# Points, lines, and planes

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

Points, lines, and planes provide foundational geometric language for position, direction, collinearity, coplanarity, and incidence.

mko-point-line-plane · concept · v0.10.0



## Mathematical statement

In a specified geometric model, a point represents a position. Lines and planes are geometric objects carrying points with their respective dimensions, and their precise properties are determined by the axioms and structure of that model.





## Representative formula (exact TeX)

P





## Assumptions


- The Euclidean, affine, or other geometric context in use is explicitly stated.





## Symbol scope


- P — representative point





## Intuition, definition, and misconceptions

A point is not the ink dot drawn on paper; the drawn size is only a visual marker. Lines and planes are likewise not defined by stroke thickness or page boundaries.

In an affine model, the roles of points and translation vectors are distinct, and lines and planes can be represented by affine subspaces of suitable dimension.

Statements about collinearity, coplanarity, and intersection depend on explicit incidence or ambient-space relations.

A common mistake is to treat the appearance of a diagram as a definition or to claim universal properties of lines and planes before specifying a geometric model.





## Adopted derivation guide

[Points, lines and planes: deciding incidence in the coordinate model](https://ocme.evemisslab.com/en/derivations/derivation-mko-point-line-plane-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites







## Verification boundary

Coordinate examples verify only instances in the specified model; they do not prove geometric axioms or general dimension statements.

not_formalized

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





## Next obligation

Choose an affine-space model and bind points, lines, planes, dimensions, and incidence statements separately.





## 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/point-line-plane.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-point-line-plane/index.md)