Krzywa regularna – krzywa kawałkami gładka.
Definicje formalne
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Parę uporządkowaną
Γ
=
(
γ
,
|
Γ
|
)
,
{\displaystyle \Gamma =(\gamma ,|\Gamma |),}
γ
:
[
α
,
β
]
→
C
,
{\displaystyle \gamma \colon [\alpha ,\beta ]\to \mathbb {C} ,}
|
Γ
|
=
γ
(
[
α
,
β
]
)
,
{\displaystyle |\Gamma |=\gamma ([\alpha ,\beta ]),}
krzywą regularną , gdy
γ
{\displaystyle \gamma }
funkcją ciągłą oraz istnieje skończony układ punktów
{
t
0
,
t
1
,
…
,
t
n
}
{\displaystyle \{t_{0},t_{1},\dots ,t_{n}\}}
α
=
t
0
<
t
1
<
t
2
<
…
<
t
n
−
1
<
t
n
=
β
{\displaystyle \alpha =t_{0}<t_{1}<t_{2}<\ldots <t_{n-1}<t_{n}=\beta }
γ
|
[
t
i
−
1
,
t
i
]
{\displaystyle \gamma |_{[t_{i-1},t_{i}]}}
[
t
i
−
1
,
t
i
]
,
{\displaystyle [t_{i-1},\ t_{i}],}
i
∈
{
1
,
2
,
…
,
n
}
,
{\displaystyle i\in \{1,2,\dots ,n\},}
ciągłą pochodną . Punkty
γ
(
α
)
,
γ
(
β
)
{\displaystyle \gamma (\alpha ),\gamma (\beta )}
Γ
,
{\displaystyle \Gamma ,}
|
Γ
|
{\displaystyle |\Gamma |}
γ
{\displaystyle \gamma }
Krzywa
L
⊂
R
n
{\displaystyle L\subset \mathbb {R} ^{n}}
regularna wtedy i tylko wtedy, gdy
∃
(
K
1
,
K
2
,
…
,
K
n
)
∀
i
∈
{
1
,
2
,
…
,
n
}
K
i
{\displaystyle \exists (K_{1},K_{2},\dots ,K_{n})\forall i\in \{1,2,\dots ,n\}K_{i}}
łukiem regularnym i koniec
K
i
{\displaystyle K_{i}}
K
i
+
1
.
:
{\displaystyle K_{i+1}.{:}}
K
n
{\displaystyle K_{n}}
K
1
{\displaystyle K_{1}}
L
{\displaystyle L}
Równoważność krzywych regularnych
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Niech
Γ
,
Γ
~
{\displaystyle \Gamma ,{\tilde {\Gamma }}}
γ
:
[
α
,
β
]
→
C
,
{\displaystyle \gamma \colon [\alpha ,\beta ]\to \mathbb {C} ,}
γ
~
:
[
α
~
,
β
~
]
→
C
.
{\displaystyle {\tilde {\gamma }}\colon [{\tilde {\alpha }},{\tilde {\beta }}]\to \mathbb {C} .}
Γ
,
Γ
~
{\displaystyle \Gamma ,{\tilde {\Gamma }}}
krzywymi równoważnymi , gdy istnieje surjekcja rosnąca (i tym samym ciągła )
σ
:
[
α
,
β
]
→
[
α
~
,
β
~
]
{\displaystyle \sigma \colon [\alpha ,\beta ]\to [{\tilde {\alpha }},{\tilde {\beta }}]}
α
=
t
0
<
t
1
<
t
2
<
…
<
t
n
−
1
<
t
n
=
β
,
{\displaystyle \alpha =t_{0}<t_{1}<t_{2}<\ldots <t_{n-1}<t_{n}=\beta ,}
i
∈
{
1
,
2
,
…
,
n
}
{\displaystyle i\in \{1,2,\dots ,n\}}
σ
|
[
t
i
−
1
,
t
i
]
{\displaystyle \sigma |_{[t_{i-1},t_{i}]}}
ciągłą pochodną i
γ
=
γ
~
∘
σ
.
{\displaystyle \gamma ={\tilde {\gamma }}\circ \sigma .}
Operacje na krzywych regularnych
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Krzywa przeciwna
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Niech
Γ
{\displaystyle \Gamma }
γ
:
[
α
,
β
]
→
C
.
{\displaystyle \gamma \colon [\alpha ,\beta ]\to \mathbb {C} .}
γ
1
{\displaystyle \gamma _{1}}
γ
1
(
t
)
=
γ
(
−
t
)
{\displaystyle \gamma _{1}(t)=\gamma (-t)}
t
∈
[
−
β
,
−
α
]
{\displaystyle t\in [-\beta ,-\alpha ]}
krzywą przeciwną do
Γ
{\displaystyle \Gamma }
−
Γ
.
{\displaystyle -\Gamma .}
Suma krzywych
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Niech
γ
1
:
[
α
1
,
β
1
]
→
C
,
{\displaystyle \gamma _{1}\colon [\alpha _{1},\beta _{1}]\to \mathbb {C} ,}
γ
2
:
[
α
2
,
β
2
]
→
C
,
{\displaystyle \gamma _{2}\colon [\alpha _{2},\beta _{2}]\to \mathbb {C} ,}
Γ
1
,
{\displaystyle \Gamma _{1},}
Γ
2
.
{\displaystyle \Gamma _{2}.}
γ
1
(
β
1
)
=
γ
2
(
α
2
)
,
{\displaystyle \gamma _{1}(\beta _{1})=\gamma _{2}(\alpha _{2}),}
γ
{\displaystyle \gamma }
γ
(
t
)
=
{
γ
1
(
t
)
dla
t
∈
[
α
1
,
β
1
]
γ
2
(
t
−
β
1
+
α
2
)
dla
t
∈
[
β
1
,
β
1
+
β
2
−
α
2
]
{\displaystyle \gamma (t)={\begin{cases}\gamma _{1}(t)&{\text{dla }}t\in [\alpha _{1},\beta _{1}]\\\gamma _{2}(t-\beta _{1}+\alpha _{2})&{\text{dla }}t\in [\beta _{1},\beta _{1}+\beta _{2}-\alpha _{2}]\end{cases}}}
nazywamy sumą tych krzywych i oznaczamy
Γ
1
+
Γ
2
{\displaystyle \Gamma _{1}+\Gamma _{2}}
Niech
a
,
b
∈
C
.
{\displaystyle a,b\in \mathbb {C} .}
Odcinkiem zorientowanym o początku w punkcie
a
{\displaystyle a}
b
{\displaystyle b}
γ
{\displaystyle \gamma }
γ
(
t
)
=
a
+
t
(
b
−
a
)
,
{\displaystyle \gamma (t)=a+t(b-a),}
t
∈
[
0
,
1
]
.
{\displaystyle t\in [0,1].}
Dodatnio zorientowanym okręgiem o środku w punkcie
a
∈
C
{\displaystyle a\in \mathbb {C} }
r
>
0
{\displaystyle r>0}
γ
{\displaystyle \gamma }
γ
(
t
)
=
a
+
r
exp
(
i
t
)
,
{\displaystyle \gamma (t)=a+r\exp(it),}
t
∈
[
0
,
2
π
]
.
{\displaystyle t\in [0,2\pi ].}
Niech dany będzie skończony ciąg punktów
z
1
,
…
,
z
n
∈
C
.
{\displaystyle z_{1},\dots ,z_{n}\in \mathbb {C} .}
Łamaną zorientowaną o początku w punkcie
z
1
{\displaystyle z_{1}}
z
n
{\displaystyle z_{n}}
I
1
+
…
+
I
n
−
1
,
{\displaystyle I_{1}+\ldots +I_{n-1},}
I
k
{\displaystyle I_{k}}
z
k
{\displaystyle z_{k}}
z
k
+
1
.
{\displaystyle z_{k+1}.}
Bibliografia
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![{\displaystyle \gamma \colon [\alpha ,\beta ]\to \mathbb {C} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5edf9ac80a0cc870a2a2779fad5d7f878586ca1)
![{\displaystyle |\Gamma |=\gamma ([\alpha ,\beta ]),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f09b4b87d5c9bbeb8211fcb0d6a9e1da08101877)
![{\displaystyle \gamma |_{[t_{i-1},t_{i}]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7de83ca5435c4339e8b4222d4480b1fa92b9b8f)
![{\displaystyle [t_{i-1},\ t_{i}],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5cede3de3e37e84623b9f9df9b8bf9955bccdbd)
![{\displaystyle {\tilde {\gamma }}\colon [{\tilde {\alpha }},{\tilde {\beta }}]\to \mathbb {C} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/496daba7b11558ca0f7bafc70d5edf1f378fc95d)
![{\displaystyle \sigma \colon [\alpha ,\beta ]\to [{\tilde {\alpha }},{\tilde {\beta }}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/674ff6eae51305fc959fa62db698f2d52510d6ec)
![{\displaystyle \sigma |_{[t_{i-1},t_{i}]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93105eeb4bd92a6b415c29ec6e267f6372817eb7)
![{\displaystyle \gamma \colon [\alpha ,\beta ]\to \mathbb {C} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7582c80f3bcfe91133519c93fd50da32c5bcf4)
![{\displaystyle t\in [-\beta ,-\alpha ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f424ffd07c9ee0581abc4b1d74477d66aae573c)
![{\displaystyle \gamma _{1}\colon [\alpha _{1},\beta _{1}]\to \mathbb {C} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf1c25d5779f53cdb5da9dd14ec1cd9415a656be)
![{\displaystyle \gamma _{2}\colon [\alpha _{2},\beta _{2}]\to \mathbb {C} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17fc917860d16c66aec62d06e6cdcc936d31f043)
![{\displaystyle \gamma (t)={\begin{cases}\gamma _{1}(t)&{\text{dla }}t\in [\alpha _{1},\beta _{1}]\\\gamma _{2}(t-\beta _{1}+\alpha _{2})&{\text{dla }}t\in [\beta _{1},\beta _{1}+\beta _{2}-\alpha _{2}]\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e91b2e129e0d1bf366291f708ff3e29c7723d225)
![{\displaystyle t\in [0,1].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bafa089d991504bb539141c6221e17f79d06d7b8)
![{\displaystyle t\in [0,2\pi ].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a66bc04b5ae0592aa1ced08b531513077c236ee2)
