Equation of Straight Line in Plane
Theorem
General Equation
A straight line $\LL$ is the set of all $\tuple {x, y} \in \R^2$, where:
- $\alpha_1 x \alpha_2 y = \beta$
where $\alpha_1, \alpha_2, \beta \in \R$ are given, and not both $\alpha_1, \alpha_2$ are zero.
Slope-Intercept Form
Let $\LL$ be the straight line in the Cartesian plane such that:
- the slope of $\LL$ is $m$
- the $y$-intercept of $\LL$ is $c$
Then $\LL$ can be described by the equation:
- $y = m x c$
such that $m$ is the slope of $\LL$ and $c$ is the $y$-intercept.
Two-Intercept Form
Let $\LL$ be a straight line which intercepts the $x$-axis and $y$-axis respectively at $\tuple {a, 0}$ and $\tuple {0, b}$, where $a b \ne 0$.
Then $\LL$ can be described by the equation:
- $\dfrac x a \dfrac y b = 1$
Normal Form
Let $\LL$ be a straight line such that:
- the perpendicular distance from $\LL$ to the origin is $p$
- the angle made between that perpendicular and the $x$-axis is $\alpha$.
Then $\LL$ can be defined by the equation:
- $x \cos \alpha y \sin \alpha = p$
Two-Point Form
Let $P_1 := \tuple {x_1, y_1}$ and $P_2 := \tuple {x_2, y_2}$ be points in a cartesian plane.
Let $\LL$ be the straight line passing through $P_1$ and $P_2$.
Then $\LL$ can be described by the equation:
- $\dfrac {y - y_1} {x - x_1} = \dfrac {y_2 - y_1} {x_2 - x_1}$
or:
- $\dfrac {x - x_1} {x_2 - x_1} = \dfrac {y - y_1} {y_2 - y_1}$
This is known as the two-point form.
Point-Slope Form
Let $\LL$ be a straight line embedded in a cartesian plane, given in slope-intercept form as:
- $y = m x c$
where $m$ is the slope of $\LL$.
Let $\LL$ pass through the point $\tuple {x_0, y_0}$.
Then $\LL$ can be expressed by the equation:
- $y - y_0 = m \paren {x - x_0}$
Homogeneous Cartesian Coordinates
A straight line $\LL$ is the set of all points $P$ in $\R^2$, where $P$ is described in homogeneous Cartesian coordinates as:
- $l X m Y n Z = 0$
where $l, m, n \in \R$ are given, and not both $l$ and $m$ are zero.