Dynamics/Kinematics/Coordinate Systems/Cylindrical

Introduction

Content from Cylindrical coordinate system

A cylindrical coordinate system with origin O, polar axis A, and longitudinal axis L. The dot is the point with radial distance ρ = 4, angular coordinate φ = 130°, and height z = 4.

A cylindrical coordinate system is a three-dimensional coordinate system 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. Other directions perpendicular to the longitudinal axis are called radial lines.

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.

They are sometimes called "cylindrical polar coordinates" and "polar cylindrical coordinates", and are sometimes used to specify the position of stars in a galaxy ("galactocentric cylindrical polar coordinates").

Definition

The three coordinates (ρ, φ, z) of a point P are defined as:

• The axial distance or radial distance ρ is the Euclidean distance 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 axial coordinate or 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 (ρ, φ ± n×360°, z) and (−ρ, φ ± (2n + 1)×180°, z), where n is any integer. Moreover, if the radius ρ is zero, the azimuth is arbitrary.

In situations where someone wants 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 spanning 360°, such as [−180°,+180°] or [0,360°].

Conventions

The notation for cylindrical coordinates is not uniform. The ISO standard 31-11 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 or s, 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.

The coordinate surfaces of the cylindrical coordinates (ρ, φ, z). The red cylinder shows the points with ρ = 2, the blue plane shows the points with z = 1, and the yellow half-plane shows the points with φ = −60°. The z-axis is vertical and the x-axis is highlighted in green. The three surfaces intersect at the point P with those coordinates (shown as a black sphere); the Cartesian coordinates of P are roughly (1.0, −1.732, 1.0).

Cylindrical coordinate surfaces. The three orthogonal components, ρ (green), φ (red), and z (blue), each increasing at a constant rate. The point is at the intersection between the three colored surfaces.

In concrete situations, and in many mathematical illustrations, a positive angular coordinate is measured counterclockwise 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 coordinates, 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,z) and Cartesian (x,y,z) are the same as for polar coordinates, namely

${\begin{aligned}x&=\rho \cos \varphi \\y&=\rho \sin \varphi \\z&=z\end{aligned}}$

in one direction, and

${\begin{aligned}\rho &={\sqrt {x^{2}+y^{2}}}\\\varphi &={\begin{cases}0&{\mbox{if }}x=0{\mbox{ and }}y=0\\\arcsin \left({\frac {y}{\rho }}\right)&{\mbox{if }}x\geq 0\\\arctan \left({\frac {y}{x}}\right)&{\mbox{if }}x>0\\-\arcsin \left({\frac {y}{\rho }}\right)+\pi &{\mbox{if }}x<0\end{cases}}\end{aligned}}$

in the other. The arcsin function is the inverse of the sine 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.

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(y,x) in the C programming language, and atan(y,x) in Common Lisp.

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:
${\begin{aligned}\rho &=r\cos \theta \\\varphi &=\varphi \\z&=r\sin \theta \end{aligned}}$	${\begin{aligned}\rho &=r\sin \theta \\\varphi &=\varphi \\z&=r\cos \theta \end{aligned}}$

Cylindrical coordinates may be converted into spherical coordinates by:

θ is elevation:	θ is inclination:
${\begin{aligned}r&={\sqrt {\rho ^{2}+z^{2}}}\\\theta &=\arctan \left({\tfrac {z}{\rho }}\right)\\\varphi &=\varphi \end{aligned}}$	${\begin{aligned}r&={\sqrt {\rho ^{2}+z^{2}}}\\\theta &=\arctan \left({\tfrac {\rho }{z}}\right)\\\varphi &=\varphi \end{aligned}}$

Position

Material taken from Vector fields in cylindrical and spherical coordinates

Vectors are defined in cylindrical coordinates by (ρ, $\phi$, z), where

• ρ is the length of the vector projected onto the xy-plane,
• $\phi$ is the angle between the projection of the vector onto the xy-plane (i.e. ρ) and the positive x-axis (0 ≤ $\phi$ < 2π),
• z is the regular z-coordinate.

(ρ, $\phi$, z) is given in cartesian coordinates by:

${\begin{bmatrix}\rho \\\phi \\z\end{bmatrix}}={\begin{bmatrix}{\sqrt {x^{2}+y^{2}}}\\\operatorname {arctan} (y/x)\\z\end{bmatrix}},\ \ \ 0\leq \phi <2\pi ,$

or inversely by:

${\begin{bmatrix}x\\y\\z\end{bmatrix}}={\begin{bmatrix}\rho \cos \phi \\\rho \sin \phi \\z\end{bmatrix}}.$

Any vector field can be written in terms of the unit vectors as:

$\mathbf {A} =A_{x}\mathbf {\hat {x}} +A_{y}\mathbf {\hat {y}} +A_{z}\mathbf {\hat {z}} =A_{\rho }\mathbf {\hat {\rho }} +A_{\phi }{\boldsymbol {\hat {\phi }}}+A_{z}\mathbf {\hat {z}}$

The cylindrical unit vectors are related to the Cartesian unit vectors by:

${\begin{bmatrix}{\boldsymbol {\hat {\rho }}}\\{\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {z}} \end{bmatrix}}={\begin{bmatrix}\cos \phi &\sin \phi &0\\-\sin \phi &\cos \phi &0\\0&0&1\end{bmatrix}}{\begin{bmatrix}\mathbf {\hat {x}} \\\mathbf {\hat {y}} \\\mathbf {\hat {z}} \end{bmatrix}}$

Note: the matrix is an orthogonal matrix, that is, its inverse is simply its transpose.

An interesting but confusing point is the abstraction of ${\boldsymbol {\hat {\phi }}}$, which can be thought of as being an orthogonal vector to ${\boldsymbol {\hat {\rho }}}$:

${\boldsymbol {\hat {\phi }}}=-\sin(\phi ){\mathsf {I}}+\cos(\phi ){\mathsf {J}}$

It is also possible to represent a position vector in Cartesian and cylindrical coordinates as follows:

${\mathsf {r}}_{P}=X_{P}{\mathsf {I}}+Y_{P}{\mathsf {J}}+Z_{P}{\mathsf {K}}=\rho {\boldsymbol {\hat {\rho }}}+Z_{P}{\mathsf {K}}$

where we can leave out the term $\phi {\boldsymbol {\hat {\phi }}}$ and describe the position fully.

Velocity

Material taken from Vector fields in cylindrical and spherical coordinates

To find out how the vector field A changes in time we calculate the time derivatives. For this purpose we use Newton's notation for the time derivative (${\dot {\mathbf {A} }}$). In Cartesian coordinates this is simply:

${\dot {\mathbf {A} }}={\dot {A}}_{x}{\hat {\mathbf {x} }}+{\dot {A}}_{y}{\hat {\mathbf {y} }}+{\dot {A}}_{z}{\hat {\mathbf {z} }}$

However, in cylindrical coordinates this becomes:

${\dot {\mathbf {A} }}={\dot {A}}_{\rho }{\hat {\boldsymbol {\rho }}}+A_{\rho }{\dot {\hat {\boldsymbol {\rho }}}}+{\dot {A}}_{\phi }{\hat {\boldsymbol {\phi }}}+A_{\phi }{\dot {\hat {\boldsymbol {\phi }}}}+{\dot {A}}_{z}{\hat {\boldsymbol {z}}}+A_{z}{\dot {\hat {\boldsymbol {z}}}}$

We need the time derivatives of the unit vectors. They are given by:

${\begin{aligned}{\dot {\hat {\mathbf {\rho } }}}&={\dot {\phi }}{\hat {\boldsymbol {\phi }}}\\{\dot {\hat {\boldsymbol {\phi }}}}&=-{\dot {\phi }}{\hat {\mathbf {\rho } }}\\{\dot {\hat {\mathbf {z} }}}&=0\end{aligned}}$

So the time derivative simplifies to:

${\dot {\mathbf {A} }}={\hat {\boldsymbol {\rho }}}({\dot {A}}_{\rho }-A_{\phi }{\dot {\phi }})+{\hat {\boldsymbol {\phi }}}({\dot {A}}_{\phi }+A_{\rho }{\dot {\phi }})+{\hat {\mathbf {z} }}{\dot {A}}_{z}$

It is also possible to represent position and velocity vectors in Cartesian and cylindrical coordinates as follows:

${\mathsf {r}}_{P}=X_{P}{\mathsf {I}}+Y_{P}{\mathsf {J}}+Z_{P}{\mathsf {K}}=\rho {\boldsymbol {\hat {\rho }}}+Z_{P}{\mathsf {K}}$

and

${\mathsf {v}}_{P}={\dot {X}}_{P}{\mathsf {I}}+{\dot {Y}}_{P}{\mathsf {J}}+{\dot {Z}}_{P}{\mathsf {K}}={\dot {\rho }}{\boldsymbol {\hat {\rho }}}+\rho {\dot {\phi }}{\boldsymbol {\hat {\phi }}}+{\dot {Z}}_{P}{\mathsf {K}}$

Acceleration

Material taken from Vector fields in cylindrical and spherical coordinates

The second time derivative is of interest in physics, as it is found in equations of motion for classical mechanical systems. The second time derivative of a vector field in cylindrical coordinates is given by:

$\mathbf {\ddot {A}} =\mathbf {\hat {\rho }} ({\ddot {A}}_{\rho }-A_{\phi }{\ddot {\phi }}-2{\dot {A}}_{\phi }{\dot {\phi }}-A_{\rho }{\dot {\phi }}^{2})+{\boldsymbol {\hat {\phi }}}({\ddot {A}}_{\phi }+A_{\rho }{\ddot {\phi }}+2{\dot {A}}_{\rho }{\dot {\phi }}-A_{\phi }{\dot {\phi }}^{2})+\mathbf {\hat {z}} {\ddot {A}}_{z}$

To understand this expression, we substitute A = P, where p is the vector (\rho, θ, z).

This means that $\mathbf {A} =\mathbf {P} =\rho \mathbf {\hat {\rho }} +z\mathbf {\hat {z}}$.

After substituting we get:

${\ddot {\mathbf {P} }}=\mathbf {\hat {\rho }} ({\ddot {\rho }}-\rho {\dot {\phi }}^{2})+{\boldsymbol {\hat {\phi }}}(\rho {\ddot {\phi }}+2{\dot {\rho }}{\dot {\phi }})+\mathbf {\hat {z}} {\ddot {z}}$

In mechanics, the terms of this expression are called:

${\begin{aligned}{\ddot {\rho }}\mathbf {\hat {\rho }} &={\mbox{central outward acceleration}}\\-\rho {\dot {\phi }}^{2}\mathbf {\hat {\rho }} &={\mbox{centripetal acceleration}}\\\rho {\ddot {\phi }}{\boldsymbol {\hat {\phi }}}&={\mbox{angular acceleration}}\\2{\dot {\rho }}{\dot {\phi }}{\boldsymbol {\hat {\phi }}}&={\mbox{Coriolis effect}}\\{\ddot {z}}\mathbf {\hat {z}} &={\mbox{z-acceleration}}\end{aligned}}$

Another form for describing the acceleration of position is given as follows:

${\mathsf {a}}_{P}={\ddot {X}}_{P}{\mathsf {I}}+{\ddot {Y}}_{P}{\mathsf {J}}+{\ddot {Z}}_{P}{\mathsf {K}}=({\ddot {\rho }}-\rho {\dot {\phi }}^{2}){\boldsymbol {\hat {\rho }}}+(\rho {\ddot {\phi }}+2{\dot {\rho }}{\dot {\phi }}){\boldsymbol {\hat {\phi }}}+{\ddot {Z}}_{P}{\mathsf {K}}$

See also Centripetal force, Angular acceleration, and Coriolis effect.