#4: CYLINDRICAL AND SPHERICAL COORDINATES

In 2-space, there are two co-ordinate systems you studied, the rectangular (x,y), and polar (r,0). In the previous worksheets, you studied surfaces represented in the rectangular coordinate system for 3-space. There are two other system that use the polar co-ordinates for some of its variables. To understand these systems, you must understand the polar coordinate system well. Review it if you need to.

CYLINDRICAL COORDINATES

If you replace the x-y plane in the rectangular coordinates (x,y,z) with the standard polar coordinate system, you have the cylindrical coordinate system. Its variables are (r, 0,z). The third coordinate of any point is the same in both systems. The conversion equations are the same as in polar coordinates:

POLAR - RECTANGULAR
r 2 = x 2 + y 2 tan0 = y/x	x = rcos0 y = rsin0

The coordinate surfaces for the rectangular coordinate system are the planes perpendicular to the coordinate axes, x = a, y = b, and z = c. The coordinate surfaces for the cylindrical coordinate system is: r = a (a cylinder), 0 = b (a plane perpendicular to the x-y plane and through the z-axis) and z = c (a plane perpendicular to the z-axis). Notice that the system is named after the r = a surface.

r = a = constant	0 = b = constant

1. Classify each of the quadratic surfaces and change the equation from rectangular coordinates to cylindrical coordinates:

1. 2x ² + 3y ² - 4z ² + x = 5
2. TRY THIS #1: 4x ² + 4y ² - 4z ² + z = 25
3. x + 2y + 3z = 8
4. TRY THIS #2: x = 2y
5. x ² + y ² - z ² = 0
6. TRY THIS #3: x ² + y ² + z ² = 16

SPHERICAL COORDINATES

The spherical coordinate system has two polar coordinates. The first is in the x-y plane just as in cylindrical coordinates. However, instead of the r coordinate, another polar coordinate system is drawn along the r-ray with the z axis as the polar axis.

The coordinate surfaces are rho = a (a sphere), 0 = b (a plane perpendicular to the x-y plane through the z-axis), and phi = c (a cone whose apex is at the origin and whose axis is along the z-axis.) The system is named after the rho = a surface.

rho = a	phi = c

The conversions are still the standard rectangular-polar conversion from cylindrical to spherical. You just have to take one more rectangular-polar conversion to get from rectangular to spherical. For the purposes of the web pages, the symbol Þ will be used for rho and the symbol ø will be used for phi.

Cylindrical - Spherical	Cylindrical - Rectangular	Spherical - Rectangular
z = Þ cosø r = Þ sinø z = z	x = r cos0 y = r sin0 z = z	x = Þ sinø cos0 y = Þ sinø sin0 z = Þ cosø
z 2 + r 2 = Þ 2 tanø = r/z	x 2 + y 2 = r 2 tan0 = y/x	x 2 + y 2 + z 2 = Þ 2 tan0 = y/x

Just as with polar coordinates, the coordinates of a point in cylindrical and spherical coordinates are not unique. You can have both positive and negative values of the radius coordinates, r and Þ. The two angel coordinates, 0 and ø, can differ by multiples of 2 pi or 360 degrees.

2. Change the equation from rectangular coordinates to spherical coordinates:

1. 2x ² + 3y ² - 4z ² + x = 5
2. TRY THIS #4: 4x ² + 4y ² - 4z ² + z = 25
3. x + 2y + 3z = 8
4. TRY THIS #5: x = 2y
5. x ² + y ² - z ² = 0
6. TRY THIS #6: x ² + y ² + z ² = 16