# Examples of Non-Orthogonal Curvilinear Coordinates

## Main Question or Discussion Point

I am beginning to study the mathematics of curvilinear coordinates and all textbooks and web sites do not have realistic examples of non-othogonal systems.

What are some examples of non-orthoganal curvilinear coordinates so that I can practice on actual systems rather than generalized examples?

All the coordinate systems that I've examined, such as parabolic cylindrical, ellipsoidal, spherical, and polar cylindrical, are all orthogonal. There must be lots of non-othogonal examples.

Orodruin
Staff Emeritus
Homework Helper
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The reason you just see orthogonal coordinate systems is that they are generally much easier to work with. It should be rather easy to construct a non-orthogonal curvilinear system though. For example, for the two-dimensional plane you could use $\rho = r$ and $\alpha = \theta + kr$, where $r$ and $\theta$ are the usual polar coordinates and $k$ is a constant.

I would say that orthogonal systems are the "actual" systems we work in!

Are you just looking for these coordinates in case you come across a scenario in which orthogonal coordinates are not the most ideal? What calculations are you looking to do?

Orodruin
Staff Emeritus
Homework Helper
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I would say that orthogonal systems are the "actual" systems we work in!
You can work in any system you like, it does not matter for the results. You will get the same results in a non-orthogonal system as you will in an orthogonal one. There is nothing more "actual" about an orthogonal system.

Are you just looking for these coordinates in case you come across a scenario in which orthogonal coordinates are not the most ideal?
The main example that comes to my mind here is light-cone coordinates on Minkowski space. There are also many generalised mechanical systems where the kinematic metric is not diagonal in the most obvious coordinate systems.

You can work in any system you like, it does not matter for the results. You will get the same results in a non-orthogonal system as you will in an orthogonal one. There is nothing more "actual" about an orthogonal system.
Yes, which is why I put it in quotes since OP said he wants to practice on "actual systems rather than generalized examples?".

Chestermiller
Mentor
An example I can provide is that of an automobile tire structure that is modeled as a membrane. One is studying the structural mechanics of the tire as it is deformed under various modes of load application such as inflation and contact with the ground. The deformations of the tire membrane can be large. The initial shape of the tire is described by a material coordinate system embedded or inscribed onto the surface of the undeformated membrane. The coordinates are initially orthogonal. However, when the tire membrane deforms, the material coordinates become non-orthogonal, and the displacements of the material points are expressed functions of the original material coordinates. This is a standard way of setting up structural deformation problems.

What calculations are you looking to do?
There are all sorts of things: covariant and contravariant vectors, metric components and the metric tensor, scale factors, differential quantities, transformations between different systems, etc., etc.

In my experience, it is better to study these things using actual examples rather than the generalized coordinates, i.e. x=x(q1, q2, q3), y=y(q1, q2, q3), and z=z(q1, q2, q3), which are universally used in textbooks and web sites.

It is straightforward to construct non-ortho systems but I was wondering if there were some actual systems used somewhere in practice.

Chestermiller
Mentor
You are interested in some actual application of this to practical problems. OK.

Here is an example of a fluid dynamics problem from Physics Forums in which non-orthogonal coordinates are used to develop the differential force balance on a free surface of a fluid, including surface tension: https://www.physicsforums.com/threads/when-to-use-which-dimensionless-number.933101/page-2 The analysis involving non-orthogonal coordinates starts at post #31.

Here is a simplified version of the tire example I alluded to in a previous post, taken from the open literature:
Miller, C., Popper, P., Gilmour, P.W., and Schaffers, W.J., Textile Mechanics Model of a Pneumatic Tire, Tire Science and Technology, 13, 4, 187-226 (1985). See the Appendix for the development of the equations.