Distance Formula

Distance Formula

The distance between two points is derived from the Pythagorean theorem and is one of the most used formulas in geometry and programming.

2D Distance

For points (x1, y1) and (x2, y2):

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

Example: Distance from (1, 2) to (4, 6):

sqrt((4-1)^2 + (6-2)^2) = sqrt(9 + 16) = sqrt(25) = 5

3D Distance

For points (x1, y1, z1) and (x2, y2, z2):

d = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2)

Example: Distance from (0, 0, 0) to (3, 4, 12):

sqrt(3^2 + 4^2 + 12^2) = sqrt(9+16+144) = sqrt(169) = 13

Manhattan Distance

Sometimes you need the "city block" distance (sum of absolute differences):

d = abs(x2 - x1) + abs(y2 - y1)

Example: From (1, 3) to (4, 7):

abs(4-1) + abs(7-3) = 3 + 4 = 7

Midpoint Formula

The midpoint between two points:

midpoint_x = (x1 + x2) / 2
midpoint_y = (y1 + y2) / 2

Example: Midpoint of (2, 8) and (6, 4):

(2+6)/2 = 4
(8+4)/2 = 6

Midpoint is (4, 6).

Programming Use Cases

• Collision detection: Check if distance between two objects < threshold
• Nearest neighbor: Find the closest point in a dataset
• Path length: Sum distances between consecutive waypoints