Daniel Suo

Scientific progress goes 'boink'

Ph.D. Candidate
Princeton University
Department of Computer Science

3D to 2D projections

We often want to know how to project 3D objects onto a 2D plane (e.g., display an image on a screen).

  • Orthographic Drop the $z$ component of any three-dimensional point $p$ to obtain the two-dimensional point $x$. We can write this transform as:

    If we use homogenous, or projective coordinates $\tilde{x}$ and $\tilde{p}$:

    Orthographic projections are fixed-depth.

  • Perspective The perspective projection divides the $x$ and $y$ components by the $z$ component, weighting them by distance to the camera. We can write this transform using homogenous coordinates as:

    where we drop the $w$ coordinate. Note that we do not divide by $z$ because homogenous coordinates are equivalent by a multiple.