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 \( 0 \leq t \leq 1 \)

 

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