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Colatitude

Colatitude is the polar angle in spherical coordinates, measured from the positive z-axis (or north pole) to a point on a sphere, ranging from 0 to π radians, and serves as the complement of geographical latitude, where latitude λ relates to colatitude φ by φ = π/2 - λ.^[1] In mathematical and physical contexts, it is denoted typically as φ and used alongside the radial distance r and azimuthal angle θ to specify positions in three-dimensional space, with Cartesian conversions given by x = r sin φ cos θ, y = r sin φ sin θ, and z = r cos φ.^[1] This coordinate system facilitates solving differential equations like the Laplace equation and computing integrals over spherical volumes or surfaces, where the volume element is r² sin φ dr dφ dθ.^[1] In astronomy, colatitude represents the angular distance from the celestial pole along a meridian, forming a key side in the astronomical triangle alongside zenith distance and polar distance, enabling calculations such as sight reduction for determining observer positions using the spherical law of cosines: cos z = cos φ sin δ + sin φ cos δ cos H, where δ is declination and H is the hour angle.^[1] Geographically and in geographic information systems (GIS), colatitude quantifies the angular separation from the pole to a boundary point along a meridian, aiding in map projections and spatial analysis on Earth's surface.^[2] Its adoption in these fields underscores its role in modeling spherical geometry, from planetary motion to geospatial data processing.^[3]

Definition

Mathematical Definition

In spherical geometry, colatitude is defined as the polar angle measured from the positive z-axis, which corresponds to the north pole, to the position vector of a point in three-dimensional space. This angle quantifies the deviation from the polar axis and serves as a fundamental component in describing positions on a sphere.^[4] There are two main notation conventions for spherical coordinates. In physics, the colatitude is typically denoted by θ and ranges from 0 to π radians (or 0° to 180°), where θ = 0 aligns with the north pole and θ = π with the south pole; the azimuthal angle is denoted by φ. In many mathematical contexts, the colatitude is denoted by φ, with θ serving as the azimuthal angle. This article primarily follows the mathematical convention, denoting colatitude by φ. Older texts and European conventions frequently align with the physics usage of θ for colatitude.^[5]^[6] Geometrically, colatitude corresponds to the shortest angular distance along a great circle arc from the north pole to the given point on the sphere's surface. This interpretation emphasizes its role as a measure of polar displacement rather than equatorial positioning. The colatitude θ (or φ, per convention) relates to the latitude angle λ—measured from the equator—via the conversion equation

\theta = \frac{\pi}{2} - \lambda,

where latitude λ ranges from -π/2 to π/2, ensuring θ remains between 0 and π.^[4]

Relation to Latitude

Colatitude is defined as the complement of latitude, calculated as 90° minus the latitude or, in radians, \pi/2 minus the latitude, where latitude is measured northward or southward from the equator.^[7]^[4] This relationship positions colatitude as the angular distance from the nearest pole rather than the equator, providing a pole-centered perspective complementary to latitude's equatorial reference.^[8] The term colatitude originated around 1790, introduced as the complement of latitude specifically in astronomical and navigational contexts to facilitate calculations involving polar distances.^[9] In geographical conventions, latitude \lambda ranges from -90° at the South Pole to +90° at the North Pole, yielding colatitude values that are always non-negative, spanning 0° at the North Pole to 180° at the South Pole.^[7] Unsigned usage predominates for consistency with polar angle measurements.^[8] For instance, at the equator (latitude 0°), colatitude measures 90°; at the North Pole (latitude 90°), it is 0°.^[4] This inverse pairing underscores colatitude's role as a straightforward transformation of latitude, often aligning with the polar angle \phi in mathematical contexts without direct dependence on equatorial metrics.^[7]

Usage in Coordinate Systems

Spherical Coordinates

In the spherical coordinate system, points in three-dimensional Euclidean space are represented by the ordered triple (r, \theta, \phi), where r \geq 0 denotes the radial distance from the origin, \theta is the colatitude (or polar angle) measured from the positive z-axis with range $0 \leq \theta \leq \pi, and \phi is the azimuthal angle in the xy-plane with range $0 \leq \phi < 2\pi. This convention aligns with the physics standard, positioning \theta = 0 at the north pole along the z-axis and \theta = \pi at the south pole.^[5] The transformation from spherical to Cartesian coordinates follows directly from projecting the point onto the axes, yielding the equations:

\begin{align*} x &= r \sin \theta \cos \phi, \\ y &= r \sin \theta \sin \phi, \\ z &= r \cos \theta. \end{align*}

These relations position the point at distance r from the origin, with the projection onto the xy-plane at radius r \sin \theta and the z-coordinate determined by the colatitude.^[10] For integration over volumes, the Jacobian determinant of the transformation introduces a scaling factor, resulting in the volume element dV = r^2 \sin \theta \, dr \, d\theta \, d\phi. The \sin \theta term accounts for the varying circumference of spherical shells at different colatitudes, ensuring the infinitesimal volume matches the geometry of the coordinate surfaces.^[11]^[12] Notation for spherical coordinates varies by discipline: the physics convention assigns \theta to colatitude and \phi to azimuth, consistent with ISO 80000-2:2019, while mathematics often interchanges them, using \phi for the polar angle and \theta for azimuth.^[13]^[14]

Geographical and Astronomical Coordinates

In geography, colatitude refers to the angular distance from the geographic north pole to a point on the Earth's surface, measured along a meridian.^[2] This measure is fundamental in geodesy, where it facilitates computations such as great-circle distances between points on the Earth's surface through spherical trigonometry; in these calculations, the colatitudes of the points serve as sides in the relevant spherical triangle.^[15] The colatitude \theta of a location is mathematically defined as \theta = 90^\circ - \lambda, where \lambda is the geographic latitude.^[4] In astronomy, the observer's co-latitude is similarly $90^\circ minus their geographic latitude and plays a key role in horizon-based coordinate systems, such as the alt-azimuth (or horizontal) system.^[16] Within this system, the co-latitude equals the zenith distance of the north celestial pole, meaning the pole's altitude above the horizon is precisely the observer's latitude.^[16] Astronomers also employ co-declination, defined as $90^\circ minus a celestial object's declination, which quantifies the object's angular distance from the north celestial pole along a great circle.^[17] This complement is particularly useful in the astronomical triangle for determining positions relative to the observer's meridian and zenith.^[17]

Applications

In celestial navigation, colatitude plays a key role in solving the navigational triangle, a spherical triangle formed by the poles, the zenith, and the geographical position of a celestial body. One side of this triangle is the colatitude, defined as 90° minus the observer's latitude, which connects the elevated pole to the zenith and facilitates computations for determining latitude from the meridian altitude of the sun or stars. When a celestial body crosses the observer's meridian, its observed altitude (after corrections) allows calculation of latitude using the formula: latitude = declination ± (90° - observed altitude), where the colatitude inherently complements the zenith distance in these reductions. This method has been essential for mariners to establish position without reliance on dead reckoning alone.^[18] A fundamental astronomical application of colatitude arises in observing the celestial poles, where the colatitude equals the zenith distance of the pole from the observer's zenith. Consequently, the altitude of the celestial pole above the horizon matches the observer's latitude; for example, at 40° north latitude, the colatitude is 50°, positioning the north celestial pole at 40° altitude. This relationship, rooted in the geometry of the celestial sphere, enables quick latitude estimation using polar stars like Polaris, which approximates the north celestial pole and serves as a navigational reference without complex computations.^[19]^[20] For positioning stars in the equatorial system, co-declination—the complement of a star's declination (90° minus declination), also known as polar distance—simplifies trigonometric calculations involving the hour angle. The hour angle measures the angular distance westward from the local meridian to the star's hour circle, and together with co-declination, it forms sides of the navigational triangle that allow solving for local sidereal time (LST = right ascension + hour angle). This approach streamlines conversions between equatorial and horizon coordinates, aiding observers in tracking stellar positions relative to sidereal time.^[21]^[22] Historically, colatitude has been integral to sight reduction procedures in nautical almanacs, with its use documented since the inaugural British Nautical Almanac of 1767, compiled by Nevil Maskelyne under the Commissioners of Longitude. These almanacs provide ephemerides and tables that incorporate colatitude to resolve the navigational triangle, enabling precise line-of-position fixes from sextant altitudes of celestial bodies. This innovation supported transoceanic voyages by standardizing computations previously reliant on cumbersome logarithmic tables.^[23]^[24]

In Physics

In physics, colatitude θ, defined as the polar angle from the positive z-axis in spherical coordinates, is essential for describing systems with spherical symmetry, enabling the separation of variables in governing differential equations. This coordinate choice facilitates analytical solutions in fields like quantum mechanics, electromagnetism, and geophysics, where physical laws often exhibit rotational invariance around a central axis.^[25] In quantum mechanics, colatitude appears prominently in the solution to the time-independent Schrödinger equation for the hydrogen atom, a foundational model for atomic structure. The equation, expressed in spherical coordinates (r, θ, φ), separates into independent radial and angular components due to the Coulomb potential's spherical symmetry. The angular part further divides into θ-dependent and φ-dependent functions, with the colatitude equation determining the θ behavior. This equation takes the form

\frac{1}{\sin \theta} \frac{d}{d\theta} \left( \sin \theta \frac{d \Theta}{d\theta} \right) + \left[ l(l+1) - \frac{m^2}{\sin^2 \theta} \right] \Theta = 0,

where Θ(θ) is the colatitude function, l is the orbital angular momentum quantum number (l = 0, 1, ..., n-1, with n the principal quantum number), and m is the magnetic quantum number (-l ≤ m ≤ l). The solutions to this ordinary differential equation are the associated Legendre functions P_l^m(cos θ), which, when combined with the φ-dependent exponential e^{imφ} and normalized, form the spherical harmonics Y_l^m(θ, φ) that describe the angular probability distribution of the electron. These functions ensure the wave function's square integrability and orthogonality, crucial for determining allowed energy levels and quantum states.^[25] In electromagnetism, colatitude is integral to multipole expansions, which approximate the fields produced by localized charge or current distributions around a spherical origin, particularly useful for far-field radiation patterns. For electric dipole radiation—the dominant term for non-relativistic accelerating charges—the time-averaged power radiated per unit solid angle exhibits an angular dependence of sin² θ, where θ is the colatitude measured from the dipole axis. This pattern arises from the vector potential's projection in spherical coordinates and results in a toroidal (doughnut-shaped) intensity distribution, with maximum radiation perpendicular to the dipole orientation (θ = 90°) and nulls along the axis (θ = 0°, 180°). Such dependence is derived from Maxwell's equations in the radiation zone, where the electric and magnetic fields are transverse and proportional to the acceleration of the dipole moment, underscoring colatitude's role in quantifying directional emission in antennas and atomic transitions.^[26] In geophysics, colatitude features in models of Earth's magnetic field, approximated as that of a bar magnet or dipole centered at the planet's core, tilted approximately 11° from the rotational axis. The field components in spherical coordinates are B_r = (2 μ_0 M cos θ)/(4π r³) for the radial part and B_θ = (μ_0 M sin θ)/(4π r³) for the colatitude part, where M is the dipole moment magnitude (about 8 × 10²² A m² for Earth) and θ is measured from the magnetic north pole. The vertical component of the field, relevant for compass behavior and auroral studies, is then B_z = B_r cos θ - B_θ sin θ ∝ (3 cos² θ - 1)/r³, reflecting the field's compression along the poles and extension at the equator. This dipole approximation captures about 90% of the observed surface field variations, aiding in paleomagnetic reconstructions and satellite navigation corrections.^[27]^[28]

References

[1]
[PDF] Laplace equation and related equations in spherical coordinates
Apr 28, 2021 · here r is the distance from the origin, and (φ, θ) are coordinates on the sphere: φ is called co-latitude, (the ordinary geographical latitude ...
[2]
Astronomical Triangle
### Definition and Use of Colatitude in Astronomical Context
[3]
Colatitude Definition | GIS Dictionary - Esri Support
The angular distance from the pole to a boundary point along a meridian.Missing: mathematics | Show results with:mathematics
[4]
Spherical Coordinates
A generic spherical coordinate system, with the radial coordinate denoted by q, the zenith (the angle from the North Pole; the colatitude) denoted by \alpha.
[5]
Colatitude -- from Wolfram MathWorld
The polar angle on a sphere measured from the north pole instead of the equator. The angle phi in spherical coordinates is the colatitude.
[6]
[PDF] Physics 103 - Discussion Notes #3
In spherical coordinates, we specify a point vector by giving the radial coordinate r, the distance from the origin to the point, the polar angle θ, the angle ...
[7]
Spherical Coordinates -- from Wolfram MathWorld
A system of curvilinear coordinates that are natural for describing positions on a sphere or spheroid.
[8]
Latitude -- from Wolfram MathWorld
The latitude delta is related to the colatitude (the polar angle in spherical coordinates) by delta=90 degrees-phi. More generally, the latitude of a point ...
[9]
Colatitude - Oxford Reference
Quick Reference. The latitudinal distance from the pole, i.e. 90° minus the latitude. From: colatitude in A Dictionary of Earth Sciences ».Missing: definition | Show results with:definition
[10]
COLATITUDE Definition & Meaning - Merriam-Webster
The meaning of COLATITUDE is the complement of the latitude.
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May 3, 2022 · The original Cartesian coordinates are now related to the spherical coordinates by... x = r sin(θ) cos(φ) y = r sin(θ) sin(φ) z ...
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The volume element in spherical coordinates. A blowup of a piece of a sphere is shown below. Using a little trigonometry and geometry, we can measure the ...
[13]
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We can see that the small volume ∆V is approximated by ∆V ≈ ρ2 sinφ∆ρ∆φ∆θ. This brings us to the conclusion about the volume element dV in spherical coordinates ...
[14]
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Oct 18, 2002 · Nearly everybody uses r and θ to denote polar coordinates. Most American calculus texts also utilize θ in spherical coordinates for the angle ...
[15]
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In the ISO convention the symbols r (instead of ρ ), ϕ (instead of θ ) and θ (instead of ϕ ) are used for spherical coordinates. distance from ...
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And we suppose the colatitude, r> of the place to be known. Then we have the spheric triangle. "hose vertices are p 7 S and whose opposite sides are n.J.r ...
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Note that the altitude of the North Celestial Pole is equal to the latitude of the observer. ... co-latitude. zenith distance. parallel of latitude. parallel of ...
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[PDF] COORDINATES, TIME, AND THE SKY John Thorstensen ...
The Astronomical Triangle. HA co-latitude co-dec zenith and the north ... and your object will be the object's co-declination, or 90 degrees minus its dec.
[19]
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in celestial navigation. All points having the ... In celestial navigation they are the assumed ... gle, is the colatitude. Arc AM of the vertical ...
[20]
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May 13, 2024 · This is equivalent to the angle between the radius of Earth through O and Earth's axis. This is by definition the (terrestrial) colatitude of O.
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However, its altitude (the angle it makes with the horizon) is not fixed, but varies according to the latitude of the observer. This is very useful for ...
[22]
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PX = co-Declination. Compliments enable formulae to be simplified because the Sine of an angle equals the Co-Sine of the compliment of that angle. This also ...
[23]
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Nov 15, 2024 · Altitude (H) is the angle between celestial body and the celestial horizon. (As opposed to declination which is measured from the celestial ...
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The Nautical Almanac and Astronomical Ephemeris for the Year 1767 was the first English nautical almanac and the earliest by far to give essential data for the ...Missing: colatitude sight
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Hydrogen Schrodinger Equation - HyperPhysics Concepts
The solutions to the colatitude equation are in a form called associated Legendre functions, and when properly normalized form part of the hydrogen ...
[27]
[PDF] antennas.pdf
For the Hertzian dipole the angular dependence of the radiation is simply sin2 𝜃 . Although this is an entirely different function from the one derived for the ...
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[PDF] Earth's Magnetic Field
Mar 1, 2017 · Magnetic Potential for a dipole field pointing South. V(r) = m • r / (4 π r3) = − m cosθ / (4 π r2) = scalar magnetic potential of dipole field.
[29]
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The dipole field equations (1) to (3) say that a dipole field is parallel to the radial direction over the poles and perpendicular to the radial direction on.Missing: approximation ∝