ARCH 655

Parametric Modeling in Design

Wei Yan, Ph.D., Associate ProfessorCollege of Architecture, Texas A&M University

Lecture 5 Matrices and Transforms

Acknowledgements: most materials in the slides are copies from Prof. Donald House’ The Digital Image, and some other materials are copies from various online resources.

Affine Maps or Warps

• General form of an affine map

coefficients aij are constants

A geometric transformation that maps points and parallel lines to points and parallel lines

Affine Maps or Warps

they can be represented in matrix form

Homogeneous Coordinate System (every point has an identical third coordinate)

Affine Maps (Transformations)

Affine Maps (Transformations)

Affine Maps (Transformations)

Affine Maps (Transformations)

Recall TRIGONOMETRY Formula:

Composing Affine Warps

Let R be a rotation, S be a scale, and T a translation. Let’s do a rotation, followed by a scale, followed by a translation.

Composing Affine Warps

This is a wonderfully compact and unified way of constructing a largevariety of useful warps in a very intuitive way – i.e. by simple composition of easy to understand operations.

Matrix representation makes it simple for us to write and understand the transforms.

Example

Commutative?

Commutative?

Perspective Warps: Non-Affine Transformations

Provides 3D feeling

Giving depth illusion

Perspective Transformation

• Step1 - matrix multiplication, after which the third coordinate of the resulting point is not 1

Perspective Transformation

• Step 2, we need to restore our points to homogeneous coordinates with w = 1. We divide each vector by its own w coordinate

Example

If P is:1          0          00          1          0a          b          1

Then the vanishing points will be (1/a, 0) and (0, 1/b).

If a=0 and b=0 there will be no vanishing points (the image will be mapped to the same as the original). If one of them is 0, then there will be 1 vanishing point.

Vanishing Points

Affine v.s. Perspective (Forward map)

Affine: only one step

Affine v.s. Perspective (Forward map)

Perspective: two steps

Affine v.s. Perspective (Forward map)

Perspective: two steps

Projective Warps: Affine, Perspective or Composite of the two

Affine and Perspective are unified here: both use the division by w. Affine becomes a special case, where w=1.