Bipolar and toroidal coordinates

[Mr. Rho]

Hi,

I'm trying to understand toroidal coordinates, that are a 3D version of bipolar coordinates. Well, I don't understand any of them, already cheked wiki and wolfram sites, but they don't give any clear explanation (in my opinion).

Is there a way to get an intuition for obtaining the cartesian coordinates x,y,z in terms of bipolar and toroidal coordinates?

Thanks for your help

PD: zorry 4 bad grammar

 

[jvicens]

Hi there,

Thank you for reaching out about your confusion with toroidal coordinates. I can understand how complex and overwhelming these concepts can be. I am happy to help you gain a better understanding of how to obtain cartesian coordinates using bipolar and toroidal coordinates.

First, let's start with bipolar coordinates. These coordinates are a 2D version of polar coordinates, which are commonly used to describe positions in a plane. Bipolar coordinates are used to describe positions in a 2D space with two focal points, instead of just one point like in polar coordinates. The two focal points are called foci and are usually labeled as (+a,0) and (-a,0) on a graph.

To obtain cartesian coordinates (x,y) from bipolar coordinates, we use the following equations:

x = a*cosh(u)*cos(v)
y = a*sinh(u)*sin(v)

Where a is the distance between the two foci and u and v are the two parameters used in bipolar coordinates.

Now, let's move on to toroidal coordinates. These are a 3D version of bipolar coordinates and are used to describe positions in a 3D space with two focal rings. Just like in bipolar coordinates, the two focal rings are labeled as (+a,0,0) and (-a,0,0) on a graph.

To obtain cartesian coordinates (x,y,z) from toroidal coordinates, we use the following equations:

x = a*cosh(u)*cos(v)
y = a*cosh(u)*sin(v)
z = a*sinh(u)

As you can see, the equations for obtaining cartesian coordinates from toroidal coordinates are very similar to those for bipolar coordinates, with the addition of the z-coordinate.

I hope this helps you gain a better understanding of how to obtain cartesian coordinates from bipolar and toroidal coordinates. If you have any further questions or need more clarification, please don't hesitate to reach out. Keep in mind that these concepts can be difficult to grasp, so don't get discouraged if it takes some time to fully understand them. Keep practicing and seeking out resources, and you will get there.

Best of luck in your studies.