Euclidean Geometry versus Analytic Geometry versus Affine Geometry?

Question

What are the relationships (connections) among:

• Euclidean (or Plane) geometry
• Analytic geometry
• Affine geometry

How do these things relate?

I know that this is a very general question, so I'm looking for a simple overview.

Thanks.

Answer 1

Felix Klein defined geometry to be the study of invariants of group actions.

Euclidean geometry looks at things unchanged by Euclidean transformations, i.e. rigid transformations, i.e. rotations, translations (and sometimes reflections).

Affine geometry looks at things unchanged by affine transformations. This usually means non-singular linear transformations, i.e. $\det M \neq 0$. An important subset, equi-affine geometry, studies linear transformations which preserve area, i.e. $\det M = 1$.

Projective geometry does the same with projective transformations.

Analytic geometry doesn't really fit in to this. For me, "analytic geometry" just means "using coordinates". All of Euclidean, affine and projective geometry can be done using coordinates.

Maybe I misunderstand "analytic geometry".

Answer 2

Euclidean Geometry is geometry studied by Euclid that is based off of shapes with no values. Analytical Geometry is geometry on planes with values. So, the difference is mainly that Euclidean doesn't use specific values