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This bachelor’s thesis gives a thorough introduction to geometric algebra (GA), an overview of conformal geometric algebra (CGA) and an application to the processing of single particle data from cryo-electron microscopy (cryo-EM).

The geometric algebra over the vector space , i.e. the Clifford algebra over an orthogonal basis of the space, is a strikingly simple algebraic construction built from the geometric product, which generalizes the scalar and cross products between vectors. In terms of this product, a host of algebraically and geometrically meaningful operations can be defined. These encode linear subspaces, incidence relations, direct sums, intersections and orthogonal complements, as well as reflections and rotations. It is with good reason that geometric algebra is often referred to as a universal language of geometry.

Conformal geometric algebra is the application of geometric algebra in the context of the conformal embedding of  into the Minkowski space . By way of this embedding, linear subspaces of  represent arbitrary points, lines, planes, point pairs, circles and spheres in . Reflections and rotations in  become conformal transformations in : reflections, rotations, translations, dilations and inversions.

The analysis of single-particle cryo-electron microscopy data leads to the common curves problem. By a variant of the Fourier slice theorem, this problem involves hemispheres and their intersections. This thesis presents a rewriting, inspired by CGA, into a problem of planes and lines. Concretely, an image in the Fourier domain is transformed by mapping points according to in suitable units. The inversive nature of this transformation causes certain issues that render its usage a trade-off rather than an unconditional advantage.