arch 655 parametric modeling in design wei yan, ph.d., associate professor college of architecture,...
TRANSCRIPT
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.