# Question 29790: Whole-Plane Classification of Uniform Scaling

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0058-29790 · q-20261006-advance-20261007-r07-geometric-transformations-0090



## Original question and givens

Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(14)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.


- G001 — Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(14)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-geometric-transformations](https://ocme.evemisslab.com/en/mko/mko-geometric-transformations/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [qf-transform-uniform-scale-class, uniform_scale_class]

After

derived = [[WHOLEplane_map_class_label_positive_length_scale]]

Conditions: none

Follow THIS bank exact keys,domain/frame,operation and answer role.STR22/App20 tuples differ from legacy scalars;retain or omit/1 by the ORIGINAL contract,not a generic visual convention.

Step identifiers and dependenciesRule: R58_EXACT_source_interface_and_answer_ROLE · Uses: G001





S002 · impliesBefore

source roles = [14, euclidean_orthonormal_equal_units]

After

derived = [14, ||cP-cQ||=|c|*||P-Q||, similarity that is not an isometry；scale factor=14]

Conditions: none

For ANY P,Q,the difference scales by c,its squared norm by c²,and the distance by|c|.At0 the whole plane collapses to the origin;|c|=1 preserves ALL distances;other nonzero c gives positive scale|c| but not isometry for distinct points.One coincident pair is not a global certificate;c=-1 is an origin180-degree rotation,not a line reflection.

Step identifiers and dependenciesRule: Transform_WHOLE_plane_distance_scaling_identity_ALL_P_Q · Uses: G001, S001









## Reverse checks


- S002 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.
- S001 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.





## Result

non-isometric similarity; scale=14

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 4ad591fa53321337ffd6b048d19cc5c654234eee1bd21858d9b9347678069e4f
- candidate record SHA-256: 3c2b0dd9f6204c777c5fcfaa63c88209f2b6897e41402dd143eabc50ee224896
- semantic review receipt SHA-256: 66bbb31c5fe644576dfd47ac71ca9e367fc23b948eac8e3eb648d96dadea05df
- question-first witness SHA-256: b1e4fbdbca8d0ac3581e2f26466374d870f3c8498fab7cf6873feca5bbec44e7
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261006-advance-20261007-r07-geometric-transformations-0090.json)