3D Points Distance

Euclidean distance between two 3D points

The Euclidean distance between P₁ = (x₁, y₁, z₁) and P₂ = (x₂, y₂, z₂) is d = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²). Think of it as the Pythagorean theorem stretched into three dimensions. The segment P₁P₂ becomes the diagonal of a rectangular box whose sides are Δx, Δy and Δz. Take a concrete case: going from (0, 0, 0) to (3, 4, 12) works out to √(9 + 16 + 144) = √169 = 13. Notice that the result never depends on which point you call first, it can't come out negative, and it only hits zero when the two points are actually the same.

Applications

It turns up all over the place. CAD and 3D modeling tools use it to measure gaps between parts. GPS in ECEF coordinates (Earth-Centered, Earth-Fixed) relies on it for satellite-to-receiver geometry. In molecular modeling it gives you bond lengths and inter-atomic distances. Game and physics engines lean on it for VR/AR occlusion, frustum culling and collision detection, and clustering methods like k-means and k-NN treat Euclidean distance as their measure of how similar two things are.

FAQ

Why is the distance always positive? Squaring each difference can't produce a negative number, and a square root of something non-negative stays non-negative. The only way to land on zero is for all three coordinates to match.

Does the order of the points matter? Not at all. Since (x₂ - x₁)² = (x₁ - x₂)², you get d(P₁, P₂) = d(P₂, P₁) either way.

What about Manhattan or Chebyshev distance? Both are different ways of measuring. Manhattan adds up the absolute differences (|Δx| + |Δy| + |Δz|), while Chebyshev just takes the largest of them. They fit grids and chess-style movement well. Euclidean is the one you want when you mean the straight line between two points.