Parameterization example

Parameterization example

Let $(x, y)$ start out from $(x_0, y_0)$ and move at a constant speed, then \begin{align} x &= x_0 + t \\ y &= y_0 + at \end{align} $(2) - a(1)$, we get \begin{align*} y - ax &= y_0 - ax_0 \\ y &= y_0 + a(x - x_0) \end{align*}

You see the parameterized form is equivalent with what we are familiar with.

You can also see it in the vector way:

\begin{align*} \begin{pmatrix} x \\ y \end{pmatrix} &= \begin{pmatrix} x_0 \\ y_0 \end{pmatrix} + \begin{pmatrix} 1 \\ a \end{pmatrix} t \end{align*}