Tangents and Curvature

Before we introduce the definition of curvature, we firs introduce the position vector ${\displaystyle \mathbf{r}(t)}$.

It is actually the vector pointing at the given point from the origin. And a function of the position vector about ${\displaystyle t}$ is the position function.

The position vector can also be presented by the sum of three independent vectors like ${\displaystyle \mathbf{p}=\mathbf{i}+\mathbf{j}+\mathbf{k}}$, where ${\displaystyle \mathbf{i},\mathbf{j},\mathbf{k}}$ are vectors in ${\displaystyle x,y,z}$ directions.

Tangents

The key property of the tangent line of the curve at a given position is its direction. This can be calculated by finding the derivative of the position function:

${\displaystyle \mathbf{v}(t)={\mathbf{r}}^{\prime }(t)}$

Where ${\displaystyle \mathbf{v}(t)}$ is the tangent vector.

Curvature

The curvature is a value that depict how sharp the curve bends at a given point. The definition of the curvature is:

${\displaystyle \kappa =|\frac{d\mathbf{T}}{ds}|}$

Where ${\displaystyle \mathbf{T}}$ is the unit tangent vector, and we have ${\displaystyle \mathbf{T}=\frac{{\mathbf{r}}^{\prime }(t)}{|{\mathbf{r}}^{\prime }(t)|}=\frac{\mathbf{v}(t)}{|\mathbf{v}(t)|}}$. ${\displaystyle s}$ is the arc length in a small interval near ${\displaystyle t}$.

And from the kinematics we can know that ${\displaystyle \frac{ds}{dt}=|\mathbf{v}(t)|}$, by finding the derivative of ${\displaystyle s}$, we have ${\displaystyle \kappa =|\frac{d\mathbf{T}}{ds}|=|\frac{d\mathbf{T}}{dt}\cdot \frac{dt}{ds}|={\mathbf{T}}^{\prime }\frac{1}{|\mathbf{v}(t)|}=\frac{|{\mathbf{T}}^{\prime }(t)|}{|{\mathbf{r}}^{\prime }t|}}$.

That is the definition of the curvature.

But it is not so clear for calculation, so we introduce a formula for calculation here:

${\displaystyle \kappa =\frac{|{\mathbf{r}}^{\prime }(t)\times {\mathbf{r}}^{″}(t)|}{|{\mathbf{r}}^{\prime }(t){|}^3}}$

Notice that there is a cross product in the numerator.

The formula above are the basic information for finding curvature.