Arc Length and Curvature
The Essentials:
The formula for arc length is:
\[
\int_a^b||\vec{r'}(t)||dt
\]
Curvature is the measure of how sharply a curve turns. Circles have a constant curvature. Bigger circles have smaller curvatures. The formula for curvature is:
\[
\kappa=\frac{||\vec{T'}(t)||}{||\vec{r'}(t)||}
\]
If the curve is three dimensional:
\[
\kappa=\frac{||\vec{r'}(t)\times\vec{r''}(t)||}{||\vec{r'}(t)||^3}
\]
If the curve is the function \( y=f(x) \):
\[
\kappa=\frac{|y''(x)|}{(1+(y')^2)^{3/2}}
\]
The principle unit normal vector is:
\[
\vec{N}(t)=\frac{\vec{T'}(t)}{||\vec{T'}(t)||}
\]
The binormal vector is:
\[
\vec{B}(t)=\vec{T}\times\vec{N}
\]