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Derived dimensions

DimensionfulAngles.jl also defines derived dimensions that include angle. These are:

This allows, among other things, dispatching on these derived dimensions.

Several units are defined for these derived dimensions, including the steradian for solid angle and RPM for angular velocity.

DimensionfulAngles.jl also provides Periodic a UnitfulEquivalences.jl Equivalence to convert between period, frequency, and angular frequency of a periodic response.

Solid Angle

Solid angle is a two-dimensional angle subtended at a point. In the SI system it has units of $m²/m²=1$ and is non-dimensional. Here, following several proposed systems, it has dimensions of angle squared, 𝐀². See Relation to proposed SI extensions. The SI unit of solid angle is the steradian, which here is defined as $sr=rad²$. The steradian takes SI prefixes and therefore defines many other units (e.g., the millisteradian DimensionfulAngles.msrᵃ). These are documented in Prefixed units.

DimensionfulAngles.SolidAngle — Type
DimensionfulAngles.SolidAngle{T, U}

A supertype for quantities and levels of dimension 𝐀 * 𝐀 with a value of type T and units U.

See also: Unitful.Quantity, Unitful.Level.

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DimensionfulAngles.srᵃ — Constant
srᵃ

The steradian, a unit of spherical angle.

There are 4π sr in a sphere. The steradian is the SI unit of solid angle. Unlike Unitful.sr, which follows SI and is therefor dimensionless, srᵃ has dimensions of Angle squared. Accepts SI prefixes.

Dimension: 𝐀²."

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Luminous flux and illuminance

Luminous flux is a measure of perceived power of light and has dimensions of $𝐉*𝐀²$. The SI unit lumen (lm) = candela x steradian is provided as DimensionfulAngles.lmᵃ.

Illuminance is luminous flux per unit surface area and has dimensions of $𝐉*𝐀²*𝐋⁻²$. The SI unit lux (lx) = lumen / meter^2 is provided as DimensionfulAngles.lxᵃ.

DimensionfulAngles.LuminousFlux — Type
DimensionfulAngles.LuminousFlux{T, U}

A supertype for quantities and levels of dimension 𝐉 * 𝐀 ^ 2 with a value of type T and units U.

See also: Unitful.Quantity, Unitful.Level.

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DimensionfulAngles.Illuminance — Type
DimensionfulAngles.Illuminance{T, U}

A supertype for quantities and levels of dimension 𝐉 * 𝐀 ^ 2 * 𝐋 ^ -2 with a value of type T and units U.

See also: Unitful.Quantity, Unitful.Level.

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DimensionfulAngles.lmᵃ — Constant
lmᵃ

The lumen, an SI unit of luminous flux.

Defined as 1 cd × sr. Accepts SI prefixes.

Dimension: 𝐉𝐀²."

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DimensionfulAngles.lxᵃ — Constant
lxᵃ

The lux, an SI unit of illuminance.

Defined as 1 lm / m^2. Accepts SI prefixes.

Dimension: 𝐉𝐀²𝐋⁻²."

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Angular velocity and acceleration

Angular velocity has dimensions of angle over time 𝐀/𝐓 and can be used to measure different quantities such as rotational velocity, rotational speed, and angular frequency of a phase angle. Two units of angular velocity are defined: the revolutions per second (RPS) and the revolutions per minute (RPM), provided as DimensionfulAngles.rpsᵃ and DimensionfulAngles.rpmᵃ respectively.

Angular acceleration is the time rate of change of angular velocity and has dimensions of angle over time squared 𝐀/𝐓². No units are defined specifically for this derived dimension.

See also: Periodic.

DimensionfulAngles.AngularVelocity — Type
DimensionfulAngles.AngularVelocity{T, U}

A supertype for quantities and levels of dimension 𝐀 * 𝐓 ^ -1 with a value of type T and units U.

See also: Unitful.Quantity, Unitful.Level.

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DimensionfulAngles.AngularAcceleration — Type
DimensionfulAngles.AngularAcceleration{T, U}

A supertype for quantities and levels of dimension 𝐀 * 𝐓 ^ -2 with a value of type T and units U.

See also: Unitful.Quantity, Unitful.Level.

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DimensionfulAngles.rpsᵃ — Constant
rpsᵃ

Revolutions per second, a unit of angular velocity defined as 2π rad / s.

This differs from Unitful.rps in that it contains units of angle. Does not accepts SI prefixes.

Dimension: 𝐀 𝐓⁻¹.

See also DimensionfulAngles.radᵃ.

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DimensionfulAngles.rpmᵃ — Constant
rpmᵃ

Revolutions per minute, a unit of angular velocity defined as 2π rad / minute.

This differs from Unitful.rpm in that it contains units of angle. Does not accepts SI prefixes.

Dimension: 𝐀 𝐓⁻¹.

See also DimensionfulAngles.radᵃ.

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Angular period, wavenumber, and wavelength

Angular wavenumber has dimensions of angle over length 𝐀/𝐋 and is the spatial analogue of (temporal) angular frequency. It is used to describe responses that are periodic in space.

The angular period (dimensions of time over angle, 𝐓/𝐀) and angular wavelength (𝐋/𝐀) are define as the reciprocal of angular frequency and angular wavenumber, respectively.

No units are defined specifically for these derived dimensions.

See also: Periodic.

DimensionfulAngles.AngularWavelength — Type
DimensionfulAngles.AngularWavelength{T, U}

A supertype for quantities and levels of dimension 𝐋 * 𝐀 ^ -1 with a value of type T and units U.

See also: Unitful.Quantity, Unitful.Level.

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DimensionfulAngles.AngularPeriod — Type
DimensionfulAngles.AngularPeriod{T, U}

A supertype for quantities and levels of dimension 𝐓 * 𝐀 ^ -1 with a value of type T and units U.

See also: Unitful.Quantity, Unitful.Level.

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DimensionfulAngles.AngularWavenumber — Type
DimensionfulAngles.AngularWavenumber{T, U}

A supertype for quantities and levels of dimension 𝐀 * 𝐋 ^ -1 with a value of type T and units U.

See also: Unitful.Quantity, Unitful.Level.

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Periodic and Dispersion equivalences

For periodic responses there are several analogous ways to measure the repeat period: period T (𝐓, s), frequency f (1/𝐓, Hz=1/s), or angular frequency ω (𝐀/𝐓, rad/s). These are related by

$f = 1/T = ω/2π$.

Analogously, spatial period and frequency are related by

$ν = 1/λ = k/2π$

between wavelength λ (𝐋, m), wavenumber ν (1/𝐋, 1/m), and angular wavenumber k (𝐀/𝐋, rad/m). Additionally an angular period and angular wavelength can be defined analogously as the reciprocal of angular frequency and angular wavenumber.

Diagram showing graphically the relationships between the various properties of harmonic waves: frequency, period, wavelength, angular frequency, and wavenumber.
image-source: Waldir, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0, via Wikimedia Commons

DimensionfulAngles.jl provides Periodic, a UnitfulEquivalences.jl Equivalence to convert between temporal or spatial period, frequency, angular frequency, and angular period of a periodic response.

It also provides Dispersion, which extends Periodic to convert between temporal and spatial values using a specific dispersion relation (or equivalently a phase velocity as used in the image above)).

DimensionfulAngles.Periodic — Type
Periodic()

Equivalence to convert between temporal or spatial period, frequency, angular frequency, and angular period.

These quantities are related by $f = ω/2π = 1/T = 1/(2πT̅)$, where

  • $f$ is the (temporal) frequency,
  • $ω$ is the (temporal) angular frequency,
  • $T$ is the (temporal) period,
  • $T̄$ is the (temporal) angular period,

and $ν = k/2π = 1/λ = 1/(2πλ̄)$, where

  • $ν$ is the (spatial) frequency (linear wavenumber),
  • $k$ is the (spatial) angular frequency (angular wavenumber),
  • $λ$ is the (spatial) period (linear wavelength), and
  • $λ̄$ is the (spatial) angular period (angular wavelength).

See also DimensionfulAngles.Dispersion

Example

julia> using Unitful

julia> using DimensionfulAngles

julia> uconvert(u"s", 10u"Hz", Periodic())
0.1 s

julia> uconvert(u"radᵃ/s", 1u"Hz", Periodic())
6.283185307179586 rad s⁻¹
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DimensionfulAngles.Dispersion — Type
Dispersion(; dispersion=nothing, dispersion_inverse=nothing)

Equivalence to convert between temporal and spatial frequencies using a specific dispersion relation.

This extends the Periodic() equivalence to convert between spatial and temporal quantities based on the provided dispersion relation.

See also DimensionfulAngles.Periodic.

Example

julia> using DimensionfulAngles, Unitful

julia> g = Unitful.gn  # gravitational acceleration
9.80665 m s⁻²

julia> deepwater = Dispersion(
        dispersion = (k -> √(g*k*θ₀)), dispersion_inverse = (ω -> ω^2/(g*θ₀))
       );

julia> uconvert(u"radᵃ/mm", 1.0u"Hz", deepwater)
0.004025678249387654 rad mm⁻¹

Some dispersion relations do not have an expressible inverse. In such cases using Roots.jl might be beneficial. For example, here is how we could use the linear water wave dispersion without deep water approximation:

julia> using DimensionfulAngles, Unitful, Roots

julia> g = Unitful.gn  # gravitational acceleration
9.80665 m s⁻²

julia> k0 = (2π)u"radᵃ/m"  # initial guess: 1m wavelength
6.283185307179586 rad m⁻¹

julia> h = 0.5u"m"  # water depth
0.5 m

julia> waterwaves = Dispersion(
        dispersion = (k -> √(k*θ₀*g*tanh(k*h/θ₀))),
        dispersion_inverse = (ω -> solve(ZeroProblem(k -> k - ω^2/(g*tanh(k*h/θ₀))/θ₀, k0)))
       );

julia> uconvert(u"Hz", 0.004025678249387654u"radᵃ/mm", waterwaves)
0.9823052153509486 Hz

julia> h = (Inf)u"m"  # water depth
Inf m

julia> waterwaves = Dispersion(
        dispersion = ( k -> √(k*θ₀*g*tanh(k*h/θ₀)) ),
        dispersion_inverse = (ω -> solve(ZeroProblem(k -> k - ω^2/(g*tanh(k*h/θ₀))/θ₀, k0)))
       );

julia> uconvert(u"Hz", 0.004025678249387654u"radᵃ/mm", waterwaves) ≈ 1u"Hz"
true
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Syntax

Contents:

Syntax provided by Unitful.jl

DimensionfulAngles.SolidAngleUnits — Type
DimensionfulAngles.SolidAngleUnits{U}

A supertype for units of dimension 𝐀 * 𝐀. Equivalent to Unitful.Units{U, 𝐀 * 𝐀}.

See also: Unitful.Units.

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DimensionfulAngles.AngularPeriodUnits — Type
DimensionfulAngles.AngularPeriodUnits{U}

A supertype for units of dimension 𝐓 * 𝐀 ^ -1. Equivalent to Unitful.Units{U, 𝐓 * 𝐀 ^ -1}.

See also: Unitful.Units.

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DimensionfulAngles.LuminousFluxUnits — Type
DimensionfulAngles.LuminousFluxUnits{U}

A supertype for units of dimension 𝐉 * 𝐀 ^ 2. Equivalent to Unitful.Units{U, 𝐉 * 𝐀 ^ 2}.

See also: Unitful.Units.

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DimensionfulAngles.IlluminanceUnits — Type
DimensionfulAngles.IlluminanceUnits{U}

A supertype for units of dimension 𝐉 * 𝐀 ^ 2 * 𝐋 ^ -2. Equivalent to Unitful.Units{U, 𝐉 * 𝐀 ^ 2 * 𝐋 ^ -2}.

See also: Unitful.Units.

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DimensionfulAngles.IlluminanceFreeUnits — Type
DimensionfulAngles.IlluminanceFreeUnits{U}

A supertype for Unitful.FreeUnits of dimension 𝐉 * 𝐀 ^ 2 * 𝐋 ^ -2. Equivalent to Unitful.FreeUnits{U, 𝐉 * 𝐀 ^ 2 * 𝐋 ^ -2}.

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Prefixed Units