# Converting between coordinate systems?

## Homework Statement

I have a bit of a general question, and I don't know whether or not the problem has a solution, but here's the idea behind it. I have two coordinate systems, let's call them CS A, and CS B. I have an infinite set of corresponding points for each system (both are 3D, so I have X, Y and Z data for each coordinate). So if I know a coordinate in CS A, I can find out the coordinate in CS B. The thing is, I don't know the actual transform that turns one system to the other (so I can use a 'function' but I don't know its definition). Given that I have an infinite set of points, is it possible to find the function definition? How would I go about doing it? (It is not a straight forward transform that is readily apparent, not just a translation... I'm sure there is rotation, might be flipping, etc).

Thanks!
Preet

Do you mean converting between catresian (x,y,z), cylindrical (r,Θ,z), and spherical (p,Θ,Φ) co-ordinate systems?

If so a point in cartesian is given by (x,y,z)

A point in cylindrical is given by (r,Θ,z)

A point in spherical is given by (p,Θ,Φ)

Conversion of cylindrical to cartesian is (rcosΘ, rsinΘ, z)

Conversion from spherical to cartesian is (pcosΦcosΘ, pcosΦsinΘ, psinΦ)

I think that's what you mean.

HallsofIvy
If you want to make a coordinate system, you need to define some base vectors first that form a right hand system, im pretty sure (not completely though). for example, you need to be able to express the unit vectors $$\vec{x}, \vec{y}, \vec{z}$$ into $$\vec{e_{1}}, \vec{e_{2}}, \vec{e_{3}}$$. To do this, however, you must know the relationship between x, y, z and the new system. If its a rotation you can do this if you have the rotation of the new system from first coordinate system and simply project the coordinates. If its not a rotation this can get complicated. Once you have all of this information, you can then go on to define the area elements and volume elements using Jacobians. I hope this is what you're wondering about. If not, let me know and i will try to clean it up.