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Geometric Distances

Geometric Distance Metrics

Euclidean Distance​

d(p,q)=∑i=1n(pi−qi)2d(p, q) = \sqrt{ \sum_{i=1}^{n} (p_i - q_i)^2 }

Explanation: Measures the straight-line (as-the-crow-flies) distance between two points in Euclidean space. Common in geometry, physics, and machine learning.

Manhattan Distance​

d(p,q)=∑i=1n∣pi−qi∣d(p, q) = \sum_{i=1}^{n} |p_i - q_i|

Explanation: Also called “city block” distance. Measures total absolute differences; better for grid-based layouts.

Minkowski Distance​

d(p,q)=(∑i=1n∣pi−qi∣r)1/rd(p, q) = \left( \sum_{i=1}^{n} |p_i - q_i|^r \right)^{1/r}

Explanation: A generalized distance metric. When r=1r=1 → Manhattan, r=2r=2 → Euclidean.

Cosine Distance​

cos_dist(p,q)=1−p⋅q∥p∥∥q∥\text{cos\_dist}(p, q) = 1 - \frac{p \cdot q}{\|p\| \|q\|}

Explanation: Measures the angle between two vectors. Commonly used in NLP, recommendation systems, and text similarity.

Hamming Distance​

d(p,q)=∑i=1n1pi≠qid(p, q) = \sum_{i=1}^{n} \mathbb{1}_{p_i \neq q_i}

Explanation: Counts how many positions two strings differ in. Used for binary strings, error detection/correction.