Geometric algebra, conformal geometry and the common curves problem

Sammanfattning: 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 Rp;q, 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 R3 into the Minkowski space R4;1. By way of this embedding, linear subspaces of R4;1 represent arbitrary points, lines, planes, point pairs, circles and spheres in R3. Reflections and rotations in R4;1 become conformal transformations in R3: 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.