Introducere [ ] Teorema 1 : Dacă curba
Γ
{\displaystyle \Gamma \!}
r
(
t
)
→
=
x
(
t
)
i
→
+
y
(
t
)
j
→
+
z
(
t
)
k
→
,
t
∈
[
a
,
b
]
{\displaystyle \overrightarrow {r(t)} = x(t) \vec i + y(t) \vec j + z(t) \vec k, \; t \in [a, b]}
atunci:
K
=
|
r
′
(
t
)
→
×
r
″
(
t
)
→
|
|
r
′
(
t
)
→
|
3
{\displaystyle K = \frac {\left | \overrightarrow {r'(t)} \times \overrightarrow {r''(t)} \right | }{\left | \overrightarrow {r'(t)} \right |^3}}
T
=
(
r
′
(
t
)
→
,
r
″
(
t
)
→
,
r
‴
(
t
)
→
)
|
r
′
(
t
)
→
×
r
″
(
t
)
→
|
2
{\displaystyle T = \frac {\left ( \overrightarrow {r'(t)}, \overrightarrow {r''(t)}, \overrightarrow {r'''(t)} \right )}{\left | \overrightarrow {r'(t)} \times \overrightarrow {r''(t)} \right |^2}}
Teorema 2 : Dacă curbura unei curbe este identic nulă, atunci curba este un segment dintr-o dreaptă.
Teorema 3 : Dacă torsiunea unei curbe este identic nulă, atunci curba este o curbă plană, planul curbei fiind planul osculator într-un punct arbitrar.
Curbura unei curbe plane [ ] Fie
Γ
⊂
E
2
{\displaystyle \Gamma \subset \mathcal E^2 \!}
r
→
=
r
→
(
s
)
,
s
∈
[
0
,
L
]
{\displaystyle \vec r = \vec r (s), \; \; s \in [0, L] \!}
M
∈
Γ
{\displaystyle M \in \Gamma \!}
r
→
(
s
)
{\displaystyle \vec r (s) \!}
s
=
l
(
Γ
Ω
M
)
,
{\displaystyle s=l (\Gamma_{\Omega M}), \!}
Ω
∈
Γ
{\displaystyle \Omega \in \Gamma \!}
s
=
0.
{\displaystyle s=0. \!}
Fie
M
′
∈
Γ
{\displaystyle M' \in \Gamma \!}
M având vectorul de poziţie
r
→
(
s
′
)
{\displaystyle \vec r(s') \!}
|
s
′
−
s
|
<
ε
,
∀
ε
>
0
{\displaystyle |s'-s| < \varepsilon, \; \forall \varepsilon >0 \!}
Considerăm tangentele la curbă în punctele
M
{\displaystyle M \!}
M
′
,
{\displaystyle M', \!}
Ox unghiurile
θ
{\displaystyle \theta \!}
θ
′
.
{\displaystyle \theta'. \!}
φ
{\displaystyle \varphi \!}
φ
=
θ
′
−
θ
.
{\displaystyle \varphi = \theta'-\theta. \!}
Curbura unei curbe plane
Definiţie
Numarul real
φ
l
(
Γ
M
M
′
)
{\displaystyle \frac{\varphi}{l(\Gamma_{MM'})} \!}
curbură medie a lui
Γ
{\displaystyle \Gamma \!}
M .
Observaţie
Lungimea arcului de curbă
l
(
Γ
M
M
′
)
=
c
o
n
s
t
,
{\displaystyle l(\Gamma_{MM'})= const, \!}
φ
{\displaystyle \varphi \!}
Γ
{\displaystyle \Gamma \!}
Definiţie .
Dacă limita
k
(
s
)
=
lim
s
′
→
s
φ
(
s
,
s
′
)
s
−
s
′
{\displaystyle k(s) = \lim_{s' \to s} \frac{\varphi (s, s')}{s-s'} \!}
curbura curbei
Γ
{\displaystyle \Gamma \!}
M .
θ
=
θ
(
s
)
{\displaystyle \theta = \theta (s) \!}
M cu axa Ox , atunci în
M
′
{\displaystyle M' \!}
θ
′
=
θ
′
(
s
′
)
⇒
φ
(
s
,
s
′
)
=
θ
′
(
s
′
)
−
θ
(
s
)
,
{\displaystyle \theta'=\theta'(s') \; \Rightarrow \; \varphi (s, s') = \theta'(s') - \theta (s), \!}
k
(
s
)
=
d
θ
d
s
{\displaystyle k(s) = \frac{d \theta}{ds} \!}
Raportul
R
M
=
1
|
k
(
s
)
|
{\displaystyle R_M = \frac{1}{|k(s)|} \!}
raza de curbură în punctul regulat M .
Observaţie .
Dacă
k
=
0
{\displaystyle k=0 \!}
Γ
{\displaystyle \Gamma \!}
Expresii ale curburii unei curbe plane pentru diferite reprezentări ale acesteia [ ] Reprezentare parametrică [ ] Dacă curba plană este dată prin reprezentarea parametrică:
{
x
=
x
(
y
)
y
=
y
(
t
)
{\displaystyle \begin{cases} x=x(y) \\ y=y(t) \end{cases} \!}
iar
M
(
x
(
t
)
,
y
(
t
)
)
{\displaystyle M(x(t), y(t))\!}
Γ
,
{\displaystyle \Gamma, \!}
M este
k
T
=
tan
θ
=
y
˙
(
t
)
x
˙
(
t
)
,
{\displaystyle k_T=\tan \theta = \frac{\dot y(t)}{\dot x(t)}, \!}
θ
=
arctan
y
˙
(
t
)
x
˙
(
t
)
.
{\displaystyle \theta = \arctan \frac {\dot y(t)}{\dot x(t)}. \!}
θ
{\displaystyle \theta \!}
s şi obţinem:
d
θ
d
s
=
d
θ
d
t
d
t
d
s
=
y
¨
(
t
)
x
˙
(
t
)
−
x
¨
(
t
)
y
˙
(
t
)
x
˙
(
t
)
2
+
y
˙
(
t
)
2
d
t
d
s
{\displaystyle \frac{d \theta}{ds} = \frac{d \theta}{dt}\frac{dt}{ds} = \frac {\ddot y(t) \dot x(t) - \ddot x(t) \dot y(t)}{\dot x(t)^2 + \dot y(t)^2} \frac{dt}{ds} \!}
unde
d
s
=
x
˙
(
t
)
2
+
y
˙
(
t
)
2
⇒
d
t
d
s
=
1
x
˙
(
t
)
2
+
y
˙
(
t
)
2
.
{\displaystyle ds = \sqrt{\dot x(t)^2 + \dot y(t)^2} \; \Rightarrow \; \frac{dt}{ds} = \frac{1}{\sqrt{\dot x(t)^2+ \dot y(t)^2}}. \!}
Astfel, formula curburii pentru o curbă plană este:
k
=
d
θ
d
s
=
y
¨
(
t
)
x
˙
(
t
)
−
x
¨
(
t
)
y
˙
(
t
)
[
x
˙
(
t
)
2
+
y
˙
(
t
)
2
]
3
/
2
{\displaystyle k= \frac{d \theta}{ds} = \frac{\ddot y(t) \dot x(t) - \ddot x(t) \dot y(t)}{[\dot x(t)^2+ \dot y(t)^2]^{3/2}} \!}
Reprezentare explicită [ ] Dacă curba plană este dată prin ecuaţia carteziană explicită:
y
=
y
(
x
)
,
{\displaystyle y=y(x), \!}
{
x
=
t
y
=
y
(
t
)
,
{\displaystyle \begin{cases} x=t \\ y=y(t) \end{cases}, \!}
{
x
˙
=
1
y
˙
=
y
˙
(
t
)
{\displaystyle \begin{cases} \dot x=1 \\ \dot y= \dot y(t) \end{cases} \!}
{
x
¨
=
0
y
¨
=
y
¨
(
t
)
,
{\displaystyle \begin{cases} \ddot x=0 \\ \ddot y= \ddot y(t) \end{cases}, \!}
k
=
y
¨
(
t
)
[
1
+
y
˙
2
(
t
)
]
3
/
2
{\displaystyle k=\frac{\ddot y(t)}{[1+ \dot y^2(t)]^{3/2}} \!}
Reprezentare implicită [ ] Dacă curba plană este dată prin ecuaţia carteziană implicită
F
(
x
,
y
)
=
0
{\displaystyle F(x, y) =0 \!}
F
x
′
+
F
y
′
⋅
y
˙
=
0
⇒
y
˙
=
−
F
x
′
F
y
′
{\displaystyle F'_x + F'_y \cdot \dot y =0 \; \Rightarrow \; \dot y = - \frac{F'_x}{F'_y} \!}
y
¨
=
−
F
x
2
″
F
y
2
″
;
{\displaystyle \ddot y = - \frac{F''_{x^2}}{F''_{y^2}}; \!}
k
=
−
(
F
z
′
)
2
⋅
F
y
2
″
−
2
⋅
F
z
′
⋅
F
y
′
⋅
F
z
y
′
+
(
F
y
′
)
2
⋅
F
z
2
″
[
(
F
z
′
)
2
+
(
F
y
′
)
2
]
3
/
2
.
{\displaystyle k = - \frac{(F'_z)^2 \cdot F''_{y^2} - 2 \cdot F'_z \cdot F'_y \cdot F'_{zy} + (F'_y)^2 \cdot F''_{z^2}}{[(F'_z)^2+(F'_y)^2]^{3/2}}. \!}
Curbura şi torsiunea unei curbe spaţiale [ ] Considerăm un punct regulat şi neinflexionar
M
=
r
→
(
t
)
∈
Γ
.
{\displaystyle M= \vec r(t) \in \Gamma. \!}
reperului Frenet în punctul M cu
t
→
(
t
)
,
n
→
(
t
)
{\displaystyle \vec t(t), \vec n(t) \!}
b
→
(
t
)
.
{\displaystyle \vec b(t). \!}
Teorema 4. (Formulele Frenet-Serret ).
t
→
˙
(
t
)
=
k
(
t
)
ν
(
t
)
n
→
(
t
)
{\displaystyle {\dot {\vec {t}}}(t)=k(t)\nu (t){\vec {n}}(t)\!}
n
→
˙
(
t
)
=
−
k
(
t
)
ν
(
t
)
t
→
(
t
)
+
τ
(
t
)
ν
(
t
)
b
→
(
t
)
{\displaystyle {\dot {\vec {n}}}(t)=-k(t)\nu (t){\vec {t}}(t)+\tau (t)\nu (t){\vec {b}}(t)\!}
b
→
˙
(
t
)
=
−
τ
(
t
)
ν
(
t
)
n
→
(
t
)
{\displaystyle {\dot {\vec {b}}}(t)=-\tau (t)\nu (t){\vec {n}}(t)\!}
unde
ν
(
t
)
=
x
˙
(
t
)
2
+
y
˙
(
t
)
2
+
z
˙
(
t
)
2
{\displaystyle \nu(t) = \sqrt{\dot x(t)^2 + \dot y(t)^2 + \dot z(t)^2} \!}
viteza curbei
Γ
,
{\displaystyle \Gamma, \!}
k
(
t
)
{\displaystyle k(t) \!}
ν
(
t
)
{\displaystyle \nu(t) \!}
curbura şi respectiv torsiunea curbei
Γ
{\displaystyle \Gamma \!}
M .
k
(
t
)
{\displaystyle k(t) \!}
k
(
t
)
=
‖
r
→
˙
(
t
)
×
r
→
¨
(
t
)
‖
‖
r
→
˙
(
t
)
‖
3
=
A
2
+
B
2
+
C
2
[
[
x
˙
(
t
)
]
2
+
[
y
˙
(
t
)
]
2
+
[
z
˙
(
t
)
]
2
]
3
{\displaystyle k(t)={\frac {\|{\dot {\vec {r}}}(t)\times {\ddot {\vec {r}}}(t)\|}{\|{\dot {\vec {r}}}(t)\|^{3}}}={\frac {\sqrt {A^{2}+B^{2}+C^{2}}}{[[{\dot {x}}(t)]^{2}+[{\dot {y}}(t)]^{2}+[{\dot {z}}(t)]^{2}]^{3}}}\!}
unde
A
,
B
,
C
{\displaystyle A, B, C \!}
|
i
→
j
→
k
→
x
˙
(
t
)
y
˙
(
t
)
z
˙
(
t
)
x
¨
(
t
)
y
¨
(
t
)
z
¨
(
t
)
|
.
{\displaystyle \begin{vmatrix} \vec i & \vec j & \vec k \\ \dot x(t) & \dot y(t) & \dot z(t) \\ \ddot x(t) & \ddot y(t) & \ddot z(t) \end{vmatrix}. \!}
Raza de curbură a curbei
Γ
{\displaystyle \Gamma \!}
M este
R
(
t
)
=
1
|
k
(
t
)
|
{\displaystyle R(t) = \frac{1}{|k(t)|} \!}
Pentru torsiunea
τ
(
t
)
{\displaystyle \tau (t) \!}
τ
(
t
)
=
r
→
˙
(
t
)
;
r
→
¨
(
t
)
;
r
→
⋯
(
t
)
‖
r
→
˙
(
t
)
×
r
→
¨
(
t
)
‖
2
=
Δ
A
2
+
B
2
+
C
2
{\displaystyle \tau (t)={\frac {{\dot {\vec {r}}}(t);{\ddot {\vec {r}}}(t);{\overset {\cdots }{\vec {r}}}(t)}{\|{\dot {\vec {r}}}(t)\times {\ddot {\vec {r}}}(t)\|^{2}}}={\frac {\Delta }{A^{2}+B^{2}+C^{2}}}\!}
unde
Δ
=
|
x
˙
(
t
)
y
˙
(
t
)
z
˙
(
t
)
x
¨
(
t
)
y
¨
(
t
)
z
¨
(
t
)
x
⋯
(
t
)
y
⋯
(
t
)
z
⋯
(
t
)
|
{\displaystyle \Delta = \begin{vmatrix} \dot x(t) & \dot y(t) & \dot z(t) \\ \ddot x(t) & \ddot y(t) & \ddot z(t) \\ \overset{\cdots} x(t) & \overset{\cdots} y(t) & \overset{\cdots} z(t) \end{vmatrix} \!}
Raza de torsiune în punctul M este
T
(
τ
)
=
1
|
τ
(
t
)
|
.
{\displaystyle T(\tau) = \frac{1}{|\tau (t)|}. \!}
Vezi şi [ ] Resurse [ ] 
![{\displaystyle {\overrightarrow {r(t)}}=x(t){\vec {i}}+y(t){\vec {j}}+z(t){\vec {k}},\;t\in [a,b]}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/93b320141994f8a543735b034852380f3f016547)
![{\displaystyle {\vec {r}}={\vec {r}}(s),\;\;s\in [0,L]\!}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/33b6d7d9a2b8dfa89c2f21dc42942638c965c0c1)
![{\displaystyle k={\frac {d\theta }{ds}}={\frac {{\ddot {y}}(t){\dot {x}}(t)-{\ddot {x}}(t){\dot {y}}(t)}{[{\dot {x}}(t)^{2}+{\dot {y}}(t)^{2}]^{3/2}}}\!}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/5560b9051c5baef58ec57561d9dae24cb8e3cbca)
![{\displaystyle k={\frac {{\ddot {y}}(t)}{[1+{\dot {y}}^{2}(t)]^{3/2}}}\!}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/248d1cd005b7471fc4f9469ca3c9c69c4db60c72)
![{\displaystyle k=-{\frac {(F'_{z})^{2}\cdot F''_{y^{2}}-2\cdot F'_{z}\cdot F'_{y}\cdot F'_{zy}+(F'_{y})^{2}\cdot F''_{z^{2}}}{[(F'_{z})^{2}+(F'_{y})^{2}]^{3/2}}}.\!}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/a2af50cbeb14191a51bf4c88aaa921c447d5240b)
![{\displaystyle k(t)={\frac {\|{\dot {\vec {r}}}(t)\times {\ddot {\vec {r}}}(t)\|}{\|{\dot {\vec {r}}}(t)\|^{3}}}={\frac {\sqrt {A^{2}+B^{2}+C^{2}}}{[[{\dot {x}}(t)]^{2}+[{\dot {y}}(t)]^{2}+[{\dot {z}}(t)]^{2}]^{3}}}\!}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/05d6b0c3166278beb853c5929f246c45ca447a27)
