## Line equations in 2D

Line is a straight path and is infinite. There are no turns.

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**2 points describes the line**

2 points are enough to describe the line.

And they are
enough to write down equations.

Equations to calculate
points of the line, or intersections point with other line etc.

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**Line equations calculation**

Lets have our 2 points on the line.

They have coordinates x1, y1 and x2, y2
And lets have x, y representing any point on the line

Than we have this equation

(x2 - x1) / (y2 - y1) = (x - x1) / (y - y1)

Cause they are common triangles. if we connect the points and then draw the horizontal and vertical lines from them.

So, lets start from
(x2 - x1) / (y2 - y1) = (x - x1) / (y - y1)

**y - y1 = (x1 - x2) (y2 - y1) / (x2 - x1) ** // Two-point form equation

Again from the beginning for the other equations

(x2 - x1) / (y2 - y1) = (x - x1) / (y - y1)

(x - x1)(y2 - y1) = (x2 - x1)(y - y1)

xy2 - xy1 - x1y2 + x1y1 = x2y - x2y1 - x1y + x1y1

x(y2 - y1) + y(x1-x2) + x2y1 - x1y2 = 0

A = y2 - y1

B = x1 - x2

C = x2y1 - x1y2

**Ax + By + C = 0 ** // General (or standard) form of the equation

By = -Ax - C

y = -A/Bx - C/B

m = -A/B = (y1 - y2)/(x1 - x2)

b = -C/B = (x1y2 - x2y1)/(x1 - x2)

**y = mx + b ** // Slope–intercept form

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