Curvature

Curvature of a smooth curve measures how fast a curve is changing direction at a given point.

Definition: Smooth Curve

• Any curve for which $\vec{r}(t)$ is continuous and $\vec{r}(t)\ne 0$ for any $t$ (except maybe endpoints)
• A helix is an example of smooth curve

Formal Definition:

Definition: Curvature

$\kappa =\left|\right|\frac{d\vec{T}}{ds}\left|\right|$ where $\vec{T}$ is the unit tangent and $s$ is the arc length

• Recall that we can reparametrize a curve to get it in terms of arc length

The formal definition can be hard to use so there are alternate formulas:

Alternate Curvature Formulas

$$\kappa =\frac{\mid \mid {\vec{T}}^{\prime }(t)\mid \mid }{\mid \mid {\vec{r}}^{\prime }(t)\mid \mid }$$ $$\kappa =\frac{\mid \mid {\vec{r}}^{\prime }(t)\times {\vec{r}}^{\prime \prime }(t)\mid \mid }{\mid \mid {\vec{r}}^{\prime }(t)\mid {\mid }^3}$$

Special case: Suppose we have a curve given by $y=f(x)$ and we want to find its curvature. We can write this as a vector function:

$$\vec{r}(x)=x\vec{i}+f(x)\vec{j}$$

This can then be rewritten using the second formula for curvature above:

$$\kappa =\frac{|f^{\prime \prime }(x)|}{1+[f^{\prime }(x{)}^2{]}^{\frac{3}{2}}}$$

I saw this in SYDE 182 Dynamics for expressing circular motion!!