Non-Linear Coordinate Systems

Section 5.1 Non-Linear Coordinate Systems

Subsection 5.1.1 Polar Coordinates

One way of thinking of transformations of \(\RR^n\) is as changes of coordinates. In \(\RR^2\text{,}\) with standard coordinates \(x,y\text{,}\) we can think of a linear transformation, say \(u = x+v\) and \(v=x-v\text{,}\) as a new system of coordinates. We can describe points, loci, and any other objects in terms of \(u\) and \(v\) just as well as in terms of \(x\) and \(y\text{.}\) The transformation tells us how to go between the two sets of coordinates.

All the linear transformations mentioned in the previous section (at least those which preserve dimension) are changes of coordinates. In this section, we look at changing to new coordinates in stranger ways; in particular, we look as non-linear transformations.

In \(\RR^2\text{,}\) the most common non-linear coordinate system is polar coordinates. Polar coordinates describe \(\RR^2\) in terms of circles and rays instead of the conventional lines of Cartesian coordinates. The system has two parameters (coordinates): \(r\) and \(\theta\text{.}\) \(r\text{,}\) the radius, is the distance of a point from the origin. We have \(r \in [0, \infty)\text{,}\) since that distance can't be negative. If we draw a ray from the origin to a point, \(\theta\) is the angle between that line and the \(x\) axis, with the convention that \(\theta \in [0, 2\pi)\text{.}\)

Figure 5.1.1. Polar Coordinates

We would like to be able to move between Cartesian and polar coordinates. To that end, we need to describe \(x\) and \(y\) in terms of \(r\) and \(\theta\text{,}\) and vice-versa. The relationships are just trigonometry.

\begin{gather*} x = r \cos \theta \\ y = r \sin \theta \end{gather*}

The reverse direction is also trigonometry.

\begin{gather*} r = \sqrt{x^2 + y^2} \\ \tan \theta = \frac{y}{x} \implies \theta = \arctan \frac{y}{x} \end{gather*}

If \(x\) is zero, then \(\tan \theta\) is undefined, but \(\theta\) will be \(\pi/2\) or \(3\pi/2\text{,}\) depending on whether \(y\) is positive or negative. If \(x\) and \(y\) are both zero, at the origin, the angle is not defined at all.

Loci in \(\RR^2\) are equations in \(x\) and \(y\text{.}\) The simplest such loci were \(x=c\text{,}\) which is a vertical line, and \(y=c\text{,}\) which is a horizontal line. We get loci in \(\RR^2\) in terms of \(r\) and \(\theta\) as well. \(r=c\) is a circle: the shape with any angle and a fixed radius. \(\theta =c\) is a ray: the shape with a fixed angle and any radius. Polar loci are show in Figure 5.1.2, Figure 5.1.3, Figure 5.1.4 and Figure 5.1.5.

To translate between loci in polar coordinates and loci in Cartesian coordinates is simply a matter of replacement. The line \(x=4\) in Cartesian coordinates becomes \(r\cos \theta = 4\) or \(\cos \theta = \frac{r}{4}\text{.}\) The circle \(x^2 + y^2 =1\) becomes \(r^2 \cos^2 \theta + r^2 \sin^2 \theta = 1\text{,}\) which is simply \(r^2 = 1 \implies r=1\text{.}\) The polar locus \(r=\theta\) becomes \(\sqrt{x^2 + y^2} = \arctan \frac{y}{x}\text{.}\)

Figure 5.1.2. \(r = \theta\)

Figure 5.1.3. \(r = |\sin \theta|\)

Figure 5.1.4. \(r = 1 + \sin \theta\)

Figure 5.1.5. \(r = 3 \sin 2 \theta\)

Subsection 5.1.2 Spherical and Cylindrical Coordinates

In \(\RR^3\text{,}\) there two are similar coordinate systems. Cylindrical coordinates use polar coordinates in the \(xy\) plane and leave the \(z\) coordinate unchanaged. The transformations are again given by trigonometry.

\begin{align*} x \amp = r \cos \theta\\ y \amp = r \sin \theta\\ z \amp = z \end{align*}

We can invert the transformation.

\begin{align*} r \amp = \sqrt{x^2 + y^2}\\ \theta \amp = \arctan \left( \frac{y}{x} \right)\\ z \amp = z \end{align*}

These are called cylindrical coordinates since the equation \(r=c\) gives rise to a infinitely tall cylinder. \(r=c\) in the \(xy\) plane is a circle, as before. The \(z\) coordinate is left free, so the circle can be located at any \(z\) value. That infinitely tall stack of circles is a cylinder.

There are a few other simple loci in cylindrical coordinates. \(\theta = c\) is a vertical half plane of all point with a fixed angle but arbitrary distance out from the origin and arbitrary height. \(z = c\) is a horizontal plane, as it would be in cartesian coordinates (since the \(z\) coordinate is unchanges). The locus \(r = |z|\) is a double cone; cylindrical coordinates is often the preferred coordinate system for cones as well.

Subsection 5.1.3 Spherical Coordinates

Spherical coordinates uses a sphere the same way that polar coordinates uses a circle. There is a radius term \(r\text{,}\) which measure the distance from a point to the origin in \(\RR^3\text{.}\) \(r\) determines the size of a sphere on which the point is located. After determining a sphere, to find a specific point on the sphere, we use a system which is similar to the system of longitude and latitude on the surface of the earth. \(\theta\) is the same as longitude, but with \(\theta \in [0, 2\pi)\) instead of counting positively in both east and west directions. The \(0\) line of longitude is the line that passes through the positive \(x\) axis. \(\phi\) is co-latitude instead of latitude: it starts at \(0\) at the top of the sphere and counts down to \(\pi\) radians at the bottom of the sphere. The transformations involve some tricky trigonometry in \(\RR^3\text{.}\)

\begin{align*} x \amp = r \sin \phi \cos \theta\\ y \amp = r \sin \phi \sin \theta\\ z \amp = r \cos \phi \end{align*}

We can invert the transformation.

\begin{align*} r \amp = \sqrt{x^2 + y^2 + z^2}\\ \theta \amp = \arctan \left( \frac{y}{x} \right)\\ \phi \amp = \arctan \left( \frac{\sqrt{x^2 + y^2}}{z} \right) \end{align*}

The equation \(r=c\) in spherical coordinates gives a sphere. \(\theta = c\) is a half plane as it was with cylindrical coordinates. \(\phi = c\) is a single cone, upward or downward depending on whether \(c\) is a colatitude below or above the equator.