Introduction to Cylindrical Coordinates

What is Cylindrical Coordinate System

The cylindrical coordinate system is an extension of the polar coordinate system used for describing the location of points in 3d space. It can be constructed by adding a perpendicular axis (z) to the polar coordinates (r, θ) that goes through the pole (origin) point.

The position of any point in the cylindrical coordinate system can be referenced by the ordered triplet (r, θ, z), where:

• (r, θ) are the polar coordinates of the point's projection in the xy-plane.
• z is the usual z-coordinate in the cartesian coordinate system.

As the name suggests, cylindrical coordinates are useful when dealing with problems involving cylinders, such as calculating the volume of a round water tank or the amount of oil flowing through a pipe.

Plotting Points Using Cylindrical Coordinates Example 1.1

Plot the point with cylindrical coordinates (4, 60°, 7).

Solution

Step 1) We identify the r, θ, and z coordinates as:

r-coordinate = 4 θ-coordinate = 60° z-coordinate = 7

Step 2) We rotate 60° from the polar axis (x) in a counterclockwise direction, as shown in the diagram below.

By convention, the counterclockwise direction is considered positive, and the clockwise direction is negative.

Step 3) We draw a line of 4 units-length from the pole in that direction. The point where the line ends is the point with polar coordinates (4, 60°).

Step 4) Now, we draw a perpendicular line of 7 units-length from that point parallel to the z-axis. The point where the line ends is the point with cylindrical coordinates (4, 60°, 7). We plot the point there, as shown in the diagram below.

Conversion Between Cylindrical and Cartesian Coordinates

1) How to convert from cylindrical to cartesian coordinates

We can convert from cylindrical to cartesian coordinates using the trigonometric functions sine and cosine as follows:

x = r cos(θ)

y = r sin(θ)

z = z

Converting Cylindrical to Cartesian Coordinates Example 2.1

Convert the cylindrical coordinates (3, 50°, 5) to cartesian coordinates.

Solution

Step 1) We identify the r, θ, and z coordinates as:

r-coordinate = 3 θ-coordinate = 50° z-coordinate = 5

Step 2) We use the previous equations to find the x, y, and z coordinates:

x = r cos(θ) = 3 cos(50°) ≈ 1.9 y = r sin(θ) = 3 sin(50°) ≈ 2.3 z = 5

The point with cylindrical coordinates (3, 50°, 5) has cartesian coordinates (1.9, 2.3, 5), as shown in the diagram below.

2) How to convert from cartesian to cylindrical coordinates

As the cylindrical coordinates are an extension of the polar coordinates, it should be noted that the coordinates (r, θ) can be expressed with an infinite number of different coordinates. For example, adding a full turn of 360° to the angular coordinate θ does not change the corresponding direction. However, if we restrict θ to values between 0 and 360° and r ≥ 0, then we can find a unique solution based on the quadrant of the xy-plane in which the original point (x,y,z) is located.

The cartesian coordinates x, y, and z can be converted to cylindrical coordinates r, θ, and z with r ≥ 0 and θ in the interval (0, 2π) by:

π is equal to 180°.

Converting Cartesian to Cylindrical Coordinates Example 2.2

Convert the cartesian coordinates (2, -3, 4) to cylindrical coordinates.

Solution

Step 1) We identify the x, y, and z coordinates as:

x-coordinate = 2 y-coordinate = -3 z-coordinate = 4

Step 2) Since the x-coordinate is positive and the y-coordinate is negative, the angular coordinate θ can be found as:

θ = arctan(y/x) + 360° = arctan(-3/2) + 360° = 303.7°

Step 3) We use the Pythagorean theorem to find the radius (r) coordinate as:

Step 4) The z-coordinate is the same in both cartesian and cylindrical coordinates. Therefore, the point with cartesian coordinates (2, -3, 4) has cylindrical coordinates (3.6, 303.7°, 4), as shown in the diagram below.

Equations in Cylindrical Coordinates

The use of cylindrical coordinates is common in fields such as physics. Physicists studying electrical charges and the capacitors used to store these charges have discovered that these systems sometimes have cylindrical symmetry. These systems have complicated modeling equations in the Cartesian coordinate system, which make them difficult to describe and analyze. The equations can often be expressed in more simple terms using cylindrical coordinates. For example, the cylinder described by equation x² + y² = 25 in the Cartesian system can be represented by the cylindrical equation r=5.

Describing Surfaces Using Cylindrical Coordinates Example 3.1

Try to describe the surfaces with the following cylindrical equations:

a) θ = 45° b) r² + z² = 9 c) z = r

Solution

a) When the angle θ is held constant while r and z are allowed to vary, the result is a half-plane (see the diagram below).

b) The equation (r² + z² = 9) represents a sphere centered at the origin with radius 3 (see the diagram below).

c) To describe the surface defined by equation z=r, it is useful to examine traces parallel to the (r, θ) plane. For example, the trace in plane z=1 is circle r=1, the trace in plane z=3 is circle r=3, and so on. Each trace is a circle. As the value of z increases, the radius of the circle also increases. The resulting surface is a cone (see the diagram below).

Frequently Asked Questions

What are cylindrical coordinates?

Cylindrical coordinates are a natural extension of the two-dimensional polar coordinates that allow us to describe the position of points in 3d space.

What is theta in cylindrical coordinates?

Theta (θ) is the angular coordinates measured from the reference direction (polar axis) that describes the direction of a point from the pole.

Who invented the cylindrical coordinate system?

The cylindrical coordinate system was developed by Sir Isaac Newton (1640-1727).