Alternative Coordinate Systems and Surfaces: Learn It 1

  • Switch between cylindrical and rectangular coordinates
  • Switch between spherical and rectangular coordinates
  • Recognize and describe different types of quadric surfaces
  • Analyze 3D surfaces by looking at their cross-sections and traces
The Cartesian coordinate system works well for many problems, but some surfaces are difficult to model with rectangular equations. Just as polar coordinates provide a useful alternative in 2D (especially for circles), we can extend coordinate systems to 3D space. Cylindrical coordinates are perfect for problems involving cylinders—like calculating the volume of a water tank or analyzing flow through a pipe. Spherical coordinates excel with sphere-related problems, such as finding volumes of domed structures.

Cylindrical Coordinates

Cylindrical coordinates provide a natural extension of polar coordinates to three dimensions. Instead of just adding a third axis like we did going from 2D to 3D Cartesian coordinates, we build on the polar coordinate foundation.

cylindrical coordinate system

In the cylindrical coordinate system, a point in space is represented by the ordered triple [latex](r,\theta,z)[/latex], where

  • [latex](r,\theta)[/latex] are the polar coordinates of the point’s projection in the [latex]xy[/latex]-plane
  • [latex]z[/latex] is the usual [latex]z[/latex]-coordinate in the Cartesian coordinate system
This figure is the first octant of the 3-dimensional coordinate system. There is a point labeled “(x, y, z) = (r, theta, z).” In the x y-plane, there is a line segment extending to underneath the point. This line segment is labeled “r.” The angle between the line segment and the x-axis is theta. There is a line segment perpendicular to the x-axis. Along with the line segment labeled r, this line segment and the x-axis form a right triangle.
Figure 1. The right triangle lies in the [latex]xy[/latex]-plane. The length of the hypotenuse is [latex]r[/latex] and [latex]\theta[/latex] is the measure of the angle formed by the positive [latex]x[/latex]-axis and the hypotenuse. The [latex]z[/latex]-coordinate describes the location of the point above or below the [latex]xy[/latex]-plane.
Think of it this way: to locate a point in cylindrical coordinates, you first find its “shadow” on the [latex]xy[/latex]-plane using polar coordinates [latex](r,\theta)[/latex], then specify how far above or below that plane the point sits using [latex]z[/latex].

Converting Between Coordinate Systems

The conversion formulas come directly from the right triangle relationships in the [latex]xy[/latex]-plane.

This figure is the first quadrant of the rectangular coordinate system. There is a point labeled “P = (x, y, 0) = (r, theta, 0).” There is a line segment from the origin to point P. This line segment is labeled “r.” The angle between the x-axis and the line segment r is labeled “theta.” There is also a vertical line segment labeled “y” from P to the x-axis. It forms a right triangle.
Figure 2. The Pythagorean theorem provides equation [latex]r^{2}=x^{2}+y^{2}[/latex]. Right-triangle relationships tell us that [latex]x=r\cos{\theta}[/latex], [latex]y=r\sin{\theta}[/latex], and [latex]\tan{\theta}=\frac{y}{x}[/latex].

conversion between cylindrical and cartesian coordinates

The rectangular coordinates [latex](x, y, z)[/latex] and the cylindrical coordinates [latex](r,\theta,z)[/latex] of a point are related as follows:

 

Cylindrical to Rectangular:

  • [latex]x = r\cos\theta[/latex]
  • [latex]y = r\sin\theta[/latex]
  • [latex]z = z[/latex]

Rectangular to Cylindrical:

  • [latex]r^2 = x^2 + y^2[/latex] (so [latex]r = \sqrt{x^2 + y^2}[/latex] for [latex]r \geq 0[/latex])
  • [latex]\tan\theta = \frac{y}{x}[/latex] (choose [latex]\theta[/latex] based on the quadrant)
  • [latex]z = z[/latex]
Finding [latex]\theta[/latex]: Just like in 2D polar coordinates, [latex]\tan\theta = \frac{y}{x}[/latex] has infinitely many solutions. Choose [latex]\theta[/latex] in [latex][0, 2\pi)[/latex] based on the signs of [latex]x[/latex] and [latex]y[/latex] to get the correct quadrant.
Plot the point with cylindrical coordinates [latex]\left(4,\frac{2\pi}3,-2\right)[/latex] and express its location in rectangular coordinates.

Convert the rectangular coordinates [latex](1, -3, 5)[/latex] to cylindrical coordinates.

Surfaces in Cylindrical Coordinates

Understanding what surfaces look like when one coordinate is held constant helps you visualize the cylindrical coordinate system.

Rectangular Coordinates (for comparison):

  • [latex]x = c[/latex]: plane parallel to [latex]yz[/latex]-plane
  • [latex]y = c[/latex]: plane parallel to [latex]xz[/latex]-plane
  • [latex]z = c[/latex]: plane parallel to [latex]xy[/latex]-plane

Cylindrical Coordinates:

  • [latex]r = c[/latex]: vertical circular cylinder of radius [latex]c[/latex] (points at fixed distance from [latex]z[/latex]-axis)
  • [latex]\theta = c[/latex]: half-plane starting at [latex]z[/latex]-axis, making angle [latex]c[/latex] with positive [latex]x[/latex]-axis
  • [latex]z = c[/latex]: plane parallel to [latex]xy[/latex]-plane (same as rectangular)
This figure has 3 images. The first image is a plane in the 3-dimensional coordinate system. It is parallel to the y z-plane where x = c. The second image is a plane in the 3-dimensional coordinate system. It is parallel to the x z-plane where y = c. the third image is a plane in the 3-dimensional coordinate system where z = c.
Figure 3. In rectangular coordinates, (a) surfaces of the form [latex]x=c[/latex] are planes parallel to the [latex]yz[/latex]-plane, (b) surfaces of the form [latex]y=c[/latex] are planes parallel to the [latex]xz[/latex]-plane, and (c) surfaces of the form [latex]z=c[/latex] are planes parallel to the [latex]xy[/latex]-plane.
This figure has 3 images. The first image is a right circular cylinder in the 3-dimensional coordinate system. It has the z-axis in the middle. The second image is a plane in the 3-dimensional coordinate system. It is vertical with the z-axis on one edge. The third image is a plane in the 3-dimensional coordinate system where z = c.
Figure 4. In cylindrical coordinates, (a) surfaces of the form [latex]r=c[/latex] are vertical cylinders of radius [latex]c[/latex], (b) surfaces of the form [latex]\theta=c[/latex] are half-planes at angle [latex]c[/latex] from the [latex]x[/latex]-axis, and (c) surfaces of the form [latex]z=c[/latex] are planes parallel to the [latex]xy[/latex]-plane.

Describe the surfaces with the given cylindrical equations.

  1. [latex]\theta=\frac{\pi}4[/latex]
  2. [latex]r^{2}+z^{2}=9[/latex]
  3. [latex]z=r[/latex]

Cylindrical coordinates shine when dealing with problems that have cylindrical symmetry. In physics, many systems involving electrical charges, capacitors, and electromagnetic fields have this symmetry.

For example, the cylinder [latex]x^2 + y^2 = 25[/latex] in rectangular coordinates becomes simply [latex]r = 5[/latex] in cylindrical coordinates—much simpler to work with!

This coordinate transformation follows the same pattern as converting from polar to rectangular coordinates in 2D. We’re just adding the unchanged [latex]z[/latex]-coordinate to extend into three dimensions.