Equation of Straight Line in Plane/Point-Slope Form/Parametric Form

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Let $\LL$ be a straight line embedded in a cartesian plane, given in point-slope form as:

$y - y_0 = \paren {x - x_0} \tan \psi$

where $\psi$ is the angle between $\LL$ and the $x$-axis.

Then $\LL$ can be expressed by the parametric equations:

$\begin {cases} x = x_0 + t \cos \psi \\ y = y_0 + t \sin \psi \end {cases}$


Let $P_0$ be the point $\tuple {x_0, y_0}$.

Let $P$ be an arbitrary point on $\LL$.

Let $t$ be the distance from $P_0$ to $P$ measured as positive when in the positive $x$ direction.

The equation for $P$ is then:

\(\ds y - y_0\) \(=\) \(\ds \paren {x - x_0} \tan \psi\)
\(\ds \leadsto \ \ \) \(\ds \dfrac {x - x_0} {\cos \psi}\) \(=\) \(\ds t\)
\(\ds \dfrac {y - y_0} {\sin \psi}\) \(=\) \(\ds t\)

The result follows.