Modeling perspective projections in 3-dimensions by rotations in 4-dimensions

We show how to represent perspective projections in 3-dimensions using rotations in 4-dimensions. This representation permits us to replace classical singular 4 × 4 matrices for perspective projection with nonsingular 4 × 4 orthogonal matrices. This approach also allows us to compute perspective projections by sandwiching vectors between two copies of a unit quaternion. In addition to deriving explicit formulas for these 4 × 4 rotation matrices for perspective projection, we also explain the geometric intuition underlying the observation that perspective projections in 3-dimensions can be represented by rotations in 4-dimensions. We show too that every rotation in 4-dimensions models either a rotation, a reflection, a perspective projection, or one of their composites in 3-dimensions.

Graphical abstractFigure optionsDownload full-size imageDownload as PowerPoint slideHighlights► How to represent perspective projections in 3-dimensions using rotations in 4-dimensions. ► How to replace classical singular 4 × 4 matrices for perspective projection with nonsingular 4 × 4 orthogonal matrices. ► How to compute perspective projections by sandwiching vectors between two copies of a unit quaternion.