposition on one hand we demonstrate the uniqueness of the mathematics involved but also the existence of parallel developments in texture analysis and in integral geometry which have not been aware of each other but can largely benefit of each other. More precise, the pole figure projection of texture analysis as a Id Radon transform of the group SO(3) is equivalent to the 1d Radon transform of even functions defined on the 3d sphere in 4d Euclidean space. Exploiting the geometry of the diffration experiment of texture analysis in terms of quater-nions, the equivalence of the approaches as well as of its results, especially of its "inversion formulae" is proven. Thus, mathematics is proven to apply usefully to problems of advances texture analysis.