Equation of Straight Line in Plane/Point-Slope Form/Parametric Form
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Theorem
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:
- $\begin {cases} x = x_0 + t \cos \psi \\ y = y_0 + t \sin \psi \end {cases}$
Proof
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.
$\blacksquare$
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $4$. Special forms of the equation of a straight line: $(1)$ Gradient forms