The exact thin plate energy functional (TPEF) for a function
f
(
x
,
y
)
is
∫
y
0
y
1
∫
x
0
x
1
(
κ
1
2
+
κ
2
2
)
g
d
x
d
y
where
κ
1
and
κ
2
are the principal curvatures of the surface mapping
f
at the point
(
x
,
y
)
.
This is the surface integral of
κ
1
2
+
κ
2
2
,
hence the
g
in the integrand.
Minimizing the exact thin plate energy functional would result in a system of non-linear equations. So in practice, an approximation that results in linear systems of equations is often used. The approximation is derived by assuming that the gradient of
f
is 0. At any point where
f
x
=
f
y
=
0
,
the first fundamental form
g
i
j
of the surface mapping
f
is the identity matrix and the second fundamental form
b
i
j
is
(
f
x
x
f
x
y
f
x
y
f
y
y
)
.
We can use the formula for mean curvature
H
=
b
i
j
g
i
j
/
2
to determine that
H
=
(
f
x
x
+
f
y
y
)
/
2
and the formula for Gaussian curvature
K
=
b
/
g
(where
b
and
g
are the determinants of the second and first fundamental forms, respectively) to determine that
K
=
f
x
x
f
y
y
−
(
f
x
y
)
2
.
Since
H
=
(
k
1
+
k
2
)
/
2
and
K
=
k
1
k
2
,
the integrand of the exact TPEF equals
4
H
2
−
2
K
.
The expressions we just computed for the mean curvature and Gaussian curvature as functions of partial derivatives of
f
show that the integrand of the exact TPEF is
4
H
2
−
2
K
=
(
f
x
x
+
f
y
y
)
2
−
2
(
f
x
x
f
y
y
−
f
x
y
2
)
=
f
x
x
2
+
2
f
x
y
2
+
f
y
y
2
.
So the approximate thin plate energy functional is
J
[
f
]
=
∫
y
0
y
1
∫
x
0
x
1
f
x
x
2
+
2
f
x
y
2
+
f
y
y
2
d
x
d
y
.
The TPEF is rotationally invariant. This means that if all the points of the surface
z
(
x
,
y
)
are rotated by an angle
θ
about the
z
-axis, the TPEF at each point
(
x
,
y
)
of the surface equals the TPEF of the rotated surface at the rotated
(
x
,
y
)
.
The formula for a rotation by an angle
θ
about the
z
-axis is
The fact that the
z
value of the surface at
(
x
,
y
)
equals the
z
value of the rotated surface at the rotated
(
x
,
y
)
is expressed mathematically by the equation
Z
(
X
,
Y
)
=
z
(
x
,
y
)
=
(
z
∘
x
y
)
(
X
,
Y
)
where
x
y
is the inverse rotation, that is,
x
y
(
X
,
Y
)
=
R
−
1
(
X
,
Y
)
T
=
R
T
(
X
,
Y
)
T
.
So
Z
=
z
∘
x
y
and the chain rule implies
In equation (2),
Z
0
means
Z
X
,
Z
1
means
Z
Y
,
z
0
means
z
x
,
and
z
1
means
z
y
.
Equation (2) and all subsequent equations in this section use non-tensor summation convention, that is, sums are taken over repeated indices in a term even if both indices are subscripts. The chain rule is also needed to differentiate equation (2) since
z
j
is actually the composition
z
j
∘
x
y
:
Z
i
k
=
z
j
l
R
k
l
R
i
j
.
Swapping the index names
j
and
k
yields
Expanding the sum for each pair
i
,
j
yields
Z
X
X
=
R
00
2
z
x
x
+
2
R
00
R
01
z
x
y
+
R
01
2
z
y
y
,
Z
X
Y
=
R
00
R
10
z
x
x
+
(
R
00
R
11
+
R
01
R
10
)
z
x
y
+
R
01
R
11
z
y
y
,
Z
Y
Y
=
R
10
2
z
x
x
+
2
R
10
R
11
z
x
y
+
R
11
2
z
y
y
.
Computing the TPEF for the rotated surface yields
Inserting the coefficients of the rotation matrix
R
from equation (1) into the right-hand side of equation (4) simplifies it to
z
x
x
2
+
2
z
x
y
2
+
z
y
y
2
.
The approximate thin plate energy functional can be used to fit B-spline surfaces to scattered 1D data on a 2D grid (for example, digital terrain model data). Call the grid points
(
x
i
,
y
i
)
for
i
=
1
…
N
(with
x
i
∈
[
a
,
b
]
and
y
i
∈
[
c
,
d
]
) and the data values
z
i
.
In order to fit a uniform B-spline
f
(
x
,
y
)
to the data, the equation
(where
λ
is the "smoothing parameter") is minimized. Larger values of
λ
result in a smoother surface and smaller values result in a more accurate fit to the data. The following images illustrate the results of fitting a B-spline surface to some terrain data using this method.
The thin plate smoothing spline also minimizes equation (5), but it is much more expensive to compute than a B-spline and not as smooth (it is only
C
1
at the "centers" and has unbounded second derivatives there).
