Game Development Reference
In-Depth Information
Bicubic Surfaces
A bicubic surface patch is defined as
r
()
T
()
T
r
()
ij Qst
,
=
SMG
s
T ,
t
()
()
where
basis matrix corre-
sponding to the class of cubic curve on which the patch is based, and r represents
one of the x , y , or z coordinates of
S
s
1,
ss s
,
23
,
,
T
t
1, ,
t t
23
,
t
, M is the 4
×
4
()
. G is the 4
××
43
array of control
Q
s t
,
ij
point coordinates.
The normal vector
()
to the surface of a bicubic patch
()
is given by
N
s t
,
Q
s t
,
ij
ij
()
()
()
N
s t
,
=
Q
s t
,
×
Q
s t
,
.
ij
ij
ij
s
t
Curvature and Torsion
The curvature
()
()
κ t of a curve
P
t
is given by
PP
P
()
t
×
′′
()
t
()
κ t
=
.
()
3
t
()
()
ρ t κ t
=
1
.
()
The torsion
τ t is defined as
d
ˆ
ˆ
()
()
()
τ t
=−
N
t
B
t
,
ds
ˆ N is the unit normal vector given by the normalized derivative of the
unit tangent direction
()
where
ˆ
ˆ
ˆ
ˆ
()
()
()
()
T
t
, and
B
t
TN .
t
t
Exercises for Chapter 11
()
1. Suppose that
2 B is a quadratic Bézier curve having the three control
points P , P , and P . That is,
( )
(
)
2
(
)
2
B
t
=−
1
t
P
+
2
t
1
t
P
+
t
P .
2
0
1
2
Determine the four control points 0
P through 3
P such that the cubic Bézier
curve