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Math Formulas: Lines in two dimensions

Line forms

Slope y-intercept form:

$$ y = mx+b $$

Two point form:

$$ y - y_1 =\frac{y_2-y_1}{x_2 - x_1} (x - x_1)$$

Point slope form:

$$ y - y_1 = m(x - x_1) $$

Intercept form

$$ \frac{x}{a} + \frac{y}{b} = 1~,~(a,b \ne 0) $$

Normal form:

$$ x\cdot \cos\Theta + y\cdot \sin\Theta = p $$

Parametric form:

$$ \begin{aligned} x &= x_1 + t\cdot \cos\alpha \\ y &= y_1 + t\cdot \sin\alpha \\ \end{aligned} $$

Point direction form:

$$ \frac{x - x_1}{A} = \frac{y - y_1}{B} $$

where $(A,B)$ is the direction of the line and $ P_1(x_1, y_1)$ lies on the line.

General form:

$$ Ax + By + C = 0~,~(A\ne 0 ~\text{or}~B \ne 0) $$

Distance

The distance from $ A\,x + B\,y + C = 0 $ to $ P_1(x_1, y_1) $ is

$$ d = \frac{|A\,x_1 + B\,y_1 + C|}{\sqrt{A^2 + B^2}} $$

Concurrent lines

Three lines

$$ \begin{aligned} A_1x + B_1y + C_1 &= 0 \\ A_2x + B_2y + C_2 &= 0 \\ A_3x + B_3y + C_3 &= 0 \end{aligned} $$

are concurrent if and only if:

$$\begin{vmatrix} A_1 & B_1 & C_1 \\ A_2 & B_2 & C_2 \\ A_3 & B_3 & C_3 \\ \end{vmatrix} = 0$$

Line segment

A line segment $P_1P_2$ can be represented in parametric form by

$$ \begin{aligned} x &= x_1 + (x_2 - x_1)t \\ y &= y_1 + (y_2 - y_1)t \\ & 0 \leq t \leq 1 \end{aligned} $$

Two line segments $P_1P_2$ and $P_3P_4$ intersect if any only if the numbers

$$ s = \frac{ \begin{vmatrix} x_2 - x_1 & y_2 - y_1 \\ x_3 - x_1 & y_3 - y_1 \end{vmatrix}} { \begin{vmatrix} x_2 - x_1 & y_2 - y_1 \\ x_3 - x_4 & y_3 - y_4 \end{vmatrix}} ~~ \text{and} ~~ t = \frac{ \begin{vmatrix} x_3 - x_1 & y_3 - y_1 \\ x_3 - x_4 & y_3 - y_4 \end{vmatrix}} { \begin{vmatrix} x_2 - x_1 & y_2 - y_1 \\ x_3 - x_4 & y_3 - y_4 \end{vmatrix}} $$

satisfy $ 0 \leq s \leq 1 $ and $ 0 \leq t \leq 1 $.

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