I couldn't really think of a good one line title for my question, so I will try to elaborate.
From what I have sort of gathered over my years, if you want to locate an arbitrary point in an n-dimensional coordinate space, you need at least n coordinates. This was sort of implied in my Linear Algebra class about linear coordinates, I think ... but I sort of felt that there was a further implication that this requirement applies to all types of coordinate systems, including angle-based ones like Polar coordinates.
For example, if I wanted to describe a 3D point, I can choose whatever coordinate system I want, but that coordinate system must require at least 3 points to be able to uniquely identify my point. That's why cartesian $(x,y,z)$, cylindrical $(r,\theta,z)$, and spherical coordinates $(r,\theta,\phi)$ work, as well as a coordinate system relying on a linear combination of three linearly independent vectors (cartesian being a special case of this where the three vectors are orthogonal unit vectors). Furthermore, if a linear combination coordinate system requires four or more coordinates but only describes points on one 3D space, that coordinate system has redundancies and may be reduced to only needing three coordinates.
I was then taught, in Physics, that a point in that 3D space was an abstract entity, and that the coordinate system is irrelevant to the nature of the point -- to describe it, all you needed was to chose a set of bases. Any bases will do -- as long as there are three of them.
I sort of took this generalization to other spaces besides just the n-dimensional real number space. For example, the human color space, I figured, has 3 dimensions, and that's why all of the current color systems (RGB, HSV, HSL, the varies CIE perceptional color spaces) all require three coordinates, and you can't describe all colors with only two coordinates, and a coordinate system with four coordinates is redundant.
Also, there is the space of all possible rectangles. A rectangle can be uniquely described by its height and width, but there are many other ways, all of which require at least two numbers.
And all "redundant" systems may be reduced to a non-redundant one (similar to how vectors that aren't linearly independent can be reduced to a set of linearly independent vectors that can combine to form the original ones)
Thank you if you are still reading at this point; I know that that was kinda long-winded. I have two questions, that are somewhat related.
Is what I am thinking true? Did I make too many vast generalizations? What is wrong about this line of thought, if anything? I'm sure I made too many jumps somewhere.
One thing that has been troubling me is the "trilateration" "coordinate system" used to find points on a 2D plane. Coordinate system is defined as: There are three reference points at arbitrary locations on the plane. The coordinate is given as $(r_1,r_2,r_3)$, where $r_n$ is the distance of the given point from the $n$th anchor point.
This bugs me because to locate a 2D point using this coordinate system, you need three coordinates. Should it not be that, because this is a 2D plane, you only need two coordinates? How does something like this exist? What makes it fundamentally different from the other coordinate systems/spaces I have mentioned previously? (Cartesian, polar, color space, rectangles, etc.)