SECTION 4.6
329
Patterns
As in a Coons patch mesh, the geometry of the tensor-product patch is described
by a surface defined over a pair of parametric variables,
u
and
v,
which vary hori-
zontally and vertically across the unit square. The surface is defined by the equa-
tion
3
3
S
(
u
,
v
) =
∑ ∑
p
ij
×
B
i
(
u
)
×
B
j
(
v
)
i
=
0
j
=
0
where
p
ij
is the control point in column
i
and row
j
of the tensor, and
B
i
and
B
j
are
the
Bernstein polynomials
B
0
(
t
) =
(
1
–
t
)
3
2
B
1
(
t
) =
3t
× (
1
–
t
)
2
B
2
(
t
) =
3t
× (
1
–
t
)
B
3
(
t
) =
t
3
Since each point
p
ij
is actually a pair of coordinates (x
ij
,
y
ij
), the surface can also
be expressed as
3
3
x
(
u
,
v
)
=
∑ ∑
x
ij
×
B
i
(
u
) ×
B
j
(
v
)
i
=
0
j
=
0
3
3
y
(
u
,
v
)
=
∑ ∑
y
ij
×
B
i
(
u
) ×
B
j
(
v
)
i
=
0
j
=
0
The geometry of the tensor-product patch can be visualized in terms of a cubic
Bézier curve moving from the bottom boundary of the patch to the top. At the
bottom and top, the control points of this curve coincide with those of the patch’s
bottom (p
00
…
p
30
) and top (p
03
…
p
33
) boundary curves, respectively. As the
curve moves from the bottom edge of the patch to the top, each of its four control
points follows a trajectory that is in turn a cubic Bézier curve defined by the four
control points in the corresponding column of the array. That is, the starting
point of the moving curve follows the trajectory defined by control points
p
00
…
p
03
, the trajectory of the ending point is defined by points
p
30
…
p
33
, and