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α β 400
)
(
400 B
C
A g
+
+
−
=
+ β
α
1
2
200
1
2
( ) ( )
A C B
g
+ = − + +
α β
+
1
2
( )
A C
+ .
@
M A C
N A C
= +
= −
1
2
1
2
( )
( )
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N
M
C
N
M
A
−
=
+
=
/ ! ( 8 :
F
9 : (
. # :
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@
D
A
D
D
A
P
B
A
B
A
B
sen sen
sen
sen
= → =
α α
DP
B
D
C
D
D
C
P
B
A
B
B
C
sen sen
sen
sen
= → =
β β
DP
B
@
D
A
A
B sen
senα
= D
C
B
C sen
senβ
% ( ," 5 0 5
: : @
M
sen
D
sen
D
senC
senA
B
A
C
B
=
⋅
=
⋅
β
α
+ 8 : #
&G
G
A C B g
+ + + + =
α β 400
B .
: ( @
( )
C B A E A
= − − − − = −
400 β α
D A D C
A
B
B
C
⋅ ⋅ = ⋅ ⋅
sen sen sen sen
β α
D A D E A
A
B
B
C
⋅ ⋅ = ⋅ − ⋅
sen sen sen( ) sen
β α
[ ]
D A D E A E A
A
B
B
C
⋅ ⋅ = ⋅ ⋅ − ⋅ ⋅
sen sen sen cos cos sen sen
β α
[ ] [ ]
sen sen sen cos sen sen cos
A D BC E D E A
A
B
B
C
⋅ ⋅ + ⋅ ⋅ = ⋅ ⋅ ⋅
β α α
tg
sen sen
sen sen cos
A
D E
D D E
B
C
A
B
B
C
=
⋅ ⋅
⋅ + ⋅ ⋅
α
β α
C E A
= −
J
" # " 8 5: : )
ˆ
200
(
ˆ A
tg
A
tg +
=
9 ! # " 5 . 5
!
6 ; # " 5 : (
. ! A
C
A
C
B
P
C
B
A
P
A
ˆ
200
ˆ
−
±
=
+
=
θ
θ
θ
θ
% 8 @
1
2
2
1
200
200
B
B
B
ó
C
B
A
B
−
=
−
−
=
−
−
=
β
α
D
A
D
D
A
P
B
A
B
A
B
sen sen
sen
sen
= → =
α α
DP
B
α
α sen
senB
D
senB
D
sen
D B
A
P
A
B
A 1
D
1
P
A =
→
=
β
β sen
senB
D
D C
B
C
B 2
D
senB2
D
sen
P
C
P
C
=
→
=
6 ; ! 5 !
A ". ! . #
P
A
P
A
A
P
P
A
P
A
A
p
D
Y
Y
sen
D
X
X
θ
θ
cos
⋅
+
=
⋅
+
=
P
C
P
C
C
P
P
C
P
C
C
p
D
Y
Y
sen
D
X
X
θ
θ
cos
⋅
+
=
⋅
+
=
4
=7
% ! ! (
( 9
# ! @
$G
)
(
)
(
)
(
r
e
C
m
i
t
H
r
e
C
m
i
t
H
r
e
C
m
i
t
H
B
P
B
P
B
P
A
P
A
P
A
P
C
P
C
P
C
P
+
+
−
+
=
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+
+
−
+
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+
+
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+
=
∆
% 8 . 8
% : 9
( # ! @
B
A
B
A
r
tgV
D
t =
+ ! + ! # '
! # ". C 5
+ 8
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C
C
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P
P
B
B
B
P
P
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A
A
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H
H
H
H
H
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H
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P
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1
1
1
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)
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1
)
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(
+
+
⋅
+
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=
6
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!
% 5 !
( @
( ) 2
2
)
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)
( A
UTM
B
UTM
A
UTM
B
UTM
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B
A Y
Y
X
X
D −
+
−
=
( ) 2
2
)
(
)
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UTM
B
UTM
C
UTM
B
UTM
UTM
C
B Y
Y
X
X
D −
+
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=
". C 5 . 5
# C
$
( + 8 (
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5
6
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67 . /67 A ! @
0 + "@
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A
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A
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P
A
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P
A
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)
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A
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A
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P
A
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A
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P
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P
A
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A
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y
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)
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)
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C
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)
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)
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C
P
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⋅
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C
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C
UTM
P
UTM
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C
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y
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)
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C
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R
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K)
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2
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A
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A
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+
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(
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# : : #
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2
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1
1
)
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A
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A
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B
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R
h
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h
k
D
D ∆
+
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D
B
A
r
P
A
r
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1
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A
r
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A
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sen
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sen
D
α
1
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5 # ; # C+ 5 +
+ ! # '
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8
R
8
2
2
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)
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B
A
B
A
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B
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h
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( B
A
g
B
A
r
B
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D
D ∆
−
=
$$
P
C
C
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P
A
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D
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+
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& , #
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;
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B
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V
A
A
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V
D
D
D
H
D
H
H
1
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)
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6
6 ' 5 67 9
: # 5 : : 8
N ' ;
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+ ! . 5
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B
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G
G
)
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200
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V
A
B
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B
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+
−
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( ) 2
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)
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UTM
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UTM
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Y
X
X
D −
+
−
=
9 # "CA. @
A
sen
D
V
sen
D
B
sen
D UTM
V
B
UTM
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A
UTM
v
A
ˆ
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)
(
=
=
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B
A
UTM
V
A D
V
sen
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sen
D )
(
ˆ
)
( =
UTM
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A
UTM
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B D
V
sen
A
sen
D )
(
)
( =
5
6
% -
67 . /67 A ! @
0 + "@
" 5A@
V
A
UTM
V
A
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V
A
UTM
V
A
UTM
O
D
senO
D
cos
)
(
)
y
)
(
)
x
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V
A
V
A
⋅
=
∆
⋅
=
∆
UTM
V
A
UTM
A
UTM
V
UTM
V
A
UTM
A
UTM
V
y
Y
Y
x
X
X
)
(
)
(
)
(
)
(
)
(
)
(
∆
+
=
∆
+
=
$3
!
0 + C@
V
B
TM
V
B
UTM
V
B
UTM
V
B
UTM
O
D
senO
D
cos
)
(
)
y
(
)
(
)
x
(
V
B
V
B
⋅
=
∆
⋅
=
∆
UTM
V
B
UTM
B
UTM
V
UTM
UTM
B
UTM
V
y
Y
Y
X
X
)
(
)
(
)
(
)
x
(
)
(
)
( V
B
∆
+
=
∆
+
=
% : ! " 5 C 8
=
% ' ! ( 9 ! :
: . 5
! " : !
% ( ( @
2
2
1
2
2
2
1
1 h
R
h
R
h
k
D
D UTM
G ∆
+
+
+
=
% ! + # @
" ! " 5C. #
# # @
A
sen
D
V
sen
D
B
sen
D
V
B
r
B
A
r
v
A
r
ˆ
=
=
B
A
r
V
A
r D
V
sen
B
sen
D
ˆ
=
B
A
r
V
B
r D
V
sen
A
sen
D =
R
8
2
2
2
r
G D
h
D +
∆
=
2
2
h
D
D G
r ∆
−
=
$F
% A ! #
V
B
V
A
V
B
B
V
V
A
A
V
V
D
D
D
H
D
H
H
1
1
1
)
(
1
)
(
+
⋅
+
⋅
=
- ( 5
- ( . )
=
9 + 5
+ " 5 : ;
!
+ 5+ # . .$ 5& <
9 ( ! ! ! 2
9 # . . $ 5& !
. 5 # * 5 1 G
G
# "+ + 5C+ +
9 ! ! # " 5C. 5
+ 8
A # " 5 C
. 5 5 # 8 8 (
$J
9 # + "C. + + ". C+ + / "C+ ! @
9 # "C+ →
AB AP
B
sen sen
1
1
=
9 # "+ + →
AP P P
1 1 2
3 5
sen sen
=
9 # C+ + →
BP P P
2 1 2
2 6
sen sen
=
9 # C+ " →
BP
A
AB
2
4
sen sen
=
9 @
AB
AP B
AP
P P
P P
BP
P B
AB
A
1
1
1 2
1 2
2
2
1
3
5
6
2
4
=
=
=
=
sen
sen
sen
sen
sen
sen
sen
sen
5 : # *
. . $. ' 1 . $. &. @
AB AP P P P B
AP P P P B AB
A
B
B
A
E
⋅ ⋅ ⋅
⋅ ⋅ ⋅
=
⋅ ⋅ ⋅
⋅ ⋅ ⋅
=
⋅ ⋅
⋅ ⋅
=
1 1 2 2
1 1 2 2
1 3 6
5 2 4
1 3 6
5 2 4
sen sen sen sen
sen sen sen sen
sen
sen
sen sen sen
sen sen sen
7
3
+
2
=
+ B
A
E
senA
senB
=
% 2 5$@
H
3
+
2 =
/ @
A B H
+ = →
= 2 + 3 B = H - A
&G
cosH)
+
(E
(senA)
=
cosA
senH
senA
E
=
senA
cosH
-
cosA
senH
senA
E
=
A)
-
sen(H
⋅
⋅
⋅
⋅
⋅
⋅
→
= E
senA
senB
tg
sen
cos
A
H
E H
=
+
B H A
= −
9 ! ( 8 :
) # : . #
: 5
. : 8 '
P A
AB B
1
1
=
⋅ sen
sen
P B
AB A
1
5
1
=
⋅ +
sen( )
sen
P A
AB B
2
6
4
=
⋅ +
sen( )
sen
P B
AB A
2
4
=
⋅ sen
sen
θ θ
A
P
A
B
A
1
5
= + +
&
θ θ
A
P
A
B
A
2
= +
θ θ
B
P
B
A
B
1
= −
θ θ
B
P
B
A
B
1
6
= − −
5
! + 5+
= >
+ : ;
+ . + . +$ 9 !
( + 8 #
. ! : :
8
+ ( ",-
. /0
.C,-
. /0. ,-
. /0 5 # α . α . α$ . β . β . β$ . :
!
" : ! 2 . ' : !
# " 5
4 # "C+ . + C+ .. ! @
&
9 # "C+ @
AB BP
A
sen sen
α1
1
=
9 # C+ + @
1
2
2
1
β
α sen
BP
sen
BP
=
9 # C+ +$@
2
3
3
2
β
α sen
BP
sen
BP
=
9 # C+$ @
3
3
β
sen
BC
C
sen
BP
=
! @
3
2
1
3
2
1
3
2
1
3
2
1
β
β
β
α
α
α sen
sen
sen
senA
BC
BP
BP
BP
senC
sen
sen
sen
BP
BP
BP
AB
⋅
⋅
⋅
⋅
⋅
=
⋅
⋅
⋅
⋅
⋅
3
2
1
3
2
1 β
β
β
α
α
α sen
sen
sen
senA
BC
senC
sen
sen
sen
AB
⋅
⋅
⋅
=
⋅
⋅
M
sen
sen
sen
AB
sen
sen
sen
BC
senC
senA
=
⋅
⋅
⋅
⋅
⋅
⋅
=
3
2
1
3
2
1
β
β
β
α
α
α
+ # ' @
A C n B
A C N
+ = − ⋅ − + + + + + +
+ =
( ) ( )
2 200 1 2 3 1 2 3
α α α β β β
sen
sen
A
C
M
A C N
=
+ =
(sen( )) sen
N A M A
− ⋅ =
(sen cos cos sen ) sen
N A N A M A
⋅ − ⋅ ⋅ =
N
M
senN
tgA
cos
1 +
=
&$
A
N
C −
=
: 5 ; : #
+ ! !
5 5 8
@
D D
A
A
P
A
B
1 1
1
=
− −
sen(200 )
sen
α
α
1
1
α
sen
A
sen
D
D B
A
P
B =
θ θ
θ θ α
A
p
A
B
B
P
B
A
A
A
1
1
1
= +
= − − −
(200 )
D D
C
B
P
B
C
3
3
=
sen
senβ
D D
C
C
P
B
C
3 3
3
=
− −
sen(200 )
sen
β
β
θ θ β
θ θ
B
P
B
C
C
P
C
B
C
C
3
3
3
= + − −
= −
(200 )
D D
D D
B
P
B
P
B
P
B
P
2 1 1
2
2 3 3
2
=
=
sen
sen
sen
sen
β
α
α
β
)
200
(
)
200
( 2
3
3
2
β
α
β
θ
θ −
−
+
−
−
+
= C
C
B
P
B
(
5 !
&&
C9M
4
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Teoria_IS_Tema_7.pdf

  • 1.
  • 2. - - ! ! " " # $ % & ' ! ( ) # $ ! & % # * + ' ,- . /0 # # ( 1 " ' ,2 0 3 4 ! 5 67 $ (
  • 3. $ $ ) # $ $ ! $ & + ' ,- . /0 ( + 8 4 $ * " ' ,2 0 $ 1 4 ! 5 67 & ( & ) # & $ ! & & + ' ,- . /0 & * " ' ,2 0 & 1 4 ! 5 67 * ( 2 * ( + 8
  • 4. & - - ! ! " " # $ % & ' ! % ( ! 5 # ( 9 7 ' : $ ; 8 ! ( # . ( # . : : ! % ( 7 !( ! ! ! : : ; . . ! + : 8 . # : * ( # 5 < 5 ! 5 = + ; # ! > = ? ( 8 > 9 ( ( : ! ! . ! . 5 : 5 !
  • 5. * 9 ( 5 # ( ! : ( ! : ! . 8 : % 7 ' ' ! ( ! . . ! : ! ! " " # . ( ! : @ @ @ @ 5 + . ! : . @ @ ! ( ! @ # : ( 8 ! ( 9 # # ; ! @ ( . ( 5
  • 6. 1 % ( # : ( ( . % ( ! ; . ! , < ; 0 5 2 ( ( A # ( 5 ! ( B . : : ! # ! ! 5 : 8 ! . : # ! 9 ( ( . : ' ! # : 8 < : ( % ( : ( # ! ! ( ! ! @ ( 8 ; ( 8 ! ( ! % ; : : ! ! " !" #$ % # & '" #( ! ) * + ! , # : ! !
  • 7. 3 " ( 8 ; ( " 5 : ! : ( . . 5 . @ C. . 5 D ; ! #: . . : ; " ' : 8 ! 9 . 5 : # ' 5 % 5 ! , 8 ; 0 # @ ( ) e e e p l ≤ + 2 2 2 ' 5 " . 5 ! @ . 9. . . C 5 D5 ! : 8 ! ) * 9 ! # ; 8 ! # . # α. # @ α = 200g n E # # : 8 !# : ;
  • 8. F 9 7 ' ! # . G 5 G G ! " !" # - ( ) # * . # ) / 0 ' ) # 11 ! . ) #% , ') # 2 # ) # , - * ' 9 ( ) . : (! . 5: ! 8 ! )5C 8 ; 8 " ) 5 9 5 # 8 8 ! ! ) " ; ! ! 5 # # ; : ! ! + ! ( ; ( ( 8 ; 5 9 . : 5 % 5 + 8 ; 8 5 : : ) + . : 8 ; ! @ = = = = 91 , 213 02 , 158 67 , 81 33 , 49 º 1 g D g C g B g R L L L L = = = 22 , 376 91 , 327 35 , 49 º 2 g F g E g R L L L ) H # : . I # = = = = = = 20 376 89 327 91 , 213 02 , 158 67 , 81 33 , 49 g F g E g D g C g B g R L L L L L L + ; ( ! ! ' @
  • 9. J A"%C69B " 6) K B . L% D 4 % B 9) 4 +4 M ". + D K 9M 9 6N ") 7 9. ) , JJ&0@ O 9 ( 4 ! " 8 ! P ' + ' A - B 1*. B ! ! JJ&. &G*F ( , # $ ! & % # - . ' / 0 1 2 3 # # ( 4 " ' / 5 3 6 ! + 7 A 5 : 5 " 5 C 9 ( @ : # " 5 ; ! @ , 0 A : # C 5 ; ! @ , 0 A 3 9 " 5C 5 8 ! , 0 " ' θA B 5 + # ! " 5 C . '
  • 10. G 5 θA B . ! ". LA B 6 ; θA V # C. θB V N # A : " 5C. V B θ 5 V A θ 9 ! ! ( : ( % : # " 5C , ! 0 9 : A # 8 # : ε" 5 εC ) # 5 ! # # Q % B 5 ' A % "A 5 CA 5 # ! 5 # ε" 5εC : < : B 5Q % "A 5 : . ' % 5 B Q CA + : Q % B % ; B5 Q % ! "A 5 8 # 2⋅ ⋅ AV B ε
  • 11. % ε" 5εC ! # 8 ; ! " 5C. " # . ; B Q5 % ! CA 5 8 # 2⋅ ⋅ BV A ε (. ; # ! ! : ! " ! : ! # . 5 . @ 2 ˆ V sen e L e a ⋅ = 9 V # G G # A G $ . & # A : ! ! # ' J1 .3FF1 9 % ! %. 5A 9 ! @ 0 ! 0 !( 0 5 # 5 : # : # # # . 5 5 B ! ! :
  • 12. : : ! # : # $G 3 9 . : ' # # . # : ! ! + ! ! , # 0 8 + # # ,G 0 . ! % # # ' . : ! C ! 8 ; a e V sen e L 2 ˆ ⋅ = 4 5 6 7 9 ( @ ",- ". /"0 C,- C. /C0 ! 8 ! ! @ A B V B B A V A L L B desde L L A desde , : , : " " 5C. θA B . 5 : 8 5 5 θA B . ! ". LA B 6 ; θA V ; C. θB V
  • 13. $ 9 . ( : ! . % ! ( ; : ' 7 ! " !" #) # 1 ! & % - * 8 ," # &) * 9 , : % & !' 0 & * - ' , " !# &# ! ) # ;'" # & 1 # , ! " . ! # ( @ A B V B V A B A B A θ θ θ θ − = − = % "C !( 5 : @ ( ) 2 2 ) ( A B A B B A Y Y X X D − + − = 9 ! : # "A 5CA % 5 9 # "CA. @ A sen D V sen D B sen D V B r B A r v A r ˆ = = B A r V A r D V sen B sen D ˆ =
  • 14. & B A r V B r D V sen A sen D = 9 # ( 9 # # 8 G G ) ˆ ˆ ( 200 ˆ 200 ˆ ˆ ˆ B A V D B A g + − = = + + # # . 5 . ! 8 : % - . / A ! @ 0 + "@ " 5A@ V A V A r V A V A r D sen D θ θ cos y x V A V A ⋅ = ∆ ⋅ = ∆ % A # @ V A A V V A A V y Y Y x X X ∆ + = ∆ + = ! 0 + C@ V B V B r V B V B r D sen D θ θ cos y x V B V B ⋅ = ∆ ⋅ = ∆ V B B V B V y Y Y X X xV B ∆ + = ∆ + = % : ! " 5 C 8 ! # . 8 5 ! B : 9 . 7 ! " !" #& ' 1 ' , : # , # 1 9 ( . : - / !
  • 15. * + @ A V A V V A Y Y X X − − = θ tan A V V A A V X X Y Y − = ⋅ − θ tan ) ( A V A A V V X Y Y X + ⋅ − = θ tan ) ( + @ B V V B B V B V B V V B X X Y Y Y Y X X − = ⋅ − − − = θ θ tan ) ( tan 8 - A @ B A V A A V B B V A V B V B A V A A V V B B V X X Y Y Y X X Y Y Y Y − + ⋅ − ⋅ = − − + ⋅ − = ⋅ − θ θ θ θ θ θ tan tan ) tan (tan tan ) ( tan ) ( V A V B B A V A A V B B V X X Y Y Y θ θ θ θ tan tan tan tan − − + ⋅ − ⋅ = 6 ;: 8 5 /. - . 5 ! : + ( ! # . : 8 , ; # 0
  • 16. 1 9 ' ; ( ( + ! 8 5 ; 8 9 @ ; " 5 ; C % 2 A ! # ! ; V B V A V B B V V A A V V D D D H D H H 1 1 1 ) ( 1 ) ( + ⋅ + ⋅ = < 6 6 ' 5 67 9 : # 5 : : 8 N ' ; ! . + ! . 5 ! : 67 . # : 8 6 ; . # ( # ! . A B V B V A B A O O B O O A − = − = % 5 ! ( ( ) 2 2 ) ( ) ( A UTM B UTM A UTM B UTM UTM B A Y Y X X D − + − = 9 # "CA. @ A sen D V sen D B sen D UTM V B UTM B A UTM v A ˆ ) ( ) ( ) ( = =
  • 17. 3 UTM B A UTM V A D V sen B sen D ) ( ˆ ) ( = UTM B A UTM V B D V sen A sen D ) ( ) ( = 9 # ( . G G ) ˆ ˆ ( 200 ˆ 200 ˆ ˆ B A V V B A g + − = = + + 5 6 % - 67 . /67 A ! @ 0 + "@ % " 5A # @ V A UTM V A UTM V A UTM V A UTM O D senO D cos ) ( ) y ) ( ) x ( V A V A ⋅ = ∆ ⋅ = ∆ UTM V A UTM A UTM V UTM V A UTM A UTM V y Y Y x X X ) ( ) ( ) ( ) ( ) ( ) ( ∆ + = ∆ + = ! 0 + C@ V B TM V B UTM V B UTM V B UTM O D senO D cos ) ( ) y ( ) ( ) x ( V B V B ⋅ = ∆ ⋅ = ∆ UTM V B UTM B UTM V UTM UTM B UTM V y Y Y X X ) ( ) ( ) ( ) x ( ) ( ) ( V B ∆ + = ∆ + = % : ! " 5 C 8 = 5 67 ' ! . ' 5 % ' ! ( 9 ! : : . 5
  • 18. F ! ". 5: < C 9 : # # # : # ! . 5: # @ R ) (D K) (0.5 m i t H 2 B A B A B A B A − + − + = ∆ " 5C. ( 67 5 ( ! # 5 9 5 ! ! @ + + ∆ − = R h R h h D K D G UTM 2 1 2 2 1 1 8 58. 5 . 5R 8 ! ) 7 . : 1$3GS % ( ( @ 2 2 1 2 2 2 1 1 h R h R h k D D UTM G ∆ + + + = % ! + # @ R 8 2 2 2 r G D h D + ∆ =
  • 19. J " ! " 5C. # # # @ A sen D V sen D B sen D V B r B A r v A r ˆ = = B A r V A r D V sen B sen D ˆ = B A r V B r D V sen A sen D = % A ! # V B V A V B B V V A A V V D D D H D H H 1 1 1 ) ( 1 ) ( + ⋅ + ⋅ = $ ( $ , # $$ ! $& . ' / 0 1 2 3 ( . ) 6 $- " ' / 5 3 $4 6 ! + 7 % ! # ' 5 ! + 8 : ! ( 2 2 h D D G r ∆ − =
  • 20. G % ! ( . 5 ( ! . 5 % +". +C. + # ! # @ " ,- ". /"0 C,- C./C0 ,-./ 0 % # @ LP A , L , L P B P C 3 % # ; : ( + 9 5 ) 5 A ! ( 9 ( # 9 8 : # T 5U 8 5 8 : " ' ! + 9 # ( # T 5U
  • 21. 9 ! , 5 : 0 # 9 . 5 @ T V U V C W G G 9 : ( % ! + ( . ; ; ; " . 5 ! : "I . CI . I "I CI I 6 "I . CI . I # ! : ! @ 2 2 ) intermedio ( mayor) ( 2 2 lado lado S e e a + ⋅ ⋅ = 9 ! @ o ! ; : o "I . CI . I: !
  • 22. o 5 o # % ! # : ! ! , 0 + # ! # 5 # 9 @ 5 : % 8 ' ! # 9 : , : ' # ! 0 # # : # : ! 5 # ' ' 9 ! # @ 0 + @ # : ! 0 % 5 @ : 1G X . 5 : 5 ' @ # : # % ! / B ! : ! , 5 : 0 # 4 5 6 7 8 ! ( 5 5 @ " ,- ". /"0 C,- C./C0 ,-./ 0 / @ LP A , L , L P B P C 4 C 9) A" Y B 9 " +4
  • 23. $ % +". +C. + # ! 9 ? 69 " 9 "% 6%4 + # T 5U T W %+ C %+ " U W %+ %+ C " ". C 5 . 5 "C 5 C 9 # C @ 9 ! # ' # " 5 + ! % . 5 ! @ C B A B B θ θ − = ˆ
  • 24. & ( + 8 ( # ;" ) ' @ ") L " "C" Z ". , JJ 0@ [ + ! + 8 @ [ ' + ' A B X &* L " JJ 9) B K B 9M "B 7 Y B . , JJ 0@ [ B ( [ ' + ' A . X&* L " JJ 9 : ! +C ! # D A D D A P B A B A B sen sen sen sen = → = α α DP B β β sen C n se D sen D C sen D C B C B B P = → = B P D +C ! @ D A A B sen senα = D C B C sen senβ % ! ( : . 5 : : ," 5 0 ! @
  • 25. * sen sen sen sen C A D D A B B C = ⋅ ⋅ β α 9 ! + # # : δ @ δ α β tan = ⋅ ⋅ sen D sen D C B B A " ! # # # ,: ( 0 . ( + @ 1 tan ˆ ˆ δ = A sen C sen A @ d c b a = 9 ! @ c d c d a b a b − + = − + 4 @ δ δ tan 1 tan 1 − + = − + senC senA senC senA 4 ( @ g 50 tan 1= δ δ δ δ tan 50 tan 1 tan 50 tan tan 1 tan 1 ⋅ − + = − + g g / ! @ b a b a b a tan tan 1 tan tan ) tan( ⋅ − + = + 9 . @
  • 26. 1 ) 50 tan( tan 50 tan 1 tan 50 tan δ δ δ + = ⋅ − + g g g 9 @ ) 50 tan( tan 1 tan 1 δ δ δ + = − + g ! : @ ) 50 tan( δ + = − + senC senA senC senA 2 ( . 5 @ 2 2 cos 2 2 cos 2 2 b a sen b a b a b a sen senb sena senb sena − + − + = − + ! @ ) ( 2 1 ) ( 2 1 cos 2 ) ( 2 1 cos ) ( 2 1 2 C A sen C A C A C A sen senC senA senC senA − + − + = − + ) ( 2 1 tan ) ( 2 1 tan 2 2 cos 2 cos 2 2 2 C A g c C A C A sen C A C A C A sen senC senA senC senA − ⋅ + = − − + + = − + % @ δ δ tan 1 tan 1 − + = − + senC senA senC senA 5 ! ) ( 2 1 tan ) ( 2 1 tan C A g c C A senC senA senC senA − ⋅ + = − + ) 50 tan( tan 1 tan 1 δ δ δ + = − + g
  • 27. 3 @ ) 50 tan( ) ( 2 1 tan ) ( 2 1 tan δ + = − ⋅ + C A c C A / @ ) 50 tan( ) ( 2 1 tan ) ( 2 1 tan δ + + = − g c C A C A 9 @ ) 50 tan( 2 tan ) ( 2 1 tan δ + + = − g C A C A 9 : 1 2 ( ) A C − . 5 : δ 9 # +"C 8 : # &G G@ A C B g + + + + = α β 400 ) ( 400 B C A g + + − = + β α 1 2 200 1 2 ( ) ( ) A C B g + = − + + α β + 1 2 ( ) A C + . @ M A C N A C = + = − 1 2 1 2 ( ) ( ) " 5 @ N M C N M A − = + = / ! ( 8 :
  • 28. F 9 : ( . # : ,"C+ 5 C +0 5 +C ! @ D A D D A P B A B A B sen sen sen sen = → = α α DP B D C D D C P B A B B C sen sen sen sen = → = β β DP B @ D A A B sen senα = D C B C sen senβ % ( ," 5 0 5 : : @ M sen D sen D senC senA B A C B = ⋅ = ⋅ β α + 8 : # &G G A C B g + + + + = α β 400 B . : ( @ ( ) C B A E A = − − − − = − 400 β α D A D C A B B C ⋅ ⋅ = ⋅ ⋅ sen sen sen sen β α D A D E A A B B C ⋅ ⋅ = ⋅ − ⋅ sen sen sen( ) sen β α [ ] D A D E A E A A B B C ⋅ ⋅ = ⋅ ⋅ − ⋅ ⋅ sen sen sen cos cos sen sen β α [ ] [ ] sen sen sen cos sen sen cos A D BC E D E A A B B C ⋅ ⋅ + ⋅ ⋅ = ⋅ ⋅ ⋅ β α α tg sen sen sen sen cos A D E D D E B C A B B C = ⋅ ⋅ ⋅ + ⋅ ⋅ α β α C E A = −
  • 29. J " # " 8 5: : ) ˆ 200 ( ˆ A tg A tg + = 9 ! # " 5 . 5 ! 6 ; # " 5 : ( . ! A C A C B P C B A P A ˆ 200 ˆ − ± = + = θ θ θ θ % 8 @ 1 2 2 1 200 200 B B B ó C B A B − = − − = − − = β α D A D D A P B A B A B sen sen sen sen = → = α α DP B α α sen senB D senB D sen D B A P A B A 1 D 1 P A = → = β β sen senB D D C B C B 2 D senB2 D sen P C P C = → = 6 ; ! 5 ! A ". ! . # P A P A A P P A P A A p D Y Y sen D X X θ θ cos ⋅ + = ⋅ + = P C P C C P P C P C C p D Y Y sen D X X θ θ cos ⋅ + = ⋅ + = 4 =7 % ! ! ( ( 9 # ! @
  • 30. $G ) ( ) ( ) ( r e C m i t H r e C m i t H r e C m i t H B P B P B P A P A P A P C P C P C P + + − + = ∆ + + − + = ∆ + + − + = ∆ % 8 . 8 % : 9 ( # ! @ B A B A r tgV D t = + ! + ! # ' ! # ". C 5 + 8 P C C C P P B B B P P A A A P H H H H H H H H H ∆ + = ∆ + = ∆ + = ) ( ) ( ) ( ! 2 + # . @ P C P B P A P C C P P B B P P A A p P D D D D H D H D H H 1 1 1 1 ) ( 1 ) ( 1 ) ( + + ⋅ + ⋅ + ⋅ = 6 # 5 : : 8 . ; ! % 5 ! ( @ ( ) 2 2 ) ( ) ( A UTM B UTM A UTM B UTM UTM B A Y Y X X D − + − = ( ) 2 2 ) ( ) ( C UTM B UTM C UTM B UTM UTM C B Y Y X X D − + − = ". C 5 . 5 # C
  • 31. $ ( + 8 ( # ;" . 5 ! # " 5 5 6 % - 67 . /67 A ! @ 0 + "@ " 5A@ P A UTM P A UTM P A UTM P A UTM O D senO D cos ) ( ) y ) ( ) x ( P A P A ⋅ = ∆ ⋅ = ∆ UTM P A UTM A UTM P UTM P A UTM A UTM P y Y Y x X X ) ( ) ( ) ( ) ( ) ( ) ( ∆ + = ∆ + = ! 0 + @ P C UTM P C UTM P C UTM P C UTM O D senO D cos ) ( ) y ( ) ( ) x ( P C P C ⋅ = ∆ ⋅ = ∆ UTM P C UTM C UTM P UTM UTM C UTM P y Y Y X X ) ( ) ( ) ( ) x ( ) ( ) ( P C ∆ + = ∆ + = % : ! " 5 8 = % ! ( 9 # @ R ) (D K) (0.5 m i t H 2 A P A P A P A P − + − + = ∆ ) ! ( ,8 ! # " 5 0 - 67 . /67 + + 5". ( 67 5 ( " : 8 . ! # ! : 5 . # : : #
  • 32. $ ". C 5 5 "C " : ; "C % ( ( @ 2 2 2 2 1 1 ) ( ) ( B A B A UTM B A g B A h R h R h k D D ∆ + + + = % ! + # @ " ! " 5C. # # # @ α sen D B sen D B A r P A r = 1 B A r P A r D sen B sen D α 1 ˆ = 5 # ; # C+ 5 + + ! # ' ! # ". C 5 + 8 R 8 2 2 2 ) ( ) ( r B A B A g B A D h D + ∆ = 2 2 ) ( ) ( B A g B A r B A h D D ∆ − =
  • 33. $$ P C C C P P B B B P P A A A P H H H H H H H H H ∆ + = ∆ + = ∆ + = ) ( ) ( ) ( ! 2 + # . @ P C P B P A P C C P P B B P P A A p P D D D D H D H D H H 1 1 1 1 ) ( 1 ) ( 1 ) ( + + ⋅ + ⋅ + ⋅ = & ( & , # &$ ! && . ' / 0 1 2 3 &- " ' / 5 3 &4 6 ! + 7 % ( : ! ! . 5 : 5 :
  • 34. $& A C % @ : # " 5 ; ! @ , 0 @ A : # A 5 ; ! @ " C 9 ( # @ ,- ". /"0. ,- C. /C0 / 8 !# ! @ "@ %" C . %" A A %A " . %A C 3 ! ! "C 5 # " T W %" A %" C : ( A ! 5 ; # U U W %A C %A " % ! ( A
  • 35. $* + ! : 8 5 : : A : ;5 : ( 9 # ! 9 # ! : ! ; ; ; 9 ! : 5 4 5 6 7 % ( # : # ( : 8 4 =7 9 ' ; ( ( + ! 8 5 ; 8 . % 2 A ! # ; V B V A V B B V V A A V V D D D H D H H 1 1 1 ) ( 1 ) ( + ⋅ + ⋅ = 6 6 ' 5 67 9 : # 5 : : 8 N ' ; ! . + ! . 5 ! : 67 . # : 8
  • 36. $1 6 ; . # ( # ! V B A B B A V A O O B O O A − = − = 9 # ( . G G ) ˆ ( 200 200 ˆ ˆ ˆ V A B D B A g + − = = + + % 5 ! ( ( ) 2 2 ) ( ) ( A UTM B UTM A UTM B UTM UTM B A Y Y X X D − + − = 9 # "CA. @ A sen D V sen D B sen D UTM V B UTM B A UTM v A ˆ ) ( ) ( ) ( = = UTM B A UTM V A D V sen B sen D ) ( ˆ ) ( = UTM B A UTM V B D V sen A sen D ) ( ) ( = 5 6 % - 67 . /67 A ! @ 0 + "@ " 5A@ V A UTM V A UTM V A UTM V A UTM O D senO D cos ) ( ) y ) ( ) x ( V A V A ⋅ = ∆ ⋅ = ∆ UTM V A UTM A UTM V UTM V A UTM A UTM V y Y Y x X X ) ( ) ( ) ( ) ( ) ( ) ( ∆ + = ∆ + =
  • 37. $3 ! 0 + C@ V B TM V B UTM V B UTM V B UTM O D senO D cos ) ( ) y ( ) ( ) x ( V B V B ⋅ = ∆ ⋅ = ∆ UTM V B UTM B UTM V UTM UTM B UTM V y Y Y X X ) ( ) ( ) ( ) x ( ) ( ) ( V B ∆ + = ∆ + = % : ! " 5 C 8 = % ' ! ( 9 ! : : . 5 ! " : ! % ( ( @ 2 2 1 2 2 2 1 1 h R h R h k D D UTM G ∆ + + + = % ! + # @ " ! " 5C. # # # @ A sen D V sen D B sen D V B r B A r v A r ˆ = = B A r V A r D V sen B sen D ˆ = B A r V B r D V sen A sen D = R 8 2 2 2 r G D h D + ∆ = 2 2 h D D G r ∆ − =
  • 38. $F % A ! # V B V A V B B V V A A V V D D D H D H H 1 1 1 ) ( 1 ) ( + ⋅ + ⋅ = - ( 5 - ( . ) = 9 + 5 + " 5 : ; ! + 5+ # . .$ 5& < 9 ( ! ! ! 2 9 # . . $ 5& ! . 5 # * 5 1 G G # "+ + 5C+ + 9 ! ! # " 5C. 5 + 8 A # " 5 C . 5 5 # 8 8 (
  • 39. $J 9 # + "C. + + ". C+ + / "C+ ! @ 9 # "C+ → AB AP B sen sen 1 1 = 9 # "+ + → AP P P 1 1 2 3 5 sen sen = 9 # C+ + → BP P P 2 1 2 2 6 sen sen = 9 # C+ " → BP A AB 2 4 sen sen = 9 @ AB AP B AP P P P P BP P B AB A 1 1 1 2 1 2 2 2 1 3 5 6 2 4 = = = = sen sen sen sen sen sen sen sen 5 : # * . . $. ' 1 . $. &. @ AB AP P P P B AP P P P B AB A B B A E ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ = 1 1 2 2 1 1 2 2 1 3 6 5 2 4 1 3 6 5 2 4 sen sen sen sen sen sen sen sen sen sen sen sen sen sen sen sen 7 3 + 2 = + B A E senA senB = % 2 5$@ H 3 + 2 = / @ A B H + = → = 2 + 3 B = H - A
  • 40. &G cosH) + (E (senA) = cosA senH senA E = senA cosH - cosA senH senA E = A) - sen(H ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ → = E senA senB tg sen cos A H E H = + B H A = − 9 ! ( 8 : ) # : . # : 5 . : 8 ' P A AB B 1 1 = ⋅ sen sen P B AB A 1 5 1 = ⋅ + sen( ) sen P A AB B 2 6 4 = ⋅ + sen( ) sen P B AB A 2 4 = ⋅ sen sen θ θ A P A B A 1 5 = + +
  • 41. & θ θ A P A B A 2 = + θ θ B P B A B 1 = − θ θ B P B A B 1 6 = − − 5 ! + 5+ = > + : ; + . + . +$ 9 ! ( + 8 # . ! : : 8 + ( ",- . /0 .C,- . /0. ,- . /0 5 # α . α . α$ . β . β . β$ . : ! " : ! 2 . ' : ! # " 5 4 # "C+ . + C+ .. ! @
  • 42. & 9 # "C+ @ AB BP A sen sen α1 1 = 9 # C+ + @ 1 2 2 1 β α sen BP sen BP = 9 # C+ +$@ 2 3 3 2 β α sen BP sen BP = 9 # C+$ @ 3 3 β sen BC C sen BP = ! @ 3 2 1 3 2 1 3 2 1 3 2 1 β β β α α α sen sen sen senA BC BP BP BP senC sen sen sen BP BP BP AB ⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ 3 2 1 3 2 1 β β β α α α sen sen sen senA BC senC sen sen sen AB ⋅ ⋅ ⋅ = ⋅ ⋅ M sen sen sen AB sen sen sen BC senC senA = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = 3 2 1 3 2 1 β β β α α α + # ' @ A C n B A C N + = − ⋅ − + + + + + + + = ( ) ( ) 2 200 1 2 3 1 2 3 α α α β β β sen sen A C M A C N = + = (sen( )) sen N A M A − ⋅ = (sen cos cos sen ) sen N A N A M A ⋅ − ⋅ ⋅ = N M senN tgA cos 1 + =
  • 43. &$ A N C − = : 5 ; : # + ! ! 5 5 8 @ D D A A P A B 1 1 1 = − − sen(200 ) sen α α 1 1 α sen A sen D D B A P B = θ θ θ θ α A p A B B P B A A A 1 1 1 = + = − − − (200 ) D D C B P B C 3 3 = sen senβ D D C C P B C 3 3 3 = − − sen(200 ) sen β β θ θ β θ θ B P B C C P C B C C 3 3 3 = + − − = − (200 ) D D D D B P B P B P B P 2 1 1 2 2 3 3 2 = = sen sen sen sen β α α β ) 200 ( ) 200 ( 2 3 3 2 β α β θ θ − − + − − + = C C B P B ( 5 !
  • 44. && C9M 4 "). ND 4 B 7. 5 9%A B . " , JFG 0 C)B Q 9) . ) D B B Q . ) 5, JF30 2 69 " +"M 4 . , JF$0@ 7 4 Z B N 69MN ") Z " 7 9L 9) 4 . , J3F0 ") L " "C" Z ". , JJ 0@ [ + ! + 8 @ [ ' + ' A B X &* L " JJ 9) B K B 9M "B 7 Y B . , JJ 0@ [ B ( [ ' + ' A . X&* L " JJ 4 L 9 ". L% , JF&0 6) 9B . LD+) 9. ] , JJ 0 A"%C69B " 6) "B . L% D 4 % B 9) 4 . + . K 9M 9 6N ") 7 9. ) , JJ&0@ [ 9 ( ! 8 ! [ ' + ' A4 % - B X1*. B ! ! JJ&. &G*F