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I am having some trouble

*categorizing*a few concepts (

__I made the one that is critical to this post to be BOLD__)

**Remote parallelism: the ability to move coordinate systems and frames around in space.**

- Euclidean Space
- Coordinate systems: Cartesian vs. cylindrical

I am aware that if space is Euclidean, then certain geometric rules hold...

For example, we convert the dot product (from its definition of the product of the norms of two vectors times the cosine of the angle between them) to an algebraic one (where we sum the products of the associated vector components.); and other things about angles to 180, parallel lines, etc.

Distinct from this, is the fact that one can move coordinate systems around. And I have understood this process to be "remote parallelism." And we can do it when the coordinate system is Cartesian. We cannot do it when the system is, say, polar or cylindrical.

So, this is where i need help. I cannot see how Euclidean space is a requirement for remote parallelism. I can see that we need Cartesian coordinates to move frames around. But I cannot see why a Euclidean assumption is required.