This book presents a deep dive into the geometry of motion, which is developed in combination with tools of numerical analysis and then applied to significant problems from robotics and biomechanics. The exposition starts with basic notions from elementary algebra and geometry, and builds increasingly complex ideas with arguments that appeal largely to intuition and visualization. The aim of the book is not to simply present a set of mathematical concepts, but to help the reader understand their most intrinsic nature, and to establish a clear sense of how and why these abstractions are used in practical, challenging situations.