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Let f(x,y) = 0 represent the equation of a curve in which both x and y are required to appear in the equation. Is it always possible to choose a new coordinate system such that only one (new) variable appears in the equation that describes the set of points on the curve?

For example, the circle (x-1)^2 + (y-2)^2 = 9, under the transformation x = 1+rcos(theta), y = 2 + rsin(theta), can be described by the equation r = 3.

In general, can this be extended to n-dimensional manifolds? Is this literally just the definition of dimensionality? Please give an intuitive answer to my question if you can

Ahdhehshdjdj
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1 Answers1

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Because any curve can be parametrized, the answer is yes. Using your example $x=1+r\cos\theta, y=2+r\sin\theta$, replacing r with 3 gives a parametrization $x=1+3\cos\theta, y=2+3\sin\theta$ of the circle. So this can be done with any curve by letting $x_n=af_n(t)$ and writing it as a=1. But I don't think you can use it to define dimensionality. For example, take the unit sphere in 3D which can also be described by the equation r=1. Because the parametrization of the spheres are $x=r\sin\theta \cos\phi, y=r\sin\theta \sin\phi, z=r\cos\theta$, what makes it 2D is not the fact it can be described as r=1 but as functions of two variables: $\theta$ and $\phi$ (since r is fixed at 1).

In general, n-dimensional manifolds have n variables in their parametrization, not fixing n variables to some value.

JonathanZ
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CyanWithPikachu
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    To address the math: The codimension is (in general) given by the number of equations, so you can still get the dimension of the submanifold from this if you know the dimension of the ambient space. Although you have to check for degeneracies; for example, $ r = 0 $ defines a point (codimension $ 2 $) in $ \mathbb R ^ 2 $, instead of a curve. (I don't think that this is what the reference to the definition of dimension in the question was about, however.) – Toby Bartels Jan 14 '24 at 04:23