Linear Algebra of Perspective Projections

by Andrew Hays Burns, Mathematics

Abstract: Imagine our eyes had the ability to take a snapshot of the space before us. The picture captures objects existing in our 3-dimensional reality, and it projects them onto a 2-dimensional plane. If we observe the snapshot, we notice that objects far from us appear small while objects near appear large. Why is this so? While taking Applied Linear Algebra, I launched into an exploration on the mathematics underlying this question. I learned that the phenomenon is modeled by perspective projections. These projections are ubiquitous in nature. They model how a realist draws a cube, how 2 highways converge into a horizon, and how a projector runs a movie in a theatre. Unraveling the math of these projections was a fun and methodical practice of compiling the natural phenomena into the language of linear algebra. Ultimately, it boils down to a point and a direction.

linear transformations, projections, parallelism, perspective

Acknowledgements: Thank you Prof. X and the rest of the UGA Mathematics faculty.

Citation Style: APA