Cylindrical coordinate system

# Cylindrical coordinate system

Overview

A cylindrical coordinate system is a three-dimensional coordinate system
Coordinate system
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...

that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis. The latter distance is given as a positive or negative number depending on which side of the reference plane faces the point.

The origin of the system is the point where all three coordinates can be given as zero.
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Encyclopedia

A cylindrical coordinate system is a three-dimensional coordinate system
Coordinate system
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...

that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis. The latter distance is given as a positive or negative number depending on which side of the reference plane faces the point.

The origin of the system is the point where all three coordinates can be given as zero. This is the intersection between the reference plane and the axis.

The axis is variously called the cylindrical or longitudinal axis, to differentiate it from
the polar axis, which is the ray that lies in the reference plane,
starting at the origin and pointing in the reference direction.

The distance from the axis may be called the radial distance or radius,
while the angular coordinate is sometimes referred to as the angular position or as the azimuth.
The radius and the azimuth are together called the polar coordinates, as they correspond to a two-dimensional polar coordinate system in the plane through the point, parallel to the reference plane.
The third coordinate may be called the height or altitude (if the reference plane is considered horizontal),
longitudinal position,
or axial position.

Cylindrical coordinates are useful in connection with objects and phenomena that have some rotational symmetry
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

about the longitudinal axis, such as water flow in a straight pipe with round cross-section, heat distribution in a metal cylinder
Cylinder (geometry)
A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...

, and so on.

It is sometimes called "cylindrical polar coordinate" and "polar cylindrical coordinate", and is sometime used to specify the position of stars in a galaxy ("galactocentric cylindrical polar coordinate").

## Definition

The three coordinates (ρ
Rho (letter)
Rho is the 17th letter of the Greek alphabet. In the system of Greek numerals, it has a value of 100. It is derived from Semitic resh "head"...

, φ
Phi
Phi may refer to:In language:*Phi, the Greek letter Φ,φ, the symbol for voiceless bilabial fricativeIn mathematics:*The Golden ratio*Euler's totient function*A statistical measure of association reported with the chi-squared test...

, z) of a point P are defined as:
• The radial distance ρ is the Euclidean distance
Euclidean distance
In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space becomes a metric space...

from the z axis to the point P.
• The azimuth φ is the angle between the reference direction on the chosen plane and the line from the origin to the projection of P on the plane.
• The height z is the signed distance from the chosen plane to the point P.

### Unique cylindrical coordinates

As in polar coordinates, the same point with cylindrical coordinates (ρ, φ, z) has infinitely many equivalent coordinates, namely and where n is any integer. Moreover, if the radius ρ is zero, the azimuth is arbitrary.

In situations where one needs a unique set of coordinates for each point, one may restrict the radius to be non-negative (ρ ≥ 0) and the azimuth φ to lie in a specific interval
Interval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...

spanning 360°, such as (−180°,+180°] or [0,360°).

### Conventions

The notation for cylindrical coordinates is not uniform. The ISO
International Organization for Standardization
The International Organization for Standardization , widely known as ISO, is an international standard-setting body composed of representatives from various national standards organizations. Founded on February 23, 1947, the organization promulgates worldwide proprietary, industrial and commercial...

standard 31-11
ISO 31-11
ISO 31-11 was the part of international standard ISO 31 that defines mathematical signs and symbols for use in physical sciences and technology...

recommends (ρ, φ, z), where ρ is the radial coordinate, φ the azimuth, and z the height. However, the radius is also often denoted r, the azimuth by θ or t, and the third coordinate by h or (if the cylindrical axis is considered horizontal) x, or any context-specific letter.

In concrete situations, and in many mathematical illustrations, a positive angular coordinate is measured counterclockwise
Clockwise
Circular motion can occur in two possible directions. A clockwise motion is one that proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back to the top...

as seen from any point with positive height.

## Coordinate system conversions

The cylindrical coordinate system is one of many three-dimensional coordinate systems. The following formulae may be used to convert between them.

### Cartesian coordinates

For the conversion between cylindrical and Cartesian coordinate systems, it is convenient to assume that the reference plane of the former is the Cartesian xy plane (with equation z = 0) , and the cylindrical axis is the Cartesian z axis. Then the z coordinate is the same in both systems, and the correspondence between cylindrical (ρ,φ) and Cartesian (x,y) are the same as for polar coordinates, namely
in one direction, and
in the other. The arcsin function is the inverse of the sine
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

function, and is assumed to return an angle in the range [−π/2,+π/2] = [−90°,+90°]. These formulas yield an azimuth φ in the range [−90°,+270°]. For other formulas, see the polar coordinate article
Polar coordinate system
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction....

.

Many modern programming languages provide a function that will compute the correct azimuth φ, in the range (−π, π], given x and y, without the need to perform a case analysis as above. For example, this function is called by atan2
Atan2
In trigonometry, the two-argument function atan2 is a variation of the arctangent function. For any real arguments and not both equal to zero, is the angle in radians between the positive -axis of a plane and the point given by the coordinates on it...

(y,x) in the C
C (programming language)
C is a general-purpose computer programming language developed between 1969 and 1973 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system....

programming language, and atan
Atan2
In trigonometry, the two-argument function atan2 is a variation of the arctangent function. For any real arguments and not both equal to zero, is the angle in radians between the positive -axis of a plane and the point given by the coordinates on it...

(y,x) in Common Lisp
Common Lisp
Common Lisp, commonly abbreviated CL, is a dialect of the Lisp programming language, published in ANSI standard document ANSI INCITS 226-1994 , . From the ANSI Common Lisp standard the Common Lisp HyperSpec has been derived for use with web browsers...

.

### Spherical coordinates

Spherical coordinates (radius r, elevation or inclination θ, azimuth φ), may be converted into cylindrical coordinates by:

 θ is elevation: θ is inclination:

Cylindrical coordinates may be converted into spherical coordinates by:

 θ is elevation: θ is inclination:

## Line and volume elements

See multiple integral for details of volume integration in cylindrical coordinates, and Del in cylindrical and spherical coordinates
Del in cylindrical and spherical coordinates
This is a list of some vector calculus formulae of general use in working with various curvilinear coordinate systems.- Note :* This page uses standard physics notation. For spherical coordinates, \theta is the angle between the z axis and the radius vector connecting the origin to the point in...

for vector calculus formulae.

In many problems involving cylindrical polar coordinates, it is useful to know the line and volume elements; these are used in integration to solve problems involving paths and volumes.

The line element
Line element
A line element ds in mathematics can most generally be thought of as the change in a position vector in an affine space expressing the change of the arc length. An easy way of visualizing this relationship is by parametrizing the given curve by Frenet–Serret formulas...

is

The volume element
Volume element
In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates...

is

The surface element
Surface element
Surface element may refer to*Volume form*Surfel in 3D computer graphics*Differential , an infinitesimal portion of a surface...

in a surface of constant radius (a vertical cylinder) is

The surface element in a surface of constant azimuth (a vertical half-plane) is

The surface element in a surface of constant height (a horizontal plane) is

The del
Del
In vector calculus, del is a vector differential operator, usually represented by the nabla symbol \nabla . When applied to a function defined on a one-dimensional domain, it denotes its standard derivative as defined in calculus...

operator in this system is written as

and the Laplace operator
Laplace operator
In mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or Δ...

is defined by

## Cylindrical harmonics

The solutions to the Laplace equation in a system with cylindrical symmetry are called cylindrical harmonics
Cylindrical harmonics
In mathematics, the cylindrical harmonics are a set of linearly independent solutions to Laplace's differential equation, \nabla^2 V = 0, expressed in cylindrical coordinates, ρ , φ , and z...

.

• List of canonical coordinate transformations
• Vector fields in cylindrical and spherical coordinates
Vector fields in cylindrical and spherical coordinates
* This page uses standard physics notation. For spherical coordinates, \theta is the angle between the z axis and the radius vector connecting the origin to the point in question. \phi is the angle between the projection of the radius vector onto the x-y plane and the x axis...

• Tricomplex number
Tricomplex number
In mathematics, the tricomplex numbers are a three-dimensional hypercomplex number system developed by Silviu Olariu in his book Complex Numbers in n Dimensions .-Definition:...