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Part II - 3D
Geometric Transformations |
Hearn and Baker – Chapter 5
Three Dimensional Coordinate Systems
Right Handed
Left Handed
Translation
Rotation
- Convention => counterclockwise rotation
- Three axes to rotate about
- Axes convention: look down axis’s positive
direction
- Rz
- Rx
- Ry
Scaling
Composite transformations
e.g. Fixed-point Scaling
Object Creation
·
Extrusion
o
Begin with points in xz
plane (x , 0 , z)
o
Create second set of
points by adding a value to y – in effect – translating y coordinates.
e.g
create a cube from a square in xy plane by adding 1 to z coordinate
|
Starting |
x |
y |
z |
|
1 |
0 |
0 |
0 |
|
2 |
0 |
1 |
0 |
|
3 |
1 |
1 |
0 |
|
4 |
1 |
0 |
0 |
|
Extruded |
|
|
|
|
5 |
0 |
0 |
1 |
|
6 |
0 |
1 |
1 |
|
7 |
1 |
1 |
1 |
|
8 |
1 |
0 |
1 |
·
Surface-of-Revolution
o
Begin with points in
plane (e.g. xy plane)
o
Rotate points about an
axis (e.g. y-axis) at angle intervals, say 20 degrees
Data
for above Figure
|
|
|
P1 |
|
|
|
P2
|
|
|
Angle |
x1 |
y1 |
z1 |
|
x2 |
y2 |
z2 |
|
0 |
0.750 |
0.250 |
0.000 |
|
0.25 |
0.750 |
0.000 |
|
60 |
0.375 |
0.250 |
-0.650 |
|
0.125 |
0.750 |
-0.217 |
|
120 |
-0.375 |
0.250 |
-0.650 |
|
-0.125 |
0.750 |
-0.217 |
|
180 |
-0.750 |
0.250 |
0.000 |
|
-0.25 |
0.750 |
0.000 |
|
240 |
-0.375 |
0.250 |
0.650 |
|
-0.125 |
0.750 |
0.217 |
|
300 |
0.375 |
0.250 |
0.650 |
|
0.125 |
0.750 |
0.217 |
|
360 |
0.750 |
0.250 |
0.000 |
|
0.25 |
0.750 |
0.000 |