Reminder

1

Reminder

Any sequence of matrix operations can be composed into a single matrix

We’ll always use an extra dimension for all vertices (x,y,w)

2

Extending to 3D

Homogeneous coordinates in 3D [x,y,z,1]T (x,y,z,w) Matrices of this form:

4x4 Matrices instead of 3x3 for 3D

111000

lkzjyix

hgzfyex

dczbyax

z

y

x

lkji

hgfe

dcba

3

Translation

1000

100

010

001

),,(l

h

d

lhdT

111000

100

010

001

lz

hy

dx

z

y

x

l

h

d

4

OpenGL

glTranslated(x, y, z)

1000

100

010

001

z

y

x

MM

Post-multiplies translation matrix onto currently selected matrix…

5

Example…

glTranslated(tox, toy, toz);

DrawBox();

6

Scaling

1000

000

000

000

),,(k

f

a

kfaS

111000

000

000

000

kz

fy

ax

z

y

x

k

f

a

How do I do auniform scale?

glScaled(x, y, z)

7

What about Rotation?

How can we convert this to 3D?

1

cossin

sincos

1100

0cossin

0sincos

yx

yx

y

x

8

Rotation about Z axis

1

cossin

sincos

11000

0100

00cossin

00sincos

z

yx

yx

z

y

x

Just keep z constant.

9

The 3 Rotation Matrices

1000

0100

00cossin

00sincos

)(

zR

1000

0cossin0

0sincos0

0001

)(

xR

1000

0cos0sin

0010

0sin0cos

)(

yR

10

OpenGL general glRotated

glRotated(angle, x, y, z) Rotates by angle (in degrees) around

the vector (x, y, z)

11

Skew or Shear

1000

0100

0010

001

)(

b

bH x

12

Example: Rotating around a point

Suppose we have an object centered on (12, 17, 32)

We want to “spin” the object around that center point

What will be the operations?

13

Homogenous Coordinates and Vectors

It is convention that Points in space are indicated with w=1 Vectors are indicated with w=0 [12, 13, 5, 1]T is a point [45, 13, 2, 0]T is a vector (point –

point?)

We often want unit-length vectors

OpenGL Transformation Composition

A global modeling transformation matrix (GL_MODELVIEW, called it M here) glMatrixMode(GL_MODELVIEW) The user is responsible to reset it if

necessary glLoadIdentity() -> M = 1 0 0 0 1 0 0 0 1

OpenGL Transformation Composition

Matrices for performing user-specified transformations are multiplied to the current matrix

For example, 1 0 1

glTranslated(1,1 0); M = M x 0 1 1

0 0 1 All the vertices defined within glBegin() /

glEnd() will first go through the transformation (modeling transformation)

P’ = M x P

Transformation Pipeline

Object Local Coordinates

Object World Coordinates

Modeling transformation

…

17

Frames

Frame – A center and three coordinate axes A coordinate system

World Frame andCamera Frame

18

World to Camera

What does gluLookAt do mathematically?

void gluLookAt( GLdouble eyex, GLdouble eyey, GLdouble eyez, GLdouble centerx, GLdouble centery, GLdouble centerz, GLdouble upx, GLdouble upy, GLdouble upz );

19

Defining a Frame(relative to another frame)

Need: Origin Vectors for X, Y, and Z axis of frame

20

Adding Orthogonal Constraint

We can get by with: Origin One axis direction and which way is

up

Z direction is negative oflook direction.

X is at right angles to Z and Up

21

Z = (eye – center) / | eye – center |

X = (up Z) / | up Z |

Y = Z X

Computing the Axis

z

x

y

Right-hand rule?

22

Making a frame the reference coordinate system

Move the center to the origin Rotate the frame axis onto (1,0,0),

(0,1,0), (0,0,1)

23

Moving the center to the origin

glTranslated(-eyex, -eyey, -eyez);

24

Rotating arbitrary axes v1,v2,v3 onto (1,0,0), (0,1,0), (0,0,1)

Notice: v1,v2,v3 must be orthogonal to each other

25

Suppose we have three ortho-normal vectors…

v1,v2,v3

Let’s build a matrix like this:

This will rotate: v1 onto the x axis, v2 onto the y axis, v3 onto the z axis

1000

0

0

0

),,(,3,3,3

,2,2,2

,1,1,1

321zyx

zyx

zyx

vvv

vvv

vvv

vvvR

26

Introduction to 3D viewing

3D is just like taking a photograph!

27

Viewing Transformation

Position and orient your camera

28

Projection Transformation

Control the “lens” of the camera Project the object from 3D world to 2D screen

29

Viewing Transformation (2)

Important camera parameters to specify Camera (eye) position (Ex,Ey,Ez) in world

coordinate system Center of interest (coi) (cx, cy, cz) Orientation (which way is up?) View-up vector

(Up_x, Up_y, Up_z)

world(cx, cy, cz)

(ex, ey, ez)view up vector(Up_x, Up_y, Up_z)

30


Transformation? Form a camera (eye) coordinate frame

Transform objects from world to eye space

world

uvn

x

y

z

Eye coordinate frame coi

31

world

uvn

x

y

z

(0,0,0) coi


Eye space?

Transform to eye space can simplify many downstream operations (such as projection) in the pipeline

(1,0,0)(0,1,0)(0,0,1)

32


In OpenGL: - gluLookAt (Ex, Ey, Ez, cx, cy, cz,

Up_x, Up_y, Up_z) - The view up vector is usually (0,1,0) - Remember to set the OpenGL matrix

mode to GL_MODELVIEW first

33


void display() { glClear(GL_COLOR_BUFFER_BIT); glMatrixMode(GL_MODELVIEW); glLoadIdentity(); gluLookAt(0,0,1,0,0,0,0,1,0); display_all(); // your display routine}

34

Suppose we have three orthogonal vectors…

v1,v2,v3

Let’s build a matrix like this:

This will rotate: v1 onto the x axis, v2 onto the y axis, v3 onto the z axis

1000

0

0

0

),,(,3,3,3

,2,2,2

,1,1,1

321zyx

zyx

zyx

vvv

vvv

vvv

vvvR

35

My Version of gluLookAt()

void mygluLookAt(Point3 eye, Point3 center, Point3 up){ Point3 cameraZ = Normalize( eye – center ); // v3 Point3 cameraX = Normalize( Cross(up, cameraZ) ); // v1 Point3 cameraY = Cross( cameraZ, cameraX ); // v2 GLdouble m[16]; // Fill the matrix in row by row m[0] = cameraX.X; m[4] = cameraX.Y; m[8] = cameraX.Z; m[12] = 0.0; m[1] = cameraY.X; m[5] = cameraY.Y; m[9] = cameraY.Z; m[13] = 0.0; m[2] = cameraZ.X; m[6] = cameraZ.Y; m[10] = cameraZ.Z; m[14] = 0.0; m[3] = m[7] = m[11] = 0.0; m[15] = 1.0; glMultMatrixd( m ); glTranslated( -eye.X, -eye.Y, -eye.Z );}

Order of transformations!

36

Messing with the camera

What if I want to PAN the camera? We need to define “PAN”

There are two possibilities

Crab Pan

37

Panning

Suppose we pan around the camera Y axis This is NOT “up” in world space. We need to change the lookAt point.

38

Operations on Points in Space

LookAt is a point in space Need these transformation:

Translate by –eye Rotate frame onto axis (using some M) Rotate around Y by pan angle Inverse rotate M Translate by eye

PP=T(eye) MT RY() M T(-eye)

39

Easier #1

Just replace the matrix using gluLookAt. Problems?

40

Easier #2

The first 2 operations are already what is done to set the camera up: M T(-eye)

We just need to add a rotate after this is done. Implies we want to pre-multiply by a

rotation matrix.

41

Easier #2

Steps:1. Read out the current matrix.2. Set the matrix to the identity matrix.3. Rotate about the y-axis the amount

we want to pan.4. Post-multiply by the matrix read out

in step #1.

42

Camera Controls

Tilt Roll Dolly Boom Zoom (same as dolly?) General camera (or entity)

movement and the user interface / control.

Eliot Lash, 2007 (from Wikipedia.org – Camera dolly)

43

Camera Controls Stationary 2 degrees of

freedom + zoom QuicktimeVR

44

Examiner Mimic holding an

object in your hand.

OpenInventor’s Examiner Viewer

Used in IRIS Explorer

Used in modeling and scientific visualization.

Image courtesy of NAG

45

Flying Camera specified

using Euler angles and position.

Many possible control strategies Acceleration /

Deceleration

46

Camera models in games Many different approaches Flying model, with a camera pan, tilt

according to the mouse position. Orthographic view with simple

translations (occasionally a examiner) Cinematic camera with view-oriented

dolly (look with mouse, move in camera look-at direction with keyboard).

47

Software Engineering Define a well-encapsulated Camera

class or base class. Separate the control of the camera from

the camera object. Separate the mouse / keyboard

handling from the control as much as possible.

I will share my Camera classes next week.

Documents

Reminder