# Plane-coordinate isometries and uniform scaling

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Translations, counterclockwise quarter-turn rotations about the origin, and specified line reflections are plane isometries; uniform scaling distinguishes zero, ±1, and other coefficients.

mko-geometric-transformations · definition · v0.10.0



## Mathematical statement

On the equal-unit orthonormal real plane, translations H(x,y)=(x+tx,y+ty), counterclockwise rotations about the origin by 0°,90°,180°,270°, and reflections across the x-axis, y-axis, y=x, and y=−x are isometries. For Hc(x,y)=(cx,cy), c=0 gives a degenerate constant map; |c|=1 gives a whole-plane isometry; any other nonzero c gives a non-isometric similarity with positive length scale |c|.





## Representative formula (exact TeX)

H(P)





## Assumptions


- Domain and codomain are the whole real plane. Exercise coordinates, translations, and uniform-scaling coefficients are safe integers.
- The rotation center is the origin and the direction counterclockwise, with turns from zero through three quarter-turns. The reflection line is explicit.
- The uniform-scaling classification concerns the whole-plane map. One coincident pair or fixed point cannot establish global isometry.





## Symbol scope


- H — explicit coordinate map on the real plane
- P — input coordinate point





## Intuition, definition, and misconceptions

A 90° counterclockwise turn sends (x,y) to (−y,x), 180° to (−x,−y), 270° to (y,−x), and 0° is the identity. The origin is fixed by each, so different turn labels at the origin do not supply substantive diversity.

Reflection across the x-axis is (x,−y), across the y-axis (−x,y), across y=x (y,x), and across y=−x (−y,−x). Central inversion (−x,−y) is a half-turn, not reflection across a single line.

Uniform scaling multiplies every distance by |c|. Zero fails to preserve distances between distinct points, while c=−1 does preserve them; a negative coefficient is not automatically non-isometric. One fixed point does not establish a property of the whole-plane map.

Common errors include omitting rotation direction or center, mixing reflection lines, treating a negative coefficient as a negative length scale, or inferring global isometry from one fixed point.





## Adopted derivation guide

[Geometric transformations: proving the basic isometries and classifying uniform scaling](https://ocme.evemisslab.com/en/derivations/derivation-mko-geometric-transformations-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Function mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) — Specify the whole-real-plane map domain.
- [Coordinate plane](https://ocme.evemisslab.com/en/mko/mko-coordinate-plane/) — Fix the origin, axes, and coordinate directions.
- [Euclidean distance in an equal-unit orthonormal frame](https://ocme.evemisslab.com/en/mko/mko-distance-formula/) — Fix global distance-preservation semantics.
- [Labeled triangle similarity with a specified positive scale](https://ocme.evemisslab.com/en/mko/mko-similarity/) — Distinguish a coordinate coefficient from a positive length scale.





## Verification boundary

Explicit integer-coordinate instances can be recomputed; general isometry/similarity maps and arbitrary-angle rotation bridges are 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

Bridge explicit isometries, line reflections, and uniform scaling on pinned EuclideanSpace R (Fin 2), not the default ordinary product metric.





## 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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/geometric-transformations.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-geometric-transformations/index.md)