Graphics Reference
In-Depth Information
•
Two unit vectors can be spherically interpolated using
sin1
sin
−
sin
sin
ˆ
w
ˆ
=
u
ˆ
+
v
0
≤
≤
1
where is the angle between
u
and
ˆ
ˆ
v
.
•
The direction cosines of a vector are the cosines of the angles between the vector and the
Cartesian axes.
•
A point P x
P
y
P
is rotated to P
x
P
y
P
using
x
P
y
P
r
11
x
P
y
P
r
12
=
·
r
21
r
22
where
r
11
r
12
are the direction cosines of the x-axis, and
r
21
r
22
are the direction cosine of the y-axis.
r
11
+
r
12
=
1 and r
21
+
r
22
=
1.
•
A point P x
P
y
P
z
P
is rotated to P
x
P
y
P
z
P
using
⎡
⎤
⎡
⎤
⎡
⎤
x
P
y
P
z
P
r
11
r
12
r
13
x
P
y
P
z
P
⎣
⎦
=
⎣
⎦
·
⎣
⎦
r
21
r
22
r
23
r
31
r
32
r
33
where
r
11
r
12
r
13
are the direction cosines of the x-axis,
r
21
r
22
r
23
are the direction cosines of the y-axis, and
r
31
r
32
r
33
are the direction cosines of the y-axis.
r
11
+
r
12
+
r
13
=
1, r
21
+
r
22
+
r
23
=
1,
r
31
+
r
32
+
r
33
=
1.
