Transformations

Transformations

Aaron BloomfieldCS 445: Introduction to Graphics

Fall 2006(Slide set originally by David Luebke)

2

Graphics coordinate systems X is red Y is green Z is blue

3

Graphics coordinate systems

If you are on the +z axis, and +y is up, then +x is to the right

Math fields have +x going to the left

4

Outline

Scaling Rotations Composing Rotations Homogeneous Coordinates Translations Projections

5

Scaling

Scaling a coordinate means multiplying each of its components by a scalar

Uniform scaling means this scalar is the same for all components:

2

6

Scaling

Non-uniform scaling: different scalars per component:

How can we represent this in matrix form?

X 2,Y 0.5

7

Scaling

Scaling operation:

Or, in matrix form:

cz

by

ax

z

y

x

'

'

'

z

y

x

c

b

a

z

y

x

00

00

00

'

'

'

scaling matrix

8

Outline


9

2-D Rotation

(x, y)

(x’, y’)

x’ = x cos() - y sin()y’ = x sin() + y cos()

10

2-D Rotation

This is easy to capture in matrix form:

3-D is more complicated Need to specify an axis of rotation Simple cases: rotation about X, Y, Z axes

y

x

y

x

cossin

sincos

'

'

11

Rotation example: airplane

12

3-D Rotation

What does the 3-D rotation matrix look like for a rotation about the Z-axis? Build it coordinate-by-coordinate

2-D rotation from last slide:

z

y

x

z

y

x

100

0)cos()sin(

0)sin()cos(

'

'

'

y

x

y

x

cossin

sincos

'

'

13

3-D Rotation

What does the 3-D rotation matrix look like for a rotation about the Y-axis? Build it coordinate-by-coordinate

z

y

x

z

y

x

)cos(0)sin(

010

)sin(0)cos(

'

'

'

14

3-D Rotation

What does the 3-D rotation matrix look like for a rotation about the X-axis? Build it coordinate-by-coordinate

z

y

x

z

y

x

)cos()sin(0

)sin()cos(0

001

'

'

'

15

Rotations about the axes

16

3-D Rotation

General rotations in 3-D require rotating about an arbitrary axis of rotation

Deriving the rotation matrix for such a rotation directly is a good exercise in linear algebra

Another approach: express general rotation as composition of canonical rotations Rotations about X, Y, Z

17

Outline


18

Composing Canonical Rotations

Goal: rotate about arbitrary vector A by Idea: we know how to rotate about X,Y,Z

So, rotate about Y by until A lies in the YZ plane Then rotate about X by until A coincides with +Z

Then rotate about Z by

Then reverse the rotation about X (by -) Then reverse the rotation about Y (by -)

Show video…

19

Composing Canonical Rotations

First: rotating about Y by until A lies in YZ Draw it…

How exactly do we calculate ? Project A onto XZ plane Find angle to X:

= -(90° - ) = - 90 ° Second: rotating about X by until A lies on Z How do we calculate ?

20

3-D Rotation Matrices

So an arbitrary rotation about A composites several canonical rotations together

We can express each rotation as a matrix Compositing transforms == multiplying matrices Thus we can express the final rotation as the

product of canonical rotation matrices Thus we can express the final rotation with a

single matrix!

21

Compositing Matrices

So we have the following matrices: p: The point to be rotated about A by Ry : Rotate about Y by Rx : Rotate about X by Rz : Rotate about Z by Rx

-1: Undo rotation about X by Ry

-1 : Undo rotation about Y by

In what order should we multiply them?

22

Compositing Matrices

Remember: the transformations, in order, are written from right to left In other words, the first matrix to affect the vector goes

next to the vector, the second next to the first, etc. This is the rule with column vectors (OpenGL); row

vectors would be the opposite So in our case:

p’ = Ry-1 Rx

-1 Rz Rx Ry p

23

Rotation Matrices

Notice these two matrices:

Rx : Rotate about X by

Rx -1: Undo rotation about X by

How can we calculate Rx -1?

Obvious answer: calculate Rx (-)

Clever answer: exploit fact that rotation matrices are orthonormal

What is an orthonormal matrix? What property are we talking about?

24

Rotation Matrices

Rotation matrix is orthogonal Columns/rows linearly independent Columns/rows sum to 1

The inverse of an orthogonal matrix is just its transpose:

jfc

ieb

hda

jih

fed

cba

jih

fed

cbaT1

25

Rotation Matrix for Any Axis

Given glRotated (angle, x, y, z) Let c = cos(angle) Let s = sin(angle) And normalize the vector so that ||(x,y,z|| == 1

The produced matrix to rotate something by angle degrees around the axis (x,y,z) is:

(1 ) (1 ) (1 ) 0

(1 ) (1 ) (1 ) 0

(1 ) (1 ) (1 ) 0

0 0 0 1

xx c c xy c zs xz c ys

yx c zs yy c c yz c xs

zx c ys zy c xs zz c c

26

Outline


27

Translations

For convenience we usually describe objects in relation to their own coordinate system

We can translate or move points to a new position by adding offsets to their coordinates:

Note that this translates all points uniformly

z

y

x

t

t

t

z

y

x

z

y

x

'

'

'

28

Translation Matrices?

We can composite scale matrices just as we did rotation matrices

But how to represent translation as a matrix? Answer: with homogeneous coordinates

29

Homogeneous Coordinates

Homogeneous coordinates: represent coordinates in 3 dimensions with a 4-vector

[x, y, z, 0]T represents a point at infinity (use for vectors) [0, 0, 0]T is not allowed Note that typically w = 1 in object coordinates

w

z

y

x

wz

wy

wx

zyx

1

/

/

/

),,(

30


Homogeneous coordinates seem unintuitive, but they make graphics operations much easier

Our transformation matrices are now 4x4:

1000

0)cos()sin(0

0)sin()cos(0

0001

xR

31




1000

0)cos(0)sin(

0010

0)sin(0)cos(

yR

32




1000

0100

00)cos()sin(

00)sin()cos(

zR

33




Performing a scale:

1000

000

000

000

z

y

x

S

S

S

S

1 0 0 0

0 2 0 02 1

0 0 1 0

0 0 0 1 1

x

yx y z

z

34

More On Homogeneous Coords

What effect does the following matrix have?

Conceptually, the fourth coordinate w is a bit like a scale factor

w

z

y

x

w

z

y

x

10000

0100

0010

0001

'

'

'

'

35

More On Homogeneous Coords

Intuitively: The w coordinate of a homogeneous point is

typically 1 Decreasing w makes the point “bigger”, meaning further

from the origin Homogeneous points with w = 0 are thus “points at

infinity”, meaning infinitely far away in some direction. (What direction?)

To help illustrate this, imagine subtracting two homogeneous points

36

Outline


37


How can we represent translation as a 4x4 matrix? A: Using the rightmost column:

Performing a translation:

1 0 0

0 1 0

0 0 1

0 0 0 1

x

y

z

T

T

T

T

1 0 0 0

0 1 0 010 1

0 0 1 10

0 0 0 1 1

x

yx y z

z

38

Translation Matrices

Now that we can represent translation as a matrix, we can composite it with other transformations

Ex: rotate 90° about X, then 10 units down Z:

w

z

y

x

w

z

y

x

1000

0)90cos()90sin(0

0)90sin()90cos(0

0001

1000

10100

0010

0001

'

'

'

'

39




w

z

y

x

w

z

y

x

1000

0010

0100

0001

1000

10100

0010

0001

'

'

'

'

40




w

z

y

x

w

z

y

x

1000

10010

0100

0001

'

'

'

'

41




w

y

z

x

w

z

y

x

10

'

'

'

'

42

Transformation Commutativity

Is matrix multiplication, in general, commutative? Does AB = BA?

What about rotation, scaling, and translation matrices? Does RxRy = RyRx?

Does RAS = SRA ?

Does RAT = TRA ?

43

Outline


44

Perspective Projection

In the real world, objects exhibit perspective foreshortening: distant objects appear smaller

The basic situation:

45


When we do 3-D graphics, we think of the screen as a 2-D window onto the 3-D world The view plane

How tall shouldthis bunny be?

46

Perspective Projection The geometry of the situation is that of similar triangles.

View from above:

What is x?

P (x, y, z)X

Z

Viewplane

d

(0,0,0) x’ = ?

47


Desired result for a point [x, y, z, 1]T projected onto the view plane:

What could a matrix look like to do this?

dzdz

y

z

ydy

dz

x

z

xdx

z

y

d

y

z

x

d

x

,','

',

'

48

A Perspective Projection Matrix

Answer:

0100

0100

0010

0001

d

M eperspectiv

49


Example:

Or, in 3-D coordinates:

10100

0100

0010

0001

z

y

x

ddz

z

y

x

d

dz

y

dz

x,,

50


OpenGL’s gluPerspective() command generates a slightly more complicated matrix:

Can you figure out what this matrix does?

2cotwhere

0100

200

000

000

y

farnear

nearfar

farnear

nearfar

fovf

ZZ

ZZ

ZZ

ZΖf

aspect

f

51

Projection Matrices

Now that we can express perspective foreshortening as a matrix, we can composite it onto our other matrices with the usual matrix multiplication

End result: can create a single matrix encapsulating modeling, viewing, and projection transforms Though you will recall that in practice OpenGL

separates the modelview from projection matrix (why?)

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Transformations