Plane-coordinate isometries and uniform scaling

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

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

Prerequisites

• Function mapping — Specify the whole-real-plane map domain.
• Coordinate plane — Fix the origin, axes, and coordinate directions.
• Euclidean distance in an equal-unit orthonormal frame — Fix global distance-preservation semantics.
• Labeled triangle similarity with a specified positive scale — 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.