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Composite Transformations

Sources:* D. Brackeen. Game Programming in Java.* D. Hearn & M. P. Baker. Computer Graphics: C edition* D. Shreiner, M. Woo, J Neider, T. Davis. OpenGL Programming

Guide: 5th Edition.

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Composite Transformations

How to combine several operations? Ex: rotate, then translate. Or the reverse: translate, then rotate.

Or two rotations: rotate this amount, then thatamount Or even more complex: scale, then rotate, then

translate

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Example

Suppose we have a point or vector v We want to rotate, then translate

Let R be rotation matrix, T be translation matrix. We would have:

v = ...;vr = R * v;vtr = T * vr;

Or, all in one step: vtr = T * (R * v) Or, more simply: vtr = T*R*v

Notice operations are applied right to leftas we read it R first Then T

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Example

Matrix multiplication is associative! Can move parentheses around however we want! So T * (R * v) is the same as (T*R) * v So what? Suppose we have lots of points to transform Instead of:

for p in pointlist:p_ = T*R*p

...do something... We do:M = T*Rfor p in pointlist:

p_ = M*p

...do something...

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Examples

Scale an object NOT at origin... Translate to origin (so it doesn't "fly away") Scale Translate back

Let S be scale matrix, T be translate matrix, T2 bereverse translation matrix

p_ = T2STp

Note: if we did it the other way: TST2v

it wouldn't work! Matrix multiplication is NOT commutative!!! Again, when we read it, it's applied in RIGHT to

LEFT order!

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Order of Transformations

Another way of looking at it: Suppose we have:

p_ = T2STp

Rewrite with parens (multiplication is associative):

p_ = (T2(S(Tp))) Compute Tp : Translate to origin Then multiply result with S: Scale it Then multiply result with T

2: Move back to original

place

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Composites

Makes a difference what order we use! Ex: Rotate, then translate

v' = TRv

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Composites

Ex: Translate, then rotatev' = RTv

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Composite Transformations

Example: rotate object 45 degrees around the point (5,7) Translate object by -5,-7 Call this matrix T Rotate 45 degrees Call this matrix R Translate object +5,+7 Call this matrix T-1

Expressed as matrices: T-1

RTv Do this for every point v that makes up the object.

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Composite Transformations

Example: scale an object around point 4,5 Translate, scale, translate Matrix notation: T-1STv

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Row Vectors

What we've just seen assumes columnvectors/points: vectors/points are treated as4x1 matrices

If we have row vectors (1x4 matrices), then wecombine like this: To rotate then translate with column vectors:

v' = TRv To rotate then translate with rowvectors:

v' = vRT Notice the order is reversed And now the operations read left-to-right, like we

might expect

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Perspective

What about our perspective transformation? If "virtual view screen" is located d units from origin and

we're looking down Z axis: to project pointp=(x,y,z) to point p'=(x',y',z'): we do:

x' = x * d / -zy' = y * d / -z //negative since looking down Z axisz' = z (save the old z coord!)

Matrix for this transformation:

[1 0 0 0

0 1 0 0

0 0 01

d

0 01

d0 ] [

x

y

z

q ]=[x

y1

d

z

d]=[

x d

zy d

z

1

z

1

]

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View Pipeline

Now we can write out our whole view pipeline: Suppose M represents the combined

transformations (any rotations, translations, scale)on the vertices of the object

Suppose P represents the perspective matrix.PM = P*Mtransformed=[]for p in points:

transformed.append( PM * p ) And now draw polygon using transformed[...]

Ignore z coordinates; just use x and y Note: will still need to do viewport transform (i.e., map

-1...1 into range 0...w-1 and 0...h-1).

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Viewport Mapping

Viewport mapping can be done with a matrix too! Screen is w x h We map x=-1...1 to range 0...w-1 We map y=-1...1 to range 0...h-1, but need to flip x' = (x+1)/2 * (w-1) = (x+1) * (w-1)/2 = x(w-1)/2 + (w-

1)/2 y' = h-1 - (y+1)/2 * (h-1) = (h-1)/2 + y(1-h)/2

[w1

2 0 0

w1

2

01h

20

h1

2

0 0 1 0

0 0 0 1] [xyz1 ]=[

xw1

2

w1

2

y1h

2

h1

2

z

1

]

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Final Operation

W=model-to-world coordinates P=perspective V=viewport mapping

M = VPW for all points:p' = Mp

draw