﻿ Analytic geometry Lines in 2d Formulas
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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 P1(x1,y1) lies on the line. General form: $$Ax + By + C = 0~,~(A\ne 0 ~\text{or}~B \ne 0)$$

Distance

The distance from Ax+By+C=0 to P1(x1,y1) 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 $$0 \leq t \leq 1$$