Projective

Look at computer vision

Homogenous Coordinates

Make one more coordinate then you need. You decide the overall scale no longer matters. The point (1,3) is the same point as (2,6). The simple model is that of a pinhole camera. If the pinhole is at (0,0,0), then all point light sources on a line stretching through the pinhole make equivalent images. It strangely turns out that these coordinates have beautiful properties.

Duality

In homogenous coordinates,

Axiomatic

Euclidean

Classical geometry. Euclidean lines and points and circles

Algebraic

https://en.wikipedia.org/wiki/Blowing_up birational

Griffiths, Phillip; Harris, Joseph (1978). Principles of Algebraic Geometry. Hartshorne, Robin Algebraic Geometry. https://math.stackexchange.com/questions/1748/undergraduate-algebraic-geometry-textbook-recommendations?noredirect=1&lq=1

Differential

https://en.wikipedia.org/wiki/Manifold https://juliamanifolds.github.io/Manifolds.jl/stable/index.html https://arxiv.org/pdf/2106.08777.pdf manopt.jl manifolds.jl “The seven major libraries for differential geometry in Python are Pymanopt (Townsend et al., 2016), Geomstats (Miolane et al., 2020), Geoopt (Kochurov et al., 2020), TheanoGeometry (Kühnel and Sommer, 2017), Jax Geometry (Kühnel and Sommer, 2017), Tensorflow RiemOpt (Smirnov, 2021), and McTorch”

Visual differential geometry Functional differential geometry

Differential forms

coordinate free abstract manifold type

type M = { 
    [(string, I , Rn -> M) ] 
    [string,string, I,I ,Rn -> Rn] I is interval of overlap
    https://en.wikipedia.org/wiki/Atlas_(topology)
}
type M = {Rn, [Rn -> R] } # intersection of constrain maps embedded in higher Rm

x : M -> R

Curvature

Discrete Geometry

Convex Geometry

See convex optimization

Automated Theorem Proving

https://www.routledge.com/Handbook-of-Geometric-Constraint-Systems-Principles/Sitharam-John-Sidman/p/book/9781498738910 handbook of geometric constraint systems

https://www.geogebra.org/?lang=en https://link.springer.com/article/10.1007/s10817-015-9326-4

A Deductive Database Approach to Automated Geometry Theorem Proving and Discovering

Automated reasoning in geometry theorem proving with Prolog

Affine geometry of collinearity and conditional term rewriting A good maude example probably

TGTP thouands of geometrical problems

basic properties of triangles isabelle

https://dblp.org/pid/18/4206.html jacques fleuriot

https://www.usna.edu/CS/qepcadweb/B/QEPCAD.html

https://github.com/lucaferranti/GeometricTheoremProver.jl https://www.youtube.com/watch?v=q_08LE4UOU8&ab_channel=TheJuliaProgrammingLanguage Ritt-Wu method. Use polynomial divisiion once? Seems kind of weak, but maybe Automated Geometric Theor omated Geometric Theorem Proving: W ving: Wu’s Method s Method

https://www.osti.gov/servlets/purl/1761359 Prove-It https://sandialabs.github.io/Prove-It// https://github.com/sandialabs/Prove-It

https://www.andrew.cmu.edu/user/avigad/Students/rojas_thesis.pdf EuclidZ3