# 14.4.6: Coordinate Systems

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

## Cartesian Coordinate System

While the rectangular (also called Cartesian) coordinate system is the most common, certain problems are easier to analyze in alternate coordinate systems. A coordinate system is a scheme that allows us to identify any point in the plane or in three-dimensional space by a set of numbers. In rectangular coordinates these numbers are interpreted, roughly speaking, as the lengths of the sides of a rectangular "box.''

## List of some other Coordinate Systems

The choice of a coordinate system depends on the shapes that are involved in the problem at hand. There are many coordinate systems1 including the Cartesian coordinate system, the cylindrical coordinate system, the spherical coordinate system, the ellipsoidal coordinate system, the prolate spheroidal coordinate system, and the oblate spheroidal coordinate system. In addition thee are coordinate systems that rather than focusing on shape are adapted to the subject. Examples of coordinate systems of this type are astronomical coordinate systems (based off spherical coordinate system), canonical coordinate system (mechanics: phase space), and Schwarzschild coordinate systems (relativistic: akin to a modified spherical coordinate system).

## Cylindrical Coordinate System

Here we will only discuss the main coordinate system the Cartesian coordinate system (which you already know), the cylindrical coordinate system, and the spherical coordinate system. First we start with the "2D cylindrical coordinate system" we call polar coordinates.

In two dimensions you may already be familiar with polar coordinates. In this system, each point in the plane is identified by a pair of numbers $$(r,\theta)$$. The number $$\theta$$ measures the angle between the positive $$x$$-axis and a vector with tail at the origin and head at the point, as shown in Figure $$\PageIndex{1}$$; the number $$r$$ measures the distance from the origin to the point. Either of these may be negative; a negative $$\theta$$ indicates the angle is measured clockwise from the positive $$x$$-axis instead of counter-clockwise, and a negative $$r$$ indicates the point at distance $$|r|$$ in the opposite of the direction given by $$\theta$$. Figure $$\PageIndex{1}$$ also shows the point with rectangular coordinates $$(1,\sqrt3)$$ and polar coordinates $$(2,\pi/3)$$, 2 units from the origin and $$\pi/3$$ radians from the positive $$x$$-axis.

¬ So as an engineer when would you likely use the cylindrical coordinate system?
Since the cylindrical coordinate system would be best for cylindrical type objects, you would use this coordinate system for pipes (since in general pipes are cylindrical) and wires (electrical engineering). These two types of objects are ubiquitous in our society. Note you do not have to use a cylindrical coordinate system, it is just a matter of convenience. You can use any coordinate system you want. Of course it might make your problem unnecessarily difficult.

¬ So as a scientist when would you likely use a spherical coordinate system?
Since a spherical coordinate system would be best for spherical type objects it would be best to use this system for astronomy as the sky is replete with spherical objects (or close to spherical objects - you might want to use one of the slightly modified spherical coordinate systems for really detailed studies). Note engineers might use a spherical coordinate system for any ball like object, say ball bearings. Of course the use of said system is only to make problems easier. You can always use whatever coordinate system you want. Just be sure to be consistent.

How about a program to make a sphere using spherical coordinate ideas? Let us try it, eh?

[theta,phi] = meshgrid(linspace(0,pi,30),linspace(0,2*pi,30));
r = 2.*ones(1,30);
surf(r.*cos(phi).*sin(theta),r.*sin(phi).*sin(theta),r.*cos(theta));

Interestingly this looks more prolate then spherical but keep on eye on the actual scales which is 2 to -2. If you run this in your own copy of octave there will be controls for you to rotate the image.

Let us really go all out and try a prolate sphere. This is beyond the discussion but it is interesting as a program example.

[nu,phi] = meshgrid(linspace(0,pi,30),linspace(0,2*pi,30));
a = 1.*ones(1,30);
mu = 0.4*ones(1,30);
surf(a.*sinh(mu).*cos(phi).*sin(nu),a.*sinh(mu).*sin(phi).*sin(nu),a.*cosh(mu).*cos(nu));
axis([-1.2 1.2 -1.2 1.2 -1.2 1.2]);

This is probably over exaggerated but it gives you a sense of a prolate sphere. The Earth is a oblate sphere, why don't you try using the code here to help you. Use actual data to make your image realistic.

That is enough for this subject in this book/course, let us move on to the next section.

## Contributors

1Here we focus on the 3D coordinate systems with some mention of 2D coordinate systems. A full explanation will also include 1D coordinate systems, etc.

14.4.6: Coordinate Systems is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.