# Spherical-esque Coordiante System

In summary, the 2D and 3D spherical coordinate system can be generalized to N-dimensions by defining an additional angle for each dimension added. This angle is the angle between the (N-1) space and the new dimension. To compute this angle, you can use the dot product. Then, to represent a point in N dimensions, you can use the formula: x = r*cos(angle)*cos(phi), y = r*cos(angle)*sin(phi), z = r*sin(angle), and so on for each added dimension. This method allows for the conversion of coordinates between different dimensions.
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?

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.

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).

## What is a Spherical-esque Coordinate System?

A Spherical-esque Coordinate System is a three-dimensional coordinate system used to describe the position of a point in space. It is similar to a spherical coordinate system, but with some differences in the way the coordinates are measured and represented.

## What are the three coordinates used in a Spherical-esque Coordinate System?

The three coordinates used in a Spherical-esque Coordinate System are radius, inclination, and azimuth. Radius is the distance from the origin to the point, inclination is the angle between the radius and the z-axis, and azimuth is the angle between the projection of the radius onto the xy-plane and the x-axis.

## What is the range of values for the coordinates in a Spherical-esque Coordinate System?

The range of values for the coordinates in a Spherical-esque Coordinate System depends on the specific system being used. In general, the radius can be any positive value, the inclination ranges from 0 to π, and the azimuth ranges from 0 to 2π.

## What are the advantages of using a Spherical-esque Coordinate System?

One advantage of using a Spherical-esque Coordinate System is that it can easily describe the position of points in a spherical or circular shape, making it useful for applications in astronomy, physics, and engineering. It also allows for a more efficient representation of 3D data compared to Cartesian coordinates.

## How does a Spherical-esque Coordinate System differ from a Cartesian coordinate system?

A Spherical-esque Coordinate System differs from a Cartesian coordinate system in the way it measures and represents coordinates. In a Cartesian system, the coordinates are measured along three perpendicular axes (x, y, and z), while in a Spherical-esque system, the coordinates are measured as distance, inclination, and azimuth from an origin point. Additionally, a Spherical-esque system is better suited for describing spherical or circular shapes, while a Cartesian system is better for describing rectangular shapes.

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