Spherical-esque Coordiante System

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The discussion focuses on generalizing spherical coordinates to N-dimensions, emphasizing the need to define angles for each additional dimension. To compute the angle for the Nth dimension, the approach involves projecting the distance onto the existing (N-1) dimensional space using cosine functions. For example, in transitioning from 3D to 4D, the angle omega is defined as the angle between the 3D space and the 4th dimension, with the projection calculated using cos(omega). This method allows for a systematic way to express coordinates in higher dimensions by prepending the cosine of the new angle to the existing coordinate equations. The conversation concludes that this generalization effectively extends the spherical coordinate system into higher dimensions.
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Is there a generalization of the 2D or 3D spherical coordinate system to N-dimensions? I want to represent points in the space as a distance r from something, and then a bunch of angles. If this works for arbitrary dimensions, what's the rule for defining the newest angle each time I add a dimension?

In case that's not clear, what I want to know is, if I have a point in 3D (x,y,z) I can also call it (r,phi,theta). But if I have a point in 4D (x,y,z,w), and I want to call it (r,phi,theta,omega), how do I compute omega?
 
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I don't exactly know for what application you need that, but to define an angle you have to first define a function between two axes. If what you mean is "convert" something N-1 D to N D coordinates, I think you take it as if the added value can be arbitrary.
 
Use the dot product.
 
LeBrad said:
But if I have a point in 4D (x,y,z,w), and I want to call it (r,phi,theta,omega), how do I compute omega?

This is one of doing it:

x=r cos(phi)
y=r sin(phi) cos(theta)
z=r sin(phi) sin(theta) cos(omega)
w=r sin(phi) sin(theta) sin(omega)
 
Ok I think I can see how it generalizes now. If I have a coordinate system for (N-1) D, I can just define an angle between (N-1) space and the Nth dimension, and then project into (N-1) space and then use the (N-1) coordinate system with the projection prepended.

So in 2D we have
x = r*cos(phi)
y = r*sin(phi).
Then when I add a third dimension, I define theta as the angle between the 2D space and the new dimension, then project r onto the 2D space with cos(theta), and get
x = r*cos(theta)*cos(phi)
y = r*cos(theta)*sin(phi). Then project onto the 3rd D to get the final piece,
z = r*sin(theta).

So for 4D I would define omega as the angle between 3D and the 4th D, and let the projection be cos(omega) into 3D to get
x = r*cos(omega)*cos(theta)*cos(phi)
y = r*cos(omega)*cos(theta)*sin(phi)
z = r*cos(omega)*sin(theta)
w = r*sin(omega)
where all I did was prepend cos(omega) to the (N-1) projections and then define the new coordinate as r*sin(omega).
 

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