What are the key differences between Euclidean and plane geometry?

[yungman]

What is the difference between the Euclidean Geometry and the simple plane geometry? They seems to work with flat planes.

 

[tiny-tim]

hi yungman!

i'm confused …

aren't they the same thing? ​

 

[micromass]

I don't necessarily associate "plane geometry" with Euclidean geometry. Something like \mathbb{R}^3 or \mathbb{R}^n are also Euclidean geometries (to me).

 

[yungman]

Thanks for the reply, I just gone on the Youtube to take a crash course into spherical trigonometry. The lectures review the basic of Euclidean geometry as an introduction to spherical geometry. Everything about Euclidean geometry sounds like just simple plane geometry I learned long time ago, but I never learn the name Euclidean geometry. I really don't know the detail, that's the reason I asked.

Thanks

Alan

 

[tiny-tim]

Hi Alan!

Everything about Euclidean geometry sounds like just simple plane geometry I learned long time ago …

Yes it is … different name, same thing!

 

[yungman]

Hi Alan!


Yes it is … different name, same thing!

Hi Tiny-Tim

Thanks for the reply. That's all I want to know.

Alan

 

[mathwonk]

to me plane geometry means two dimensional geometry, either euclidean or hyperbolic, while euclidean geometry means essentially the geometry of R^n, i.e. a geometry of any finite dimension in which triangles have angle sum 180 degrees.

to people who have not studied hyperbolic plane geometry, the term plane geometry probably means the more familiar euclidean plane geometry. i do not consider the hyperbolic plane to be flat however.

 

[yungman]

I guess I consider plane geometry can be extended to 3D as long as all the surfaces are flat. Just like a cube composes of six planes. All the trig functions apply.

The spherical surface is totally different where circumference is not 2\piR as the surface is not flat.

 

[SteamKing]

I believe geometries (2D or 3D) are referred to as Euclidean if the parallel postulate holds. In Euclidean geometry, a 2D triangle has a total internal angle of 180 degrees or pi radians. In non-Euclidean geometries, there is no parallel postulate, and the total internal angle of a 2D triangle is not equal to 180 degrees.