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

()

()

The radius of curvature is

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

Search Nedrilad ::

Custom Search