Daniel Suo

Scientific progress goes 'boink'

Ph.D. Candidate
Princeton University
Department of Computer Science

2D and 3D Transformations

2D Linear transformations

  • Closed under combinations of scale, rotation, sheer, mirror (interestingly, not translation)
  • Linear transformations preserve
    • Origin
    • Lines
    • Parallelism
    • Ratios

Affine transformations

Affine transformations are a superset of linear transformations; they relax the requirement that the origin is preserved. Affine transformations, like linear transformations, are still closed under composition and preserve

  • Lines
  • Parallelism
  • Ratios

We can tell if a matrix represents an affine transformation if its last row contains all zeros except for a 1 in the last column. Physically, we interpret affine transformations as some combination of a linear transformation and a translation.

Projective transformations

Projective transformations are a superset of affine transformations. Parallel lines do not necessarily remain parallel and ratios are not preserved. We can tell if a matrix represents a projective transformation if its last row is not $[0,0,\ldots,1]$ as is required for an affine transformation.

Summary of transformations

From Szeliski’s book.

transformations.png