1. Affine Transformations
Jim Van Verth
Lars M. Bishop

2. Affine Transformations
Extension of linear transforms
Lines preserved
Distances and angles can change
Transform in an affine space
Almost all apply to both vectors and points – exception is translation
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3. Affine Transformations
The most common ones are translation, scale and rotation
Translation moves an object along a direction vector
Scale shrinks/grows the object size
Rotation rotates the object around an axis vector
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4. Affine Transformations
Can’t use 3x3 matrix for R3 R3 affine xforms
Perform linear transforms, not affine transforms
Need another representation
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5. Essential Math for Games
What’s Wrong With A 3x3?
Can’t construct a 3x3 matrix that will translate the point (0,0,0)
The zeroes in the coordinate vector cancel out all the coefficients!
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6. Homogeneous Coordinates
One way is to use homogeneous coordinates
Initially developed in the 40s for use in geometry
Later found their way into computer graphics
Allow us to perform affine transforms via matrices
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7. Homogeneous Coordinates
Convert point in Rn to one of equivalent points in homogeneous space RPn
Point in RPn has n+1 coordinates
So point in R3 (x y z) has corresponding points in RP3 (x’ y’ z’ w)
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8. Homogeneous Coordinates
How to convert an R3 coordinate to its homogeneous representation
Pick a number for the fourth element, w, and scale the real coordinate by it
(x y z) => (xw yw zw w)
Standard value for w is 1, so:
(x y z) => (x y z 1)
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9. Homogeneous Coordinates
Map back to R3 by dividing out the w
(x' y' z' w) => (x'/w y'/w z'/w)
Usually assume w = 1, so just drop the w
(x' y' z' 1) => (x' y' z')
But not always, so be careful!
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10. Essential Math for Games
Homogeneous Matrices
N-dimensional affine transformation is n+1 dimensional linear transformation
A 3D homogeneous transformation matrix is a 4x4 matrix
Treat 3D homogeneous point as a 4D vector
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11. Translation Revisited
Remember translating the point (0,0,0)?
No problem; right column is translation vector
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12. Translation Txyz(a,b,c)
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13. Essential Math for Games
Transforming Point
Can represent affine transform as block matrix multiplication
Transform point P = (x, 1), get
So y is translation part, what does A represent?
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14. Essential Math for Games
Transforming Vectors
Represent vector with w = 0
(x, y, z) => (x, y, z, 0)
Can also write as
v => (v, 0)
Think of as point at infinity
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15. Essential Math for Games
Transforming Vectors
Apply affine transform to vector, get
Applies linear transform A to vector
So linear transforms are subset of affine transforms
Note no translation: makes sense, vectors have no position!
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16. Other Affine Transformations
Remaining affine transformations are also linear transformations
Can be applied to vectors as well
Notice that the top-left 3x3 block in the translation example was the identity?
Just place the linear transform in the upper-left 3x3 block, as follows…
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17. Essential Math for Games
Scaling Sxyz(a,b,c)
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18. Rotating About the x-axis Rx()
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19. Rotating About the y-axis Ry()
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20. Rotation About the z-axis Rz()
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21. Essential Math for Games
Rotation Matrices
Have some useful properties
Are orthogonal, i.e., R-1 = RT
Column vectors are transformed axis of old coordinate system
We’ll see how we can use this later
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22. Concatenation of Transformations
Transforms concatenated into a single matrix by multiplying them together, in the order they are to occur
Can then use single matrix to perform transforms on many points
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23. Essential Math for Games
Concatenation
Can perform a series of transformations
v’ = M1  v
v’’ = M2  v’
w = M3  v’’
Substitute to get
w = M3  M2  v’
w = M3  M2  M1  v
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24. Essential Math for Games
Concatenation
Can combine series into one matrix that performs all of them
M = M3  M2  M1
w = M  v
Example: Euler rotations
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25. Essential Math for Games
Euler Angles
Common representation for 3D orientation (common  best)
Three Euler angles:
Yaw = shaking your head, “no”
Pitch = nodding your head, “yes”
Roll = cocking your head sideways
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26. Essential Math for Games
The Euler Transform
Each angle corresponds to a rotation about an axis (dependent on coordinate system) z x y
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27. Essential Math for Games
The Euler Transform
Looking down the x-axis:
Yaw = Rz
Pitch = Ry
Roll = Rx z x y
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28. Essential Math for Games
The Euler Transform
For this coordinate system, the Euler Transform E
E(y,p,r) = Rx(r) Ry(p) Rz(y)
Again, E is orthogonal (it’s a pure rotation matrix)
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29. Concatentation of Transforms
Can concatenate any sequence of scale, rotation, translate
Matrix products are non-commutative, so the order matters
Translating the point (0, 0, 0) and then rotating will not yield the same point as rotating (0, 0, 0) then translating
(rotation of (0, 0, 0) does nothing)
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30. Concatentation of Transforms
Similarly, scaling and rotation are non-commutative
Example:
rotate (1 0 0) by 90° around z, scale by (sx, sy, sz), get (0 sy 0)
scale (1 0 0) by (sx, sy, sz), rotate (1 0 0) by 90° around z, get (0 sx 0)
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31. Essential Math for Games
Transformation Order
To perform scale, then rotation, then translation (generally the order desired), use
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32. Essential Math for Games
Matrix Breakdown
Rotation, scale in upper left 3x3
Translation rightmost column
Does rotation/scale first, then translate
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33. Essential Math for Games
Better Format
Problem: want to
Translate object in space and change rotation and scale arbitrarily
Handle rotation/scale separately
Can’t do easily with 4x4 matrix format
Involves SVD, Polar decomposition
Messy, but we can do better using…
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34. Essential Math for Games
Rigid Body Transforms
Any sequence of translation and rotation transformations
Not scale or shear
Object shape is not affected (preserves angles and lengths)
Usually include uniform scale despite this
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35. Essential Math for Games
Better Format
Scale – one uniform scale factor s
Rotation – 3x3 matrix R
Translation – single vector t
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36. Essential Math for Games
Better Format
Want to concatenate transforms T1, T2 in this form, or
Do this by
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37. Essential Math for Games
Better Format
Advantages
Clear what each part does
Easier to change individual elements
Concat: 39 mult 27 add vs. 48 mult 32 add
Disadvantages
Eventually have to convert to 4x4 matrix anyway for renderer/video card
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38. Essential Math for Games
Better Format
Matrix conversion
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39. Inverting Rigid Body Xforms
We can easily invert our rigid body transforms symbolically:
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40. Inverting Rigid Body Xforms (2)
In fact, the result itself may be written as a rigid body transform:
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41. Essential Math for Games
References
Rogers, F. David and J. Alan Adams, Mathematical Elements for Computer Graphics, 2nd Ed, McGraw-Hill, 1990.
Watt, Alan, 3D Computer Graphics, Addison-Wesley, Wokingham, England, 1993.
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