Math 2.0

Geophysics Definition

Deflection of the vertical

Builds on Reference ellipsoidGeodetic coordinatesGeopotential

Definition

Let PP be a point near the Earth's surface. Let Φ\Phi and Λ\Lambda be the astronomical latitude and longitude of PP. They give the direction of the gravity vector ∇W\nabla W of the geopotential at PP. Let φ\varphi and λ\lambda be the geodetic latitude and longitude of PP, which give the direction of the normal to the reference ellipsoid through PP. The deflection of the vertical at PP is the angle θ\theta between these two directions. Its north–south component ξ\xi and east–west component η\eta are ξ=Φ−φ,η=(Λ−λ)cos⁡φ,\xi = \Phi - \varphi, \qquad \eta = (\Lambda - \lambda)\cos\varphi, and for small angles θ2≈ξ2+η2\theta^2 \approx \xi^2 + \eta^2.

Remarks

Example: if Φ=45∘0′6′′\Phi = 45^\circ 0' 6'', φ=45∘0′0′′\varphi = 45^\circ 0' 0'' and Λ=λ\Lambda = \lambda, then ξ=6′′\xi = 6'', η=0\eta = 0, and the deflection is 6′′6'' in the north–south direction. Non-example: the difference Λ−λ\Lambda - \lambda alone is not the east–west component; it must be multiplied by cos⁡φ\cos\varphi, because meridians converge towards the poles. The east–west component also links astronomical and geodetic azimuths: the Laplace equation A−α=ηtan⁡φ=(Λ−λ)sin⁡φA - \alpha = \eta\tan\varphi = (\Lambda - \lambda)\sin\varphi holds to first order for a line of sight close to the horizon. It is used to control the azimuths of triangulation networks. Some authors use the opposite sign convention, with ellipsoid direction minus gravity direction.

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Geodetic coordinates, concept, no evidence levelGeodetic coordinatesconceptGeopotential, concept, no evidence levelGeopotentialconceptReference ellipsoid, concept, no evidence levelReference ellipsoidconceptDeflection of the vertical, definition, no evidence levelDeflection of the ver…definition
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  1. Geodetic coordinates concept
  2. Geopotential concept
  3. Reference ellipsoid concept
  4. Deflection of the vertical definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Vertical_deflection
About this page
TagPPPSMTJMJP
URIm2://PPPSMTJMJP
Kinddefinition
WritingW0 stub
MSC86A30
DefinesDeflection of the vertical
UsesReference ellipsoid, Geodetic coordinates, Geopotential
Changed3 November 2026

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