Методическое пособие по решению задач геометрического моделирования в системе MathCAD. Найханов В.В. - 29 стр.

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29
λ
i
1 μ
i
−:=μ
i
h
i1−
h
i
h
i1−
+
:=i1M1−..:=
h
i
T
i1+
T
i
−:=i0M1−..:=
T
i
T
i1−
X
i
X
i1−
−
()
2
Y
i
Y
i1−
−
()
2
++:=i1M..:=T
0
0:=
Y
i
sin 3 G
i
⋅
()
:=X
i
sin 2 G
i
⋅
()
:=G
i
i
2π
M
⋅:=i0M..:=
b1 v() v
2
− v
3
+:=b0 v() v 2v
2
− v
3
+:=a1 v() 3v
2
2v
3
−:=a0 v() 1 3v
2
− 2v
3
+:=
M30:=ORIGIN 0:=
c
M
3
X
1
X
0
−
h
0
()
2
X
M
X
M1−
−
h
M1−
()
2
+
⎡
⎢
⎢
⎣
⎤
⎥
⎥
⎦
⋅:=
A
MM,
2
h
M1−
:=A
MM1−,
1
h
M1−
:=A
M1,
1
h
0
:=A
M0,
2
h
0
:=c
0
0:=A
0M,
1−:=A
00,
1:=
c
i
3 μ
i
X
i1+
X
i
−
h
i
⋅λ
i
X
i
X
i1−
−
h
i1−
⋅+
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
⋅:=A
ii 1+,
μ
i
:=A
ii,
2:=A
ii1−,
λ
i
:=i1M1−..:=
A
ij,
0:=j0M..:=i0M..:=
i0M..:= j0M..:= B
ij,
0:=
i1M1−..:= B
ii1−,
λ
i
:= B
ii,
2:= B
ii 1+,
μ
i
:= d
i
3 μ
i
Y
i1+
Y
i
−
h
i
⋅λ
i
Y
i
Y
i1−
−
h
i1−
⋅+
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
⋅:=
B
00,
1:= B
0M,
1−:= d
0
0:= B
M0,
2
h
0
:= B
M1,
1
h
0
:= B
MM1−,
1
h
M1−
:= B
MM,
2
h
M1−
:=
d
M
3
Y
1
Y
0
−
h
0
()
2
Y
M
Y
M1−
−
h
M1−
()
2
+
⎡
⎢
⎢
⎣
⎤
⎥
⎥
⎦
⋅:=
pX A
1−
c⋅:= pY B
1−
d⋅:=
xt() E 0←
z
tT
i
−
h
i
←
EX
i
a0 z()⋅ X
i1+
a1 z()⋅+ pX
i
b0 z()⋅ h
i
⋅+ pX
i1+
b1 z()⋅ h
i
⋅+←
T
i
t≤ T
i1+
<if
i0M1−..∈for
E
:=
                                                                                                                        29

ORIGIN:= 0                 M := 30
                           2              3                                         2              3                                           2            3                         2           3
a0( v ) := 1 − 3v + 2v                             a1( v ) := 3v − 2v b0( v ) := v − 2v + v                                                                     b1( v ) := −v + v

                                                                                (              )                        (                  )
                                         2π
i := 0 .. M        G := i ⋅                                X := sin 2 ⋅ G                              Y := sin 3 ⋅ G
                       i                 M                    i                              i             i                           i

T := 0
 0
                  i := 1 .. M                      T := T
                                                       i                 i− 1
                                                                                +           ( Xi − Xi−1)2 + (Yi − Yi−1)2
i := 0 .. M − 1                h := T                  −T
                                 i            i+ 1                 i

                                                  h
                                                      i− 1
i := 1 .. M − 1                μ i :=                                         λi := 1 − μ i
                                          h +h
                                              i             i− 1


i := 0 .. M        j := 0 .. M                A             := 0
                                                   i, j
                                                                                                                                                    ⎛           X         −X                          X −X
                                                                                                                                                                                                                     i− 1 ⎞
                                                                                                                              c := 3 ⋅ ⎜ μ i ⋅                                                                            ⎟
                                                                                                                                                                  i+ 1            i                       i
i := 1 .. M − 1 A                         := λi A                        := 2              A               := μ i                                                                     + λi ⋅
                               i , i− 1                           i, i                         i , i+ 1                            i                ⎜                h                                        h              ⎟
                                                                                                                                                    ⎝                    i                                        i− 1       ⎠
                                                                                                               2                                        1                                         1                                             2
A          := 1 A                    := −1                 c := 0                       A              :=                   A                  :=                A                     :=                           A                 :=
    0, 0               0, M                                   0                             M,0                h                M,1                 h                M , M −1                 h                          M,M                h
                                                                                                                   0                                    0                                         M −1                                          M −1
                                                   ⎡ X1 − X0 XM − XM −1 ⎤
                                          c := 3 ⋅ ⎢        +             ⎥
                                           M
                                                   ⎢ (h )2    ( h M − 1) ⎦
                                                                        2 ⎥
                                                   ⎣    0



i := 0 .. M        j := 0 .. M                    B             := 0
                                                       i, j
                                                                                                                                                                     ⎛           Y            −Y                         Y −Y
                                                                                                                                                                                                                                        i− 1 ⎞
                                                                                                                                                        d := 3 ⋅ ⎜ μ i ⋅                                                                     ⎟
                                                                                                                                                                                      i+ 1            i                      i
i := 1 .. M − 1                 B                 := λi                       B            := 2                B              := μ i                                                                      + λi ⋅
                                     i , i− 1                                       i, i                           i , i+ 1                                 i        ⎜                    h                                      h              ⎟
                                                                                                                                                                     ⎝                        i                                      i− 1       ⎠
                                                                                                   2                                   1                                         1                                               2
B          := 1    B             := −1 d := 0 B                                             :=                 B              :=                    B                :=                           B                 :=
 0, 0                  0, M                            0                      M,0                  h               M,1                 h                M , M −1             h                     M,M                   h
                                                                                                       0                                   0                                     M −1                                        M −1

                                                                          ⎡ Y1 − Y0                        Y
                                                                                                               M
                                                                                                                       −Y
                                                                                                                              M −1 ⎤
                                              d            := 3 ⋅ ⎢                                    +                                   ⎥
                                                  M
                                                                          ⎢
                                                                          ⎣     ( h 0)      2
                                                                                                               (hM −1)          2          ⎥
                                                                                                                                           ⎦

                  −1                                       −1
 pX := A               ⋅c                pY := B                       ⋅d

       x( t) :=        E← 0
                       for i ∈ 0 .. M − 1
                           if T ≤ t < T
                                     i                  i+ 1
                                              t−T
                                                            i
                                  z←
                                                   h
                                                       i
                                  E ← X ⋅ a0( z) + X                                         ⋅ a1( z) + pX ⋅ b0( z) ⋅ h + pX                                             ⋅ b1( z) ⋅ h
                                                   i                                 i+ 1                               i                           i            i+ 1                         i
                       E

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