Computer Graphics

3D Transformations

3D Translation

• Remembering 2D transformations -> 3x3 matrices, take a wild guess what happens to 3D transformations.

1000

100

010

001

,,

100

10

01

,

z

y

x

z

y

x

zyx

y

x

y

xyx

t

t

t

t

t

t

z

y

x

tttT

t

t

t

t

y

xttT

T=(tx, ty, tz)

3D Scale

1000

000

000

000

*

00

00

00

,,

100

00

00

*0

0,

z

y

x

z

y

x

zyx

y

x

y

xyx

s

s

s

z

y

x

s

s

s

sssS

s

s

y

x

s

sssS

S=(sx, sy, sz)

3D Rotations

R=(rx, ry, rz, )What does a rotation in 3D mean?

Q: How do we specify a rotation?

A: We give a vector to rotate about, and a theta that describes how much we rotate.

Q: Since 2D is sort of like a special case of 3D, what is the vector we’ve been rotating about in 2D?

Rotations about the Z axis

R=(0,0,1,)

What do you think the rotation matrix is for rotations about the z axis?

1000

0100

00cossin

00sincos

),1,0,0(

100

0cossin

0sincos

cossin

sincos

R

R

Rotations about the X axis

R=(1,0,0,)

Let’s look at the other axis rotations

1000

0cossin0

0sincos0

0001

),0,0,1(

R

Rotations about the Y axis

R=(0,1,0,)

1000

0cos0sin

0010

0sin0cos

),0,1,0(

R

Rotations around a principle axis

1000

0100

00cossin

00sincos

),1,0,0(

R

1000

0cos0sin

0010

0sin0cos

),0,1,0(

R

1000

0cossin0

0sincos0

0001

),0,0,1(

R

xyzyx RRRRRR 11

Rotations for an arbitrary axis

Steps:1. Normalize vector u2. Compute 3. Compute 4. Create rotation matrix

u

Vector Normalization

• Given a vector v, we want to create a unit vector that has a magnitude of 1 and has the same direction as v. Let’s do an example.

V

VVNormalized _

Computing the Rotation Matrix

• 1. Normalize u ( the vector we are rotating around )

• 2. Compute Rx(x)

• 3. Compute Ry(y)

• 4. Generate Rotation Matrix

xy

y

yx

z

z

zx

zy

z

zy

zyx

dd

duu

uu

u

u

d

u

sin

cos

sin

'

'cos

),,0('

)1,0,0(

),,(

22

• Unit vectors are made of direction cosines:

Rotations for an arbitrary axis

Rotations for an arbitrary axis

xy

y

yx

z

z

zx

zy

z

zy

zyx

dd

duu

uu

u

u

d

u

sin

cos

sin

'

'cos

),,0('

)1,0,0(

),,(

22

Rotations for an arbitrary axis

xy

y

yx

z

z

zx

zy

z

zy

zyx

dd

duu

uu

u

u

d

u

sin

cos

sin

'

'cos

),,0('

)1,0,0(

),,(

22

Rotation Matrix

1000

00

0010

00

1000

00

00

0001

11

d

d

R

dd

ddR

RRRRRR

x

x

yy

zy

yz

xx

xxyyzyyxx

Axis-Angle Rotations in OpenGL

glRotatef( angle, x, y, z);

Euler Angles

More intuitive: represent rotations by 3 angles, one for each axis

glRotatef(anglex,1,0,0);glRotatef(angley,0,1,0);glRotatef(anglez,0,0,1);

Think: if we have a torus unstranformed at the origin, what will the torus look like if you have anglex=90, angley=90, and anglez=90

Transformed

Gimbal LockRotate around these

How can we gain altitude here?

Initial orientation (x=blue, y=green, z = red)

Gimbal Lock in OpenGL

What is the problem here?

glRotatef(anglex,1,0,0);glRotatef(angley,0,1,0);glRotatef(anglez,0,0,1);

Avoiding Gimbal Lock: Quaternions

• Quaternions represent 3D rotations in 4D using imaginary numbers – a 4-tuple Q = (w,x,y,z)

• Convert Angle-axis -> Quaternion

• (,x,y,z) -> ( cos( /2), xsin(/2), ysin(/2), zsin(/2) )

• Why quaternions?

• No gimbal lock

• Smooth interpolation between rotations (for animation)

Quaternions

• Quaternions have rules for multiplication, inversion, etc

• See: http://www.j3d.org/matrix_faq/matrfaq_latest.html

• Typical usage:

• Convert from euler, matrix, or angle-axis

• Do rotations

• Convert to angle-axis or matrix

What is going on here?

frame1 frame2 frame3

gluLookAt

Orients and positions the “camera”

gluLookat(eyex, eyey, eyez,centerx, centery, centerz, upx,upy, upz);

eye – the position of the camera in world coordinatescenter – the camera is pointed at this pointup – the direction defined to be up for the camera

gluLookAt – in matrix form

Cross Products

Given two vectors, the cross product returns a vector that is perpendicular to the plane of the two vectors and with magnitude equal to the area of the parallelogram formed by the two vectors.

u

Different Phases:Model Definition

Different Phases:Transformations

Different Phases:Projection

Different Phases:Projection

Different Phases:Rasterization

Different Phases:Scan Conversion

What are the steps needed?

Let’s Examine the Camera

• If I gave you a world, and said I want to “render” it from another viewpoint, what information do I have to give you?– Position

– Which way we are looking

– Which way is “up”

– Aspect Ratio

– Field of View

– Near and Far

Camera

View Direction

View UpView Right

View Normal

Camera

View Up

View Right

ViewUp

ViewRightoAspectRati

What are the vectors?

Graphics Pipeline So FarObject

Object Coordinates

Transformation

Object -> World

World

World Coordinates

Projection Xform

World -> Projection

Camera

Projection Coordinates

Screen

Device Coordinates

Normalize Xform & Clipping

Projection -> Normalized

Viewport

Normalized Coordinates

Viewport Transform

Normalized -> Device

Transformation World->Camera

View Direction

View UpView Right

View Normal

Transformation World->Camera

TRM

nnn

vvv

uuu

R

vvvunv

uuuNV

NVu

nnnN

Nn

camera

camera

camera

T

VCWC

z

y

x

1000

0

0

0

),,(

),,(

),,(

1000

100

010

001

321

321

321

321

321

321

View Up = VView Right = u

View Direction = -N