2D Matrix Transformations Lecture Slides

2D Matrix Transformations
Consider a drawing made of lines and
polygons:
CS-184: Computer Graphics
Prof. James O’Brien
Consider another one:
Reuse of Shapes
Some butterflies same as others but moved
Others distorted, but mostly the same
Painful to do each one from scratch
Wastes memory
Representing Lines and Polygons
Representing Points
Just a list of points
May use indexed vertex lists
Points are vectors in
2
R
or
3
R
Relative to some origin and some coordinate axes
2
p = [2,4]
4
Origin, 0
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Basic Transformations
Uniform/isotropic
Non-uniform/anisotropic
Rotations
Cos(
θ ) − Sin(
θ )
p'
=
p
Sin(
θ ) Cos( θ )
Rotate
Scale
Rotate
Translate
Shear -- not really “basic”
Rotations
Rotations
Rotations are counter-clockwise
Right hand rule
Could do the other way, but we don’t
Preserve lengths/distance to origin
Orthonormal matrices
Det == 1, not –1
Note: rotate by zero degrees gives identity matrix
Note: rotate +/- 360 degrees stays the same
Scales
Scale
Uniform/isotropic
Non-uniform/anisotropic
sx
0
p'
= p
0 s
y
Scales
If Sx and Sy are the same scale is uniform
Scaling by Sx or Sy < 0 causes problems
If both are –1, it looks like 180 degree rotation
2

Shears
Translation ( method 1)
Shear
1 H
yx
p'
= p
H
xy
1
Translate
t
p' = p +
t
x
y
True, but not the right way to think about shears….
Does not look like the others…..
Arbitrary matrix transformation
For everything but translation we have:
p ' = Ap
In a short while, we will make translation fit in too.
What does an arbitrary A matrix mean
Singular Value Decomposition
(or polar decomposition)
For any matrix A, we can write SVD:
A = QSR
where Q and R are orthonormal and S is diagonal
Also write Polar Decomposition:
A = QRSR
(Different Q! Still orthonormal.)
Decomposing matrices
We can force Q and R to have Det=1 so they are
rotations.
Any matrix is now:
– Rotation:Rotation:Scale:Rotation
– That means shear is not primitive (in some sense)
Multiple transformations
What does something like the following mean
p ' = BAp
“Apply A to p and then apply B to the result.”
p ' = B(
Ap)
= ( BA)
p = Cp
Lots of transformations all get composted into one.
But translations are being left out…
3

Translations again
We could do this:
p ' = B(
Ap + t)
= BAp + Bt = Cp + u
Ugly and awkward
Some systems do it anyhow.
Every transformation would be a matrix and a
vector…
Homogeneous coordinates
Take a vector that represents a point and
append a 1 at the end
p
p =
p
x
y
p
p~
=
p
1
x
y
Homogeneous translation
1
p
~ ' =
0
0
0
1
0
p
~ ~
' = Ap
~
tx
p
t
y
p
1
1
x
y
The tildes are for clarity. If only using homogeneous coordinates,
we would omit them.
Other transformations
~ A
A =
0
0
Now everything looks the same…
0
0
1
Composing matrices
Example
When we rotate, we rotate about the origin
Same for scale
How could we do it about another point
versus
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Example
Example
Step 1: Translate point you want to
rotate about to the origin
Step 2: Rotate as desired
Translate (-C)
Rotate (θ)
Example
Step 3: Put back where it was
Scaling about an arbitrary axis
•Diagonal matrices are only axis-aligned scales
Translate (C)
p
~
' = ( −T)
RTp
~
= Ap
~
Scaling about an arbitrary axis
1. Translate so that axis passes through origin
Scaling about an arbitrary axis
2. Rotate axis to align with X or Y axis
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Scaling about an arbitrary axis
3. Scale
Scaling about an arbitrary axis
4. Un-do rotation
5. Un-do translation
Trivia: scale matrices (w/o translation) are symmetric
R SR
Scene Hierarchy
Point Vectors and Direction Vectors
Linearity
Ray equations
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