Orthogonal Coordinate Systems

[bolbteppa]

Hey guys, I'd really love it if you could post little essays explaining your intuition on how to derive the x, y & z coordinates from all/any of the orthogonal coordinate systems in this list, how you think about, say, bipolar coordinates if you had to re-derive the coordinate system on a desert island etc...

Apart from spherical & cylindrical I have no idea how to remember the others, these two I remember because I can think of the picture & re-derive how to express x, y & z in terms of r, θ & z etc... but the others are completely crazy geometrically & I have no intuition on them, & I need to at least learn how to represent x, y & z in terms of each system (so I can get grad, div, curl etc...) along with intuition on when to use them.

One interesting example of what I'm hoping for is with toroidal coordinates whose wiki is incomprehensible yet apparently there is an insanely simply way (page 114, also in this link) to think about these coordinates, [though the formulas for x, y & z in the links don't agree, which isn't a good sign, apparently they're a bit different but that's not a happy realization ] Thanks!

 

[jvicens]

Hi there,

I completely understand your struggle with remembering and deriving the coordinates for different coordinate systems. It can be overwhelming and confusing, especially when there are so many different types to keep track of. But don't worry, with a little bit of intuition and practice, you'll be able to easily derive and use the coordinates for any system.

First, let's start with the basics. The Cartesian coordinate system, also known as the rectangular coordinate system, is probably the most familiar to you. It uses three perpendicular axes (x, y, and z) to define a point in three-dimensional space. To derive the coordinates for this system, you simply need to measure the distance along each axis from the origin point.

Next, let's look at the polar coordinate system, which is essentially a two-dimensional version of the cylindrical coordinate system. This system uses a distance (r) from the origin and an angle (θ) to define a point in two-dimensional space. To derive the coordinates for this system, you use trigonometry to find the x and y coordinates from the given r and θ values.

Now, for the other orthogonal coordinate systems, such as spherical, cylindrical, and toroidal, it helps to think about them in terms of their geometric shapes. For example, the spherical coordinate system is based on a sphere, with the origin at the center. To derive the coordinates for this system, you use the distance from the origin (r), the polar angle (θ), and the azimuthal angle (φ) to define a point on the sphere.

Similarly, the cylindrical coordinate system is based on a cylinder, with the origin at the center of the circular base. To derive the coordinates for this system, you use the distance from the origin (r), the polar angle (θ), and the height (z) to define a point on the cylinder.

Now, for the toroidal coordinate system, which is based on a torus (donut-shaped object), it may seem a bit more complicated at first glance. But, as you mentioned, there is a simple way to think about it. The toroidal coordinate system uses two angles (θ and φ) and a distance (ρ) from the center of the torus to define a point. To derive the coordinates for this system, you can think of it as a combination of the spherical and cylindrical coordinates. The angle θ represents the polar angle on the torus, while the angle φ represents the azimuthal