# 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. 