How can geometric objects and operations be represented in such a way that they can be processed through the simplest possible algebraic manipulations? This is the guiding question of this book, which is explored from various perspectives over the course of 12 chapters. From this viewpoint, introductions to projective geometry, geometric invariant theory, Euclidean geometry (with a special focus on complex numbers), Möbius geometry, and Lie's circle geometry are provided. The emphasis is on the elegance of the methods, which often leads automatically to elegant algorithmic approaches. For the reader, the book serves as a bridge from fundamental knowledge in linear algebra to modern (and classical) approaches to geometry. Each chapter is supplemented with numerous exercises, illustrations, and interactive visualizations available online.