Spatial Alignments

This chapter considers the positioning of panoramic sensors or panoramas in 3D space, defining and using world or local (e.g., camera or sensor) coordinate systems. The chapter also specifies coordinates and locations of capturing surfaces or panoramic images. The chapter starts with a few fundamentals for metric spaces, coordinate system transforms, or projections from 3D space into panoramic images, briefly recalling mathematical subject areas such as linear algebra, projective geometry, and surface geometry.

3.1 Mathematical Fundamentals

It is assumed that the reader already knows basic mathematical concepts such as typically taught in first or second year university classes. This section recalls such concepts and prepares for the formal discussion of panoramic sensors, but will not start at basic mathematical levels. For example, it is assumed that the reader is already familiar with basic vector and matrix algebra, or homogeneous coordinates (introduced in the 19th century by A. F. Möbius, and today of crucial importance, for example, in computer graphics or vision).

This section also specifies the (default) notation used in this book.1 For example, by default we use Greek letters for angular values, and bold letters for points, vectors, or matrices. is the set of all reals and is the set of all integers.

3.1.1 Euclidean Spaces and Coordinate Systems

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