Cartesian, Euclidean, and Gaussian... Coordinate System?

Is there such thing as Cartesian, Euclidean, and Gaussian Coordinate System? In my mind, I have these concepts described in the following way:

Cartesian Coordinate System: a coordinate system that specifies each point uniquely in a plane like this; (x,y)

Euclidean Coordinate System: a coordinate system that specifies each point uniquely in a plane like this; (x,y,z)

Gaussian Coordinate System: a coordinate system that specifies each point uniquely in a plane like this; (r,theta)

I recently participated in a Math Competition and we were divided into three groups: Cartesians, Euclideans, and Gaussians; and now I am trying to figure out what do these three adjectives have in common besides of coming from really great mathematicians.

Easy Math Editor

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`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
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Comments

The Cartesian co-ordinate system just means a system of rectangular co-ordinates, where the Euclidean notion of perpendicularity (zero Euclidean scalar product) implies that each co-ordinate axes are mutually perpendicular. A more general co-ordinate system would be the affine co-ordinate system; here, no notion of perpendicularity is assumed on the co-ordinate axes and armed only with the notion of parallelism it's more like a system of parallelogram co-ordinates, as opposed to rectangular ones in the Cartesian case. As for the circular/spherical/cylindrical co-ordinate systems, the best way to frame them would be in terms of projective co-ordinates (as opposed to the traditional "distance"/"angle" formulation).

A Former Brilliant Member - 2 years, 9 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$