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Cylindrical & Spherical Coordinates: Definition, Equations & Examples

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  • 0:04 Mapping 3D Space
  • 1:03 Polar Coordinate Systems
  • 1:48 Cylindrical Coordinates
  • 3:54 Spherical Coordinates
  • 5:59 Application: Celestial Maps
  • 6:41 Lesson Summary
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Lesson Transcript
Instructor: Aaron Miller

Aaron teaches physics and holds a doctorate in physics.

In this lesson, we introduce two coordinate systems that are useful alternatives to Cartesian coordinates in three dimensions. Both cylindrical and spherical coordinates use angles to specify the locations of points, a feature they share with 2-D polar coordinates.

Mapping 3D Space

When you look into the night sky, you can see about five thousand of the more than one hundred billion stars in our Milky Way Galaxy. Humans have charted the stars as long as they have been mapping large-scale features on Earth's surface. So why are maps of Earth's features so widely available, while maps of the locations of nearby stars in our galaxy are hard to come by?

There are likely a few reasons. One big reason is that humans use land and sea maps regularly for navigation, but star maps? Not so much. Another reason is that accurately and meaningfully representing locations of features and relative distances in a three-dimensional space is not easy. The map itself needs to be 3-D if you don't want to lose information by suppressing one of the dimensions. Maps of Earth typically suppress elevation, which is why Earth's surface can be represented on a 2-D map. Let's see how spherical coordinates provide a natural way of representing the locations of stars in our local region of the galaxy.

Polar Coordinate Systems

The idea behind cylindrical and spherical coordinates is to use angles instead of Cartesian coordinates to specify points in three dimensions. Sometimes, employing angles can make mathematical representations of functions simpler. Polar coordinates represent points in the coordinate plane, not with the usual Cartesian ordered pair (x, y), but with two different coordinates (r, phi) that are functionally related to (x, y). Specifically, for a given point P, r is the absolute distance from the origin to P. The angle phi is the angular position of P, with angle measured from the positive x-axis. Cylindrical and spherical coordinate systems are extensions of 2-D polar coordinates into a 3-D space.

Cylindrical Coordinates

Cylindrical coordinates are most similar to 2-D polar coordinates. Let's consider a point P that has coordinates (x, y, z) in a 3-D Cartesian coordinate system. The same point can be represented in cylindrical coordinates (r, phi, z) where r and phi are the 2-D polar coordinates of P's image in the xy plane (z = 0), and z is exactly the same as P's Cartesian z-coordinate. Here is the relationship between a point's Cartesian and cylindrical coordinates on a graph:


Fig. 1: Cylindrical coordinates r, phi and z


The coordinate transformations to go from Cartesian x and y coordinates to cylindrical r and phi coordinates are as follows:


Eqs. 1: Cartesian to Cylindrical Coordinate Transformation


These are the same as the transformation to 2-D polar coordinates.

There are a few features of this transformation to notice. First, the coordinate r under this transformation is always a positive number. r is interpreted as the smallest distance from P to the z axis. Also, phi, expressed in radians, will always be between -pi and pi. The z coordinate keeps the same value as you transform from one system to the other. The inverse transformation from (r, theta, z) to (x, y, z) may also be familiar from 2-D polar coordinates as well.


Eqs. 2: Cylindrical to Cartesian Coordinate Transformation


Let's consider the following question as an example of applying the coordinate transformation: what are the Cartesian coordinates (x, y, z) of the point P specified by cylindrical coordinates (2, -pi/6, 1)?

In moving from cylindrical to Cartesian coordinates, the z-coordinate does not change. z is conventionally the third value in the ordered triplet, therefore, z = 1 in both cylindrical and Cartesian coordinates. Now, to find x and y, we should plug in values r = 2 and phi = -pi/6 into the transformation equations.


cylind_example


Therefore, the Cartesian coordinates of this point are (sqrt(3), -1, 1).

Spherical Coordinates

Cylindrical coordinates are not the only way to specify a point in a 3-D space using an angle. Spherical coordinates are another generalization of 2-D polar coordinates. However, in this coordinate system, there are two angles, theta and phi.

Let's consider a point P that is specified by coordinates (x, y, z) in a 3-D Cartesian coordinate system. The same point can be represented in spherical coordinates as (r, theta, phi,) where r, theta, and phi are functionally related to x, y, and z, as we will see.

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