681 Introduction to Computer Graphics

1
Object Representation
Define object in world space Object space data
Translation Rotation Scale
Desired operations Interpolation concatenation
Handle rotation, translation, scale independently
2
Orientation Representation
orientation
3
Orientation Representation
Rotation Matrix
Fixed Angles
Euler Angles
Axis-Angle
Quaternion
4
Interpolation
5
Concatenation
6
Transformation Matrices
7
Transformation Matrices
0
0
-1
0
0
0
1
0
0
1
0
0
0
-1
0
0
0
0
0
1
0
0
0
1
1
0
0
0
1
0
0
0
8
Fixed Angles
E.g., (Z,Y,X)
Rx(q1). Ry(q2). Rz(q3). P
9
Fixed Angles
E.g., (0,90,0)
Y
Y
X
X
Z
Z
10
Fixed Angles
E.g., (-45,90,0)
Y
Y
X
X
Z
Z
11
Gimbal Lock
E.g., (0,90,0)
E.g., (e,90,0)
Y
E.g., (0,90e,0)
X
E.g., (0,90,e)
Z
12
Fixed Angles
E.g., (-45,90,0)
E.g., (0,90,0)
Y
Y
Z-axis rotation
X
X
Z
Z
13
Fixed Angle Interpolation
(0,90,0) to (-45,90,0)
14
Euler Angles
(e.g., z,y,x)
Y
y
Z
z
x
X
15
Euler Angles
(z,y,x)
Rz(q1).P
Rz(q1).Ry(q2). Rz(-q1).P
Rz(q1). Ry(q2). Rz(-q1). Rz(q1).P
Rz(q1). Ry(q2). P
Rz(q1). Ry(q2). Rx(q3). Ry(- q2). Rz(-q1).P
Rz(q1). Ry(q2). Rx(q3). Ry(- q2). Rz(-q1).
Rz(q1). Ry(q2). P
Rz(q1). Ry(q2). Rx(q3). P
16
Axis-Angle
(Ax,Ay,Az,q)
A
q
Y
Z
X
Eulers rotation theorem
17
Axis-Angle
(Ax,Ay,Az,q)
A1
q1
Y
A2
q2
Z
X
Eulers rotation theorem
18
Quaternions
q s,v
s,x,y,z
A
q
(cos(q/2),sin(q/2)A)
19
Quaternions
s1,v1 s2,v2 s1s2,v1v2
s1,v1 s2,v2 s1s2-v1.v2,s1v2s2v1v1Xv2

q sqrt(ss xx yy zz)
q 1,0,0,0 q
q-1 -s,v/q2
q q-1 1,0,0,0
20
Quaternions - rotate a point
v (x,y,z) gt 0,v
Rotq(v) v q 0,v q-1
21
Quaternions - composite transformations
Rotq(Rotp(v)) Rotq( p 0,v p-1 )
Rotq(Rotp(v)) q p 0,v p-1 q -1
Rotq(Rotp(v)) q p 0,v (qp) -1
Rotq(Rotp(v)) Rotqp(v))
22
Unit Quaternion
q
q
s,v -s, -v