Geography Reference
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2 + λ
x 0 ) 2 +( y
y 0 ) 2
r 0 }
L ( x, y, λ ):=
X-x
{
( x
=min
x,y, λ
(23.150)
1000
α =10°, k 0 =1/500,
k ´ 0 =0, x 0 =0, y 0 =0, s 0 =0
800
600
11
7
6
400
200
-600
-400
-200
200
400
Fig. 23.30. Local representation of the circle by ( 23.145 ), κ 0 =1 / 500 0 =0 0 =10 ,x 0 = y 0 =0 ,s 0 =0,
various degrees of approximation
The necessary condition
x, y, λ =0 , ∂L
∂y
x, y, λ =0 , ∂L
∂λ
x, y, λ =0
∂L
∂x
or
(X − x )+ λ ( x − x 0 )=0
y )+ λ ( y
(23.151)
( Y
y 0 )=0
x 0 ) 2 + λ ( y
y 0 ) 2 = r 0
( x
lead to three normal equations we are going to solve
x (1 + ˆ
x 0 ˆ
=X , y (1 + ˆ
y 0 ˆ
x 0 ) 2 +( y
y 0 ) 2 = r 0
λ
)
λ
λ
)
λ
=Y , ( y
, y = Y y 0 λ
1+ λ
x = X x 0 λ
1+ λ
x 0 ) 2 +( Y + y 0 λ
y 0 ) 2 =( X + x 0 λ
x 0 ) 2 +( y
r 0 =( x
y 0 ) 2
λ
1+ ˆ
1+ λ
y 0 ) 2 = r 0 1+ ˆ
2
x 0 ) 2 +(Y
(X
λ
(X
1
r 0
1+ ˆ
x 0 ) 2 +(Y
y 0 ) 2
λ
=
±
(23.152)
 
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