Euclidean distance in an equal-unit orthonormal frame

Mathematical statement

For P=(x1,y1) and Q=(x2,y2) in the same equal-unit orthonormal Cartesian frame, d(P,Q)=√((x2−x1)²+(y2−y1)²)≥0. Its square is the sum under the root; two such distances can be compared by their squares.

Representative formula (exact TeX)

d(P,Q)

Assumptions

• Both points share an orthonormal frame whose axes use the same length unit. Skew coordinates or unequal axis scales do not directly give Euclidean distance by this formula.
• Exercise coordinates are integers. Squared distances and comparisons are exact; direct integer-distance questions additionally require a perfect-square sum.
• A candidate distance must be nonnegative and have the required square. Coincident points legitimately have distance zero.

Symbol scope

• d — plane Euclidean distance
• P — first coordinate point
• Q — second coordinate point

Intuition, definition, and misconceptions

The intended metric is Euclidean L2, not the default product metric on an ordinary pair of real numbers. A formal bridge must therefore identify an appropriate EuclideanSpace model rather than assume that every product distance is this formula.

Exchanging endpoints, translating both points, exchanging axes, or reversing coordinate signs preserves the appropriate squared-distance data; these cosmetic changes do not supply substantive diversity. Non-perfect-square distances are not replaced by floating-point guesses.

A negative number whose square is correct is still not a distance. The earlier Pythagorean side-model Evidence is not automatically a proof for arbitrary coordinate points.

Common errors include accepting a negative square root as distance, using unequal axis scales without conversion, or confusing the ordinary product metric with Euclidean L2.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Coordinate plane — Specify one shared coordinate frame.
• Natural-number powers and principal square roots — Use the sum of squares and its nonnegative principal root.
• Euclidean length — Fix the meaning of distance and length units.

Verification boundary

Finite squared distances and eligible integer roots can be recomputed; the coordinate-to-Euclidean-space bridge is not newly formalized.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Bind EuclideanSpace R (Fin 2), the equal-unit orthonormal frame, and the squared-norm identity.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.