# What is the difference between Euclidean and Cartesian spaces?

For me those are references to the same thing.

On Wikipedia there are references to both but I still don't see the difference. http://en.wikipedia.org/wiki/Cartesian_coordinate_system#Cartesian_space

Is there any difference? If yes, is there a simple way to describe it?

Point in Euclidean plane can be written in many ways: either using Cartesian coordinate system, or polar coordinate system. That is same point $p$ can be written in two ways... If we are saying Euclidean plane, It simply means that we are giving some axioms and using theorem based on that axioms. But if we are saying Cartesian plane, it means that with euclidean axiom we are giving some method of representing of points.

This means: Euclidean Plane means we have only some set of axiom

Cartesian plane means Euclidean plane+ One fixed method of representing points.

• Thank you. Now it's clear for me. Feb 23, 2012 at 9:39
• @AalexGabi & @ zapkm, What did he/you mean by "But if we are saying Cartesian plane, it means that with euclidean axiom we are giving some method of representing of points."? I didn't understand that. If possible, could you please elaborate on that? ... Also, does it mean: "Cartesian plane does not exist without Euclidean Plane"? I'm still super confused! Thanks in advance. Aug 22, 2020 at 18:56
• @AalexGabi & @ zapkm In fact, it would be great and very helpful if you can mention some good resources for further reading! Aug 22, 2020 at 18:57
• @AalexGabi, you mentioned log-polar coordinates also belong to Euclidean plane, is that correct?
– BiMo
Feb 1 at 10:19

The Cartesian system is Euclidean space with coordinates. The Cartesian Coordinate System unified geometry and algebra into one system of analytic geometry.

If you know MATLAB, A weak way of explaining it is:

clf;
ezplot('sin(x)'); %Euclidean
grid ON; %Cartesian


Euclidean

Cartesian

• I don't have matlab, if the above produces a kind of shape I would love to see it! Oct 23, 2015 at 10:36