Cylindrical Coordinates

Cylindrical coordinates are ordered triples in the cylindrical coordinate system that are used to describe the location of a point. Cylindrical coordinates are a natural extension of polar coordinates in 3D space. These coordinates combine the z coordinate of cartesian coordinates with the polar coordinates in the xy plane.

The radial distance, azimuthal angle, and the height from a plane to a point are denoted using cylindrical coordinates. The cylindrical coordinate system is useful in analyzing systems that exhibit rotational symmetry. In this article, we will learn more about cylindrical coordinates, various coordinate conversions, and certain associated examples.

Cylindrical coordinates can be defined as a set of three coordinates that are used to locate a point in the cylindrical coordinate system. In two dimensions, the location of a point can be denoted by both cartesian and polar coordinates. When the polar coordinates are extended to a three-dimensional plane, an additional z coordinate is added. These three measures together form cylindrical coordinates. The coordinates describe two distances and one angle.

r is used to denote the normal polar coordinate. It is the distance from the origin to Q. Here, Q is the projection of point P in the xy plane.

θ is the angle between the x-axis and the line segment that is drawn from the origin to Q. This is known as the azimuthal angle and is measured in radians.

z represents the third cylindrical coordinate. It is the same as the z cartesian coordinate and denotes the signed distance of P to the xy plane.

Thus, the cylindrical coordinates of P are (r, θ, z).

A three-dimensional coordinate system that is used to specify a point's location by using the radial distance, the azimuthal, and the height of the point from a particular plane is known as a cylindrical coordinate system. This coordinate system is useful in dealing with systems that take the shape of a cylinder. For example, the flow of water through a pipe. In the cylindrical coordinate system, if each coordinate is held constant then the surfaces traced out are as follows:

• If r is constant then a cylindrical surface will be obtained.
• If θ is held constant then a vertical plane can be traced out.
• If z is constant then a flat circular plane is obtained.

Many three-dimensional coordinate systems exist such as the spherical and the cylindrical coordinate system. With the help of trigonometry, coordinates can be converted from one system to another. Some important conversions are listed below.

Cylindrical Coordinates to Cartesian Coordinates

Cartesian coordinates can also be referred to as rectangular coordinates. To convert cylindrical coordinates (r, θ, z) to cartesian coordinates (x, y, z), the steps are as follows:

When polar coordinates are converted to cartesian coordinates the formulas are,

x = rcosθ

y = rsinθ

Cylindrical coordinates are polar coordinates extended into three-dimensional space by adding the z cartesian coordinate. Thus, cylindrical coordinates can be expressed as cartesian coordinates using the equations given below:

x = rcosθ

y = rsinθ

z = z

Cartesian Coordinates to Cylindrical Coordinates

The equations that can be used to convert cartesian coordinates to cylindrical coordinates are as follows:

r² = x² + y²

tanθ = y / x

z = z

Spherical Coordinates to Cylindrical Coordinates

The conversions from cartesian to cylindrical coordinates are used to derive a relationship between spherical coordinates (ρ,θ,φ) and cylindrical coordinates (r, θ, z).

By using the figure given above and applying trigonometry, the following equations can be derived.

r = ρsinφ

z = ρcosφ

θ = θ (θ will be the same in both coordinate systems).

Cylindrical Coordinates to Spherical Coordinates

To convert cylindrical coordinates to spherical coordinates the following equations are used.

\(\rho =\sqrt{r^{2}+z^{2}}\)

θ = θ

φ = \(cos^{-1}\left ( \frac{z}{\sqrt{r^{2}+z^{2}}} \right )\)

Cylindrical coordinate systems are best used when dealing with cylindrical systems such as the volume of a water tank. Other applications of cylindrical coordinates are given below:

• In computer animation, cylindrical coordinates are used to locate positions on a curved cylindrical surface.
• When an object is moving on a curved cylindrical trajectory then physicists and engineers use cylindrical coordinates to analyze the motion.
• Cylindrical coordinates are used to design applications that make use of alternating currents.

Related Articles:

• Coordinate Geometry
• Coordinate Plane
• 3D Geometry

Important Notes on Cylindrical Coordinates

• Cylindrical coordinates are ordered triples that used the radial distance, azimuthal angle, and height with respect to a plane to locate a point in the cylindrical coordinate system.
• Cylindrical coordinates are represented as (r, θ, z).
• Cylindrical coordinates can be converted to cartesian coordinates as well as spherical coordinates and vice versa.

What are Cylindrical Coordinates?

Cylindrical coordinates are part of the three-dimensional cylindrical coordinate system and are used to locate a point in this system. They are represented as (r, θ, z).

How Do You Find Cylindrical Coordinates?

To find the cylindrical coordinates of a point P project it down to a point Q on the xy plane. The polar coordinates of Q will be the first and second cylindrical coordinates of P. The third coordinate will be the distance of P from the xy plane. Thus, the cylindrical coordinates of P are (r, θ, z).

What are the Three Coordinates of a Cylindrical Coordinate System?

The three cylindrical coordinates are given as follows:

• r represents the radial distance from the origin to the projection of the point on the xy plane.
• θ is the azimuthal angle between the x axis and the line from the origin to the projection point.
• z is the signed distance from the plane to the point.

How Do You Convert Cylindrical Coordinates?

Using the equations x = rcosθ, y = rsinθ and z = z, cylindrical coordinates can be converted to rectangular coordinates. Furthermore, cylindrical coordinates can be converted to spherical coordinates using the equations, \(\rho =\sqrt{r^{2}+z^{2}}\), θ = θ and φ = \(cos^{-1}\left ( \frac{z}{\sqrt{r^{2}+z^{2}}} \right )\).

Why Do We Use Cylindrical Coordinates?

Cylindrical coordinates are used to draw inferences about systems that display some form of rotational symmetry. They are also used to analyze oscillatory motions.

What are Spherical and Cylindrical Coordinates?

Spherical coordinates are used in the spherical coordinate system. These coordinates are represented as (ρ,θ,φ). Cylindrical coordinates are a part of the cylindrical coordinate system and are given as (r, θ, z). Cylindrical coordinates can be converted to spherical and vise versa.

What are Polar Cylindrical Coordinates?

Cylindrical coordinates are an extension of two-dimensional polar coordinates (r, θ) into three-dimensional space. This is done by adding the z cartesian coordinate to get (r, θ, z).