8. Properties of Curves

Position Vector

\(\vec{r}(t)=(x(t),y(t),z(t))\)

Velocity Vector

\(\vec{v}(t)=\dfrac{d\vec{r}}{dt} =\left\langle \dfrac{dx}{dt},\dfrac{dy}{dt},\dfrac{dz}{dt}\right\rangle\)

Acceleration Vector

\(\vec{a}(t)=\dfrac{d\vec{v}}{dt}=\dfrac{d^2\vec{r}}{dt^2} =\left\langle \dfrac{d^2x}{dt^2},\dfrac{d^2y}{dt^2},\dfrac{d^2z}{dt^2}\right\rangle\)

Jerk Vector

\(\vec{j}(t)=\dfrac{d\vec{a}}{dt}=\dfrac{d^2\vec{v}}{dt^2}= \dfrac{d^{3}\vec{r}}{dt^{3}} =\left\langle \dfrac{d^3x}{dt^3},\dfrac{d^3y}{dt^3},\dfrac{d^3z}{dt^3}\right\rangle\)

Speed

\(\dfrac{ds}{dt}=|\vec{v}|=\sqrt{\left(\dfrac{dx}{dt}\right)^2+ \left(\dfrac{dy}{dt}\right)^2+\left(\dfrac{dz}{dt}\right)^2}\)

Arclength (from \(A=\vec{r}(a)\)
to \(B=\vec{r}(b)\))

\(\displaystyle L=\int_{A}^{B} ds=\int_{a}^{b} |\vec{v}|\,dt\)

Unit Tangent Vector

\(\hat{T}=\dfrac{\vec{v}}{|\vec{v}|}\)

Velocity \(\times\) Acceleration



and its length

\(\vec{v}\times\vec{a}= \begin{vmatrix} \hat{\imath} & \hat{\jmath} & \hat{k} \\ \dfrac{dx}{dt} & \dfrac{dy}{dt} & \dfrac{dz}{dt} \dfrac{\frac{}{}}{\frac{}{}} \\ \dfrac{d^2x}{dt^2} & \dfrac{d^2y}{dt^2} & \dfrac{d^2z}{dt^2} \end{vmatrix}\)

\(|\vec{v}\times\vec{a}|\)

Unit Binormal Vector

\(\hat{B}=\hat{T}\times\hat{N} =\dfrac{\vec{v}\times\vec{a}}{|\vec{v}\times\vec{a}|}\)

Unit Normal Vector

\(\begin{aligned} \hat{N} &=\dfrac{\text{proj}_{\bot \vec{v}}\vec{a}}{\left|proj_{\bot \vec{v}}\vec{a}\right|} =\dfrac{|\vec{v}|}{|\vec{v}\times\vec{a}|} \left(\vec{a}-\,\dfrac{\vec{a}\cdot\vec{v}}{|\vec{v}|^2}\vec{v}\right) \\ &=\hat{B}\times\hat{T}=\dfrac{\hat{T}'(t)}{|\hat{T}'(t)|} \end{aligned}\)

Curvature and
Radius of Curvature

\(\kappa=\dfrac{1}{R}=\dfrac{1}{|\vec{v}|}\left|\dfrac{d\hat{T}}{dt}\right| =\dfrac{|\vec{v}\times\vec{a}|}{|\vec{v}|^{3}}\)

Torsion

\(\tau=-\,\dfrac{d\hat{B}}{ds}\cdot\hat{N}= \dfrac{\vec{v}\times\vec{a}\cdot\vec{j}}{|\vec{v}\times\vec{a}|^2}\)

Tangential Acceleration

\(a_{T}=\vec{a}\cdot\hat{T}=\dfrac{d}{dt}|\vec{v}|\)

Normal Acceleration

\(a_{N}=\vec{a}\cdot\hat{N}=\kappa|\vec{v}|^2=\dfrac{|\vec{v}|^2}{R}\)