Geography Reference
In-Depth Information
Table 22.10
Basis functions: bivariate harmonic polynomial up to degree 10
r
=0
1
r
=1
q
l
q
2
−
l
2
r
=2
2
ql
q
3
−
3
ql
2
3
q
2
l
l
3
r
=3
−
q
4
−
6
q
2
l
2
+
l
4
4
q
3
l
4
ql
3
r
=4
−
q
5
−
10
q
3
l
2
+5
ql
4
5
q
4
l
10
q
2
l
3
+
l
5
r
=5
−
r
=6
q
6
−
15
q
4
l
2
+30
q
2
l
4
−
l
6
6
q
5
l
20
q
3
l
3
+6
ql
5
−
r
=7
q
7
−
21
q
5
l
2
+ 210
q
3
l
4
−
7
ql
6
7
q
6
l
35
q
4
l
3
+21
q
2
l
5
−
l
7
−
r
=8
q
8
−
28
q
6
l
2
+ 1680
q
4
l
4
−
8
q
7
l
56
q
5
l
3
+
−
8
q
7
l
56
q
5
l
3
+
+56
q
3
l
5
−
8
ql
7
−
r
=9
q
9
−
36
q
7
l
2
+ 15120
q
5
l
4
−
9
q
8
l
84
q
6
l
3
+ 126
q
4
l
5
−
−
504
q
3
l
6
+9
ql
8
256
q
2
l
7
+
l
9
−
−
r
=10
q
10
−
45
q
8
l
2
+ 15120
q
6
l
4
−
10
q
9
l
120
q
7
l
3
+36
q
5
l
5
−
−
5040
q
4
l
6
+90
q
2
l
8
−
l
10
120
q
3
l
7
+10
ql
9
−
−
Meanwhile we determined the basis functions which span as bivariate Taylor polynomi-
als of degree r (rank r) the space of harmonic functions. Accordingly the general structure
of those functions which represent
harmonic maps
is of type Eqs.(
22.78
)and(
22.79
)of
Table
22.11
given as an approximation of the order N. We say also Eq.(
22.78
),
x
=
ξ
(
q,l
),
and Eq.(
22.79
),
y
=
η
(
q,l
), is a bandlimited representation (band 0, 1, ..., N) of the general
harmonic map.
{
α
0
,α
1
,α
2
,...,α
N−
2
,α
N−
1
,α
N
}
,
{
γ
0
,γ
1
,γ
2
...,γ
N−
2
,γ
N−
1
,γ
N
}
are the sets of
unknown coe
cients
for the even polynomial terms, while
{
β
0
,β
1
,β
2
,...,β
N−
2
,β
N−
1
,β
N
}
and
{
for the odd polynomial term. Let us summarize the representation
of a fundamental solution of the vector-valued
Laplace-Beltrami
equation by
δ
0
,δ
1
,δ
2
,...,δ
N−
2
δ
N−
1
,δ
N
}
Theorem 22.3 (fundamental solution of Laplace-Beltrami equation, bivariate harmonic polyno-
mials).
A fundamental solution of the
Laplace-Beltrami equation
(
22.53
) in isometric coordinates (con-
formal coordinates of
Mercator type,
isothermal coordinates) which generate harmonic map
{
is Eqs.(
22.78
)and(
22.79
)ofTable
22.11
. Such a fundamental solution is in
the space of
bivariate harmonic Taylor polynomials.
x
(
q,l
)
,y
(
q,l
)
}
End of Theorem.
22-32 Solving the Characteristic Vector-Valued Boundary Value
Problem of the Vector-Valued Laplace-Beltrami Equation
How to determine the unknown coecients
{
α
0
,α
1
,β
1
,α
2
,β
2
,...
}
,
{
γ
0
,γ
1
,δ
1
,γ
2
,δ
2
,...
}
of the fun-
damental solution
of the two-dimensional Laplace-Beltrami with respect
to a conformally flat metric (
Mercator type
) on the ellipsoid of revolution which generate har-
monic maps? Indeed we have to fix all the unknown coecients by a properly chosen
vector-valued
boundary problem
which is formulated in the language of map projections. We have chosen
four
conditions.
{
Eqs.(
22.78
), (
22.79
)
}
Search WWH ::
Custom Search