Coordinates in 3-Space

This post was inspired by a calculus student and will be in three parts: This on the development of different coordinate systems, one with calculus and one without. We are taught in school that the volume of a sphere with radius r is V = \frac{4}{3}\pi r^3. In this post we shall look at a development in calculus that will not only give us the volume of a sphere, but will also give us the conversion formula between rectangular, cylindrical, and spherical coordinates.Recall in the plane we have rectangular and polar coordinates. For rectangular coordinates we express a point by (horizontal distance, vertical distance) or (x,y). For polar coordinates we express a point by (distance from the center, counterclockwise angle of rotation from the polar axis) or (r,\theta).Polar1We see in this picture a depiction of polar coordinates. We can also, from this picture and the next, quickly obtain the conversion formulas for polar coordinates.


So, by using right triangle trig we can take a point (r,\theta) and convert it to rectangular coordinates (r cos \theta, r sin \theta). We can also convert back, but there are certain intricacies we would need to observe. This is not the goal today however as we have our sights set on 3-space. We often think of the polar plane as infinitely many cocentric circles (circles that have the same center). We also often think of 3-space (\mathbb{R}^3) as infinitely many planes stacked on top of each other. But notice that we don’t care what kind of planes since anytime a point has unique coordinates in two planes we can always convert between them (this may seem familiar from linear algebra as a change of basis). So, if we have planes stacked on top of each other in the obvious way, then all we need to do to obtain cylindrical coordinates is add how far up or down we need to go from the plane at level 0. That is, in cylindrical coordinates a point has a unique representation as (polar coordinates, height) or (r, \theta, z). Quickly then we can convert this to rectangular coordinates by x = r cos\theta, y = r sin \theta, and z = z.

Notice that we chose to tack on height to obtain cylindrical coordinates. We could instead rotate again by some angle \phi. And as long as we keep 0 \leq \phi \leq \pi we see that each point in 3-space has a unique representation as (\rho, \theta, \phi) where \rho is the distance from the origin, \theta is the “azimuthal angle”, and \phi is the polar angle as indicated in the following picture.


Now we will find some conversions. Suppose we have a point P in spherical coordinates. This means that the point has a representation as  (\rho, \theta, \phi). Let us convert this to cylindrical coordinates first. Note that from the picture and a little geometry, in particular parallel lines cut by the transversal (brown line) and alternate interior angles tells us that the angle between the blue and brown line is indeed \phi. So with this and the knowledge that the light blue line represents r we see that r = \rho sin \phi. Using some more right triangle trig we see that z = cos \phi. And of course, \theta = \theta. Next, if we were going to further convert this to rectangular coordinates we simply apply the cylindrical conversions that we have already made. And so we end with this,

x = r cos \theta = \rho sin\phi cos\theta;

y = r sin \theta = \rho sin\phi sin \theta; and

z = r cos \phi.

I encourage the interested reader to attempt to derive other conversion formulas (Hint: Use the pictures and reverse solve the equations already set forth!)


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