# 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$$A\,x + B\,y + C = 0$$ 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 P1P2 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 P1P2 and P3P4 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 $$0 \leq t \leq 1$$