Math Insight

Tangent lines to parametrized curves

For a curve parametrized by $\dllp(t)$, the derivative $\dllp'(t)$ is a vector that is tangent to the curve. We can use this fact to derive an equation for a line tangent to the curve.

Fix a time $t_0$. The line through point $\dllp(t_0)$ in the direction parallel to the tangent vector $\dllp'(t_0)$ will be a tangent line to the curve.

A parametrization of the line through a point $\vc{a}$ and parallel to the vector $\vc{v}$ is $\vc{l}(t) = \vc{a} + t\vc{v}$. Setting $\vc{a} = \dllp(t_0)$ and $\vc{v} = \dllp'(t_0)$, we obtain a parametrization of the tangent line: $\vc{l}(t) = \dllp(t_0) + t \dllp'(t_0).$

However, we typically want the line given by $\vc{l}(t)$ to pass through $\dllp(t_0)$ when $t=t_0$. So we usually change the parametrization slightly to \begin{align*} \vc{l}(t) = \dllp(t_0) + (t-t_0) \dllp'(t_0). \end{align*}

You can see some examples here.