Minimum Distance between a Point and a Line

The Equations

The equation:

P = P1 + u (P2 - P1)

Denotes how to find the point of where p3 would cross the line P1,P2.

u is worked out by the equation:

u = ((x3 - x1)(x2 - x1) + (y3 - y1)(y2 - y1)) / (||P2 - P1||^2)

This makes the two equations for the intersection:

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

This makes working out the distance between P3 and the line (P1,P2) equal the same as working out the distance between P3 and (x,y).

Using the distance formula:

distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Where x1 and y1 are the P3's x and y coordinates. And x2 and y2 are P's x and y coordinates.

Worked Example

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Point 1: x: y:
Point 2: x: y:
Point 3: x: y:

Using the above equations and coordinates we get the following equations:

u = ((45 - 33)(100 - 33) + (100 - 10)(50 - 10)) / (||P2 - P1||^2)

u = 4404 / (||P2 - P1||^2)

u = 4404 / 6089.000000000001

u = 0.7232714731483001

This makes the position of P:

x = 33 + u (100 - 33)
y = 10 + u (50 - 10)

x = 81.4591887009361
y = 38.930858925932
		

Using these coordinates we can work out the distance between P3 and the line now:

distance = sqrt((81.4591887009361 - 45)^2 + (38.930858925932 - 100)^2)

distance = 71.12462606056278