parametric equations

\(x(t),~y(t),~z(t)\)

position vector function

\(\vec{r}(t) = x(t)\vhat{i} + y(t)\vhat{j} + z(t)\vhat{k}\)

vector function in terms of the arc length parameter, s

\(\vec{r}(s) = x(s)\vhat{i} + y(s)\vhat{j} + z(s)\vhat{k}\)

Note - When the variable s is used on this page, it refers to the arc length parameter.

velocity

\(\vec{v}(t) = \vec{r}'(t)\)

basic acceleration

\(\vec{a}(t) = \vec{v}'(t) = \vec{r}''(t)\)

unit tangent vector

\(\displaystyle{ \vhat{T}(t) = \frac{\vec{r}'(t)}{ \| \vec{r}'(t) \| } }\)

\(\displaystyle{ \vhat{T}(t) = \frac{\vec{v}(t)}{ \| \vec{v}(t) \| } }\)

principal unit normal vector

\(\displaystyle{ \vhat{N}(t) = \frac{d\vhat{T}/dt}{ \| d\vhat{T}/dt \| } }\)

acceleration vector

\( \vec{a}(t) = a_{\vhat{T}}\vhat{T} + a_{\vhat{N}}\vhat{N}\)

tangential component of acceleration

\(a_{\vhat{T}} = \vec{a} \cdot \vhat{T} \)

\(\displaystyle{a_{\vhat{T}} = \frac{\vec{a} \cdot \vec{v}}{\|\vec{v}\|} }\)

\(a_{\vhat{T}} = \| \vec{v} \|' \)

normal component of acceleration

\(a_{\vhat{N}} = \vec{a} \cdot \vhat{N} \)

\(a_{\vhat{N}} = \|\vec{v}\| \|\vhat{T}'\|\)

\(\displaystyle{a_{\vhat{N}} = \frac{\|\vec{v} \times \vec{a}\|}{\|\vec{v}\|} }\)

\(a_{\vhat{N}} = \sqrt{\|\vec{a}\|^2 - a_{\vhat{T}}^2}\)

curvature

\(\displaystyle{ K(t) = \frac{1}{\|\vec{v}\|} \left\| \frac{d\vec{T}}{dt} \right\| = \frac{\| \vec{T}'(t) \|}{\|\vec{r}'(t)\|} }\)

\(\displaystyle{ K=\frac{\|\vec{v} \times \vec{a} \|}{\|\vec{v}\|^3} }\)