1. Three-Dimensional Graphics
A 3D point (x,y,z) – x,y, and Z coordinates
We will still use column vectors to represent points
Homogeneous coordinates of a 3D point
(x,y,z,1)
Transformation will be performed using 4x4 matrix T y x z

2. Right hand coordinate system
X x Y = Z ; Y x Z = X; Z x X = Y; Y y z x +z x
Left hand coordinate system
Right hand coordinate system

3. 3D transformation Very similar to 2D transformation Translation
x’ = x + tx; y’ = y + ty; z’ = z + tz
X’ tx X
Y’ ty Y
Z’ tz Z
OpenGL - glTranslated(tx, ty, tz); =
homogeneous coordinates

4. 3D transformation Scaling X’ = X * Sx; Y’ = Y * Sy; Z’ = Z * Sz
X’ Sx X
Y’ Sy Y
Z’ Sz Z
OpenGL - glScaled(Sx, Sy, Sz); =

5. 3D transformation look down negative axis)
3D rotation is done around a rotation axis
Fundamental rotations – rotate about x, y, or z axes
Counter-clockwise rotation is referred to as a positive rotation (when you
look down negative axis) x y z +

6. 3D transformation Rotation about Z – similar to 2D rotation
x’ = x cos(q) – y sin(q)
y’ = x sin(q) + y cos(q)
z’ = z y x +
cos(q) -sin(q)
sin(q) cos(q) z
OpenGL - glRotatef(q, 0,0,1)

7. 3D transformation Rotation about y z’ = z cos(q) – x sin(q)+
Rotation about y
z’ = z cos(q) – x sin(q)
x’ = z sin(q) + x cos(q)
y’ = y
cos(q) sin(q) 0
-sin(q) 0 cos(q) 0 z x y +
OpenGL - glRotatef(q, 0,1,0)

8. 3D transformation Rotation about x y’ = y cos(q) – z sin(q)+
Rotation about x
y’ = y cos(q) – z sin(q)
z’ = y sin(q) + z cos(q)
x’ = x
cos(q) -sin(q) 0
sin(q) cos(q) 0 y z x +
OpenGL - glRotatef(q, 1,0,0)

9. 3D transformation Arbitrary rotation axis (rx,ry,rz)
Text pp. 193 explains how to do it
We omit the detail here
Use OpenGL:
glRotatef(angle, rx, ry, rz)
Can fill in all 9 entries of the rotation matrix. x z y
(rx, ry, rz)

10. 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 =
0 1 0
0 0 1

11. OpenGL Transformation Composition
Matrices for performing user-specified transformations are multiplied to the current matrix
For example,
glTranslated(1,1 0); M = M x
All the vertices defined within glBegin() / glEnd() will first go through the transformation (modeling transformation)
P’ = M x P

12. Transformation Pipeline
Object
Local Coordinates
Object
World Coordinates
Modeling
transformation …