Question about the nature of coordinate systems

Question

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.

1. 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.

2. 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.)

Answer 1

Good question! Just what is three-dimensional about a three-dimensional space, and why are three coordinates required to locate a point in it? Could you possibly locate one with two coordinates, or just one?

It turns out that it is possible to locate a point in three-dimensional space with only a single coordinate, but any way of doing this will have some problems with it. One simple way to do this is by taking the digits and interleaving them. If there is a point at $(1.2345, 2.2222, 5.5555)$, we can shuffle all of these numbers together into a single number, like $125.225325435525$. This number can be turned back into the original three coordinates, so we can say that it is a single-coordinate representation of the point. Let's say that $f(x,y,z)$ is the interleaving of the numbers $x$, $y$ and $z$.

One problem with this method is that sometimes, a real number has two different decimal expansions. Specifically, if a number has a finite decimal expansion, it also has a decimal expansion ending in nines forever: $1.25$ is the same number as $1.24999\ldots$, but written differently. We can get around this, to some extent, by insisting on using the finite decimal expansions of $x$, $y$ and $z$ instead of the nines-forever decimal expansion.

However, even after we "fix" the method that way, it still has a significant problem: it is not continuous. Suppose we take the point $(1, 1, 1)$ and change the $z$-coordinate just a little bit. The new $f$ coordinate, instead of changing just a little bit, will jump between $111$ and $110.009009009\ldots$. That's not a property we like; a point's coordinates ought to change smoothly when the point changes smoothly. The function is not continuous.

What if we try to make the function $f(x,y,z)$ continuous? The only way to do this is by making it something like $f(x,y,z) = x + y + z$. But now this is even worse: the points (1, 2, 2) and (2, 0, 3) would both be represented by the coordinate 5. This function is not injective.

And so there is one key property that makes a three-dimensional space three-dimensional: even though there is a function taking points in three-dimensional space and returning points on a plane or a line, no such function is both continuous and injective. This is a result of topology, the study of continuity and related concepts.

As for your other question, I can't give you a good reason why the trilateration coordinate system requires three coordinates in order to identify a point on the plane, apart from simply that it's not a great coordinate system! There are many other systems that require three coordinates to identify a point on a plane: for example, we could say that a point's coordinates are the absolute value of its $x$-coordinate, the absolute value of its $y$-coordinate, and its angle around the origin. The only thing (as far as I know) that coordinate systems like these have in common is that they represent points inefficiently.

You always can represent a point on the plane with two coordinates, by using a coordinate system like the Cartesian one. The fact that it's possible to do a "bad job", and create a coordinate system that requires extra coordinates, is not so important.

Answer 2

You musings appear to be valid and coherent.

I think the concept you're grasping for is that of a (differentiable) manifold, which is -- very roughly! -- a topological space that comes with a selection of abstract coordinate systems, such that each coordinate system locally approximates a linear coordinate system in a sufficiently small neighborhood of each point it can describe. Under appropriate assumptions it can then be proved that all of these coordinate systems must have the same dimension.

Your problem with trilateration is that the coordinates are not free -- most arbitrary triples of numbers do not define any point at all, so the system does not actually work like coordinates are usually supposed to. It is more of a (distorted) embedding of the plane into $\mathbb R^3$ than it is a coordinate system for it.