Mathematical Foundations of VR/AR

Abstract

In Virtual Reality and Augmented Reality, mathematical methods offer fundamental principles to model three-dimensional space. This makes it possible to provide exact information and perform calculations, e.g., to determine distances or to describe the effects of transformations such as rotations or translations exactly. This chapter compiles the most important mathematical methods, especially from linear algebra, that are frequently used in VR and AR. For this purpose, the term vector space is defined and extended to a Euclidean space. Afterwards, some basics of analytic geometry are introduced, especially the mathematical description of lines and planes. Finally, changes of coordinate systems as well as affine transformations are discussed and their computation with matrices in homogeneous coordinates is explained.

Dedicated website for additional material: vr-ar-book.org