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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 $0 \leq t \leq 1$.