Curvature

A numerical measure of bending

Notated as $\kappa$

Equations

$$\kappa (s)=\Vert \frac{\text{d}\hat{T}(t)}{\text{d}s}\Vert$$ $$\kappa (t)=\frac{\Vert {\hat{T}}^{\prime }(t)\Vert }{\Vert \vec{r}(t)\Vert }$$ $$\kappa (t)=\frac{\Vert {\vec{r}}^{\prime }(t)\times {\vec{r}}^{\prime \prime }(t)\Vert }{\Vert {\vec{r}}^{\prime }(t){\Vert }^3}$$

Osculating Plane

• The plane that a curve bends in at a particular point
• Spanned by $\hat{T}$ and $\hat{N}$
• If $\vec{r}\prime \prime$ is not parallel with $\vec{r}\prime$ then it is also spanned by them
• $\vec{n}=\vec{r}\prime \prime (P_0)\times \vec{r}\prime (P_0)$

Osculating Circle

• The quadratic approximation of a curve (like how the tangent line is the linear approximation)
• Properties
  • Through a point $P_0$ on the curve
  • Lies on the osculating plane
  • $\rho =\kappa (P_0{)}^{-1}$
  • $\overrightarrow{P_{0}C}=\frac{\hat{N}(P_0)}{\kappa (P_0)}=\rho \hat{N}$
  • $\vec{r}(t)=r_0+{\rho }_0(1-\cos t){\hat{N}}_0+{\rho }_0\sin t{\hat{T}}_0,t\in [0,2\pi ]$

Planar Curves

$$\kappa =\frac{|x^{\prime }{y}^{\prime \prime }-x^{\prime \prime }{y}^{\prime }|}{{\left[(x^{\prime }{)}^2+(y^{\prime }{)}^2\right]}^{3/2}}$$ $$\kappa (x)=\frac{|f^{\prime \prime (x)|}}{(1+f^{\prime }(x{)}^2{)}^{3/2}}$$ $$\kappa (\theta )=\frac{|2f^{\prime }(\theta {)}^2-f(\theta )f^{\prime \prime }(\theta )+f(\theta {)}^2|}{{\left[f^{\prime }(\theta {)}^2+f(\theta {)}^2\right]}^{3/2}}$$

• Optimizing
  • $\kappa$ is at a maximum whenever $f(x)$ is at an extrema and $f\prime \prime (x)\ne 0$
  • If the numerator of $\kappa$ constant, $\kappa$ has a max where the denominator has a min and visa versa
  • Can always just do standard derivative optimization

Application: Mechanics

Note: $v=\Vert \vec{v}\Vert$

• $\vec{a}=\frac{\text{d}}{\text{d}t}\left[v\hat{T}\right]=v^{\prime }\hat{T}+v\hat{T}=v^{\prime }\hat{T}+\kappa v^2\hat{N}$
• $\vec{a}=a_T\hat{T}+a_N\hat{N}$
• $a_T=v^{\prime }=\frac{\vec{v}(t)\cdot \vec{a}(t)}{\Vert \vec{v}(t)\Vert }$
• $a_N=\kappa v^2=\frac{v^2}{\rho }=\frac{\Vert \vec{v}(t)\times \vec{a}(t)\Vert }{\Vert \vec{v}(t)\Vert }$
  • That is the circular motion equation for centripetal acceleration