GeoGridUnitDistance
✖
GeoGridUnitDistance
gives the actual geo distance corresponding to a unit distance on the geo grid obtained with projection proj, evaluated in the limit of small displacement from location loc in direction α.
Details and Options

- GeoGridUnitDistance describes the local distortion of distance induced by a geo projection around a given location.
- The inverse of geo grid unit distance is also known as point scale or particular scale.
- GeoGridUnitDistance combines the global nominal scale (the shrinking factor of the geo model to the reference model of the map, traditionally denoted as 1:125000 and similar) and the local distortion of scale induced by the geo projection.
- The result of GeoGridUnitDistance[…] corresponds to a ratio of Quantity geo distances on the geo model (of Earth or any other body) and dimensionless distances of the projected geo grid.
- Points of maps including large regions ("large-scale maps") correspond to large values of geo distance scale, and points of "small-scale maps" correspond to small values of geo distance scale.
- If geo grid unit distance is independent of azimuth at a point, then it is said to be isotropic at that point. A geo projection is conformal if and only if geo grid unit distance is isotropic at all points, though its value may still vary from point to point.
- A geo projection can be given as a named projection "proj" with default parameters or as {"proj",params}, where "proj" is any of the entities of GeoProjectionData and params are parameter rules like "StandardParallels"->{33,60}. GeoProjectionData["proj"] gives the default values of the parameters for the projection "proj".
- The location loc can be given as a coordinate pair {lat,lon} in degrees, a geo position object like GeoPosition[…] or GeoGridPosition[…] or as a geo entity Entity[…].
- The bearing or azimuthal direction α is an angle measured clockwise from true north. It can be given as a Quantity angle, as a number in degrees or as a named compass direction like "North", "NE" or "NEbE".
- GeoGridUnitDistance threads over its location and direction arguments.
- Possible options of GeoGridUnitDistance include:
-
GeoModel Automatic model of Earth or a celestial body UnitSystem $UnitSystem unit system to use in the result
Examples
open allclose allBasic Examples (2)Summary of the most common use cases
Compute the unit geo distance induced by the Mercator projection at Copenhagen in the northeast direction:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-f5nanx

Compute the final position of a geodesic of that length starting from Copenhagen in the northeast direction:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-xftk7i

Compare the geo path joining those locations with the map units and the scale bar:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-hbz7qu

If map units are made to correspond to inches, the traditional scale notation is 1:2468243, as given by the following:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-oeyhck

Use the Mollweide projection to construct a flat map based on a reference sphere of radius 6371:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-oa9muq

Then a unit of projected distance at London corresponds to a geo distance between these two values:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-hzu04k


https://wolfram.com/xid/0ixhk5yjlv9ebyq-iom3dr

Scope (9)Survey of the scope of standard use cases
Compute the geo grid unit distance for a geo projection at your current geo location in the northward direction:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-7o46v8

These are the default values of the parameters of the "Mollweide" projection:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-ijio5v

Specify other values for the parameters of the projection:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-fg4xfv


https://wolfram.com/xid/0ixhk5yjlv9ebyq-jhoi12

Specify a location using a pair {lat,lon} in degrees:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-oblue2

Use locations with geo position heads:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-mxtpox


https://wolfram.com/xid/0ixhk5yjlv9ebyq-mv5nny


https://wolfram.com/xid/0ixhk5yjlv9ebyq-g9eq9s

Specify a location using a geo Entity object:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-h5tq64

Compute the geo distance scale for a list of locations, all along the same direction:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-j6wzoq

Convert the QuantityArray output into its normal form:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-rkhe1g

Specify the azimuth as a number of degrees:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-scuzy8

Specify the same azimuth as a Quantity angle:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-n0ibp8


https://wolfram.com/xid/0ixhk5yjlv9ebyq-z6kxia

Compute the geo grid unit distance for a list of different azimuths at the same location:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-ims07

The input can also be given as a QuantityArray object:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-mgw46x

Compute the range of possible values of geo grid unit distance at a given location:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-sj6lob

Compare with the MinMax of values for each integer degree azimuth:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-wlkltx

GeoGridUnitDistance can efficiently process values for large numbers of locations:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-pcm2kl

https://wolfram.com/xid/0ixhk5yjlv9ebyq-faqg5p

https://wolfram.com/xid/0ixhk5yjlv9ebyq-k3gk85

Select the same reference model and geo model to eliminate the effect of nominal scale:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-fwnfs2
The inverses of geo grid unit distance along meridians and parallels are traditionally denoted as h and k:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-htrcfo
The behavior for these cylindrical projections is identical along parallels, but different along meridians:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-tafaa7


https://wolfram.com/xid/0ixhk5yjlv9ebyq-8zz0dh

Options (2)Common values & functionality for each option
GeoModel (1)
By default, GeoGridUnitDistance returns values for Earth:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-pmybot

Performing the same computation on the corresponding point of the Moon returns smaller scales:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-fr2cnl

Choose a spherical model of specific radius:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-2rl179

UnitSystem (1)
Select the unit system used to return the distance scale:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-vie8gu


https://wolfram.com/xid/0ixhk5yjlv9ebyq-1ig4cs

They are the same value, but in different units:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-4p4n6w

Properties & Relations (11)Properties of the function, and connections to other functions
Take the Mercator projection on the default ellipsoidal model of Earth, a location and a direction:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-h40rt9
Geo grid unit distance at p in direction α is the limit of the quotient of true and projected distances between p and a nearby point in direction α:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-z3jc9n

https://wolfram.com/xid/0ixhk5yjlv9ebyq-i4l89i


https://wolfram.com/xid/0ixhk5yjlv9ebyq-ii4iz1

Compare with the computed value:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-b5wyn0

GeoGridUnitDistance is periodic in azimuth with a period of 180 degrees:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-y5haw0

https://wolfram.com/xid/0ixhk5yjlv9ebyq-5gr5k1

https://wolfram.com/xid/0ixhk5yjlv9ebyq-5rwht5

Find the positions of one minimum and one maximum:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-ypmtbi


https://wolfram.com/xid/0ixhk5yjlv9ebyq-sf0fpv

Those correspond to the semiaxes of this Tissot ellipse:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-svezuy

Geo distance scale can vary strongly with azimuth at a given point:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-z06mtu


https://wolfram.com/xid/0ixhk5yjlv9ebyq-5axoth

These are the minimum and maximum values attained:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-vftf3v

Geo grid unit distance can vary strongly from point to point for the same projection and azimuth:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-g7vgfh


https://wolfram.com/xid/0ixhk5yjlv9ebyq-e8rgc5

The result varies by more than two orders of magnitude:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-kfhy6p


https://wolfram.com/xid/0ixhk5yjlv9ebyq-oxi563

Geo distance scale is proportional to the geo model parameter:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-s228m4

Geo grid unit distance is inversely proportional to the reference model and central scale parameters:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-ltgc4g


https://wolfram.com/xid/0ixhk5yjlv9ebyq-yu17ek

For an ellipsoidal projection, geo grid unit distance depends only slightly on the choice of datum or ellipsoid:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-2yezzu

Equidistant projections have constant geo grid unit distance along special paths on the map:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-j10xlt

For conic and cylindrical projections, this usually happens along meridians, at any location:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-9nnyrx


https://wolfram.com/xid/0ixhk5yjlv9ebyq-w1ucy5

For the azimuthal equidistant projection, this happens for all directions from its centering:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-kjt3yl

For short distances, GeoDistance can be approximated as a product of projected distance by geo grid unit distance:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-8gep29

https://wolfram.com/xid/0ixhk5yjlv9ebyq-07px8h

Compute projected distance in a given projection:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-hujh8p

https://wolfram.com/xid/0ixhk5yjlv9ebyq-2v11mi

Multiply by geo grid unit distance in the direction from p to q:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-9eo9lr

The difference with the true result is smaller than 1%:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-n5c2ka

Compute geo distance along a meridian using any projection:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-ssdr2w

https://wolfram.com/xid/0ixhk5yjlv9ebyq-6blgvt

Extract the projection selected by GeoGraphics and compute the projected points:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-3nrwjl


https://wolfram.com/xid/0ixhk5yjlv9ebyq-ralw3w
Here is the geo grid unit distance along the meridian, as a function of the projected y coordinate:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-v5aqm3
Compute the distance through a numerical integration:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-ky642s

Compare with the unprojected geo distance:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-bu13l3

Compare intervals of geo grid unit distance for different projections at the same point:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-9j3a8w

https://wolfram.com/xid/0ixhk5yjlv9ebyq-whi95o

https://wolfram.com/xid/0ixhk5yjlv9ebyq-h0x1fh

The geo grid unit distance in conformal projections, like Mercator, is isotropic (does not depend on azimuth):

https://wolfram.com/xid/0ixhk5yjlv9ebyq-licmak

https://wolfram.com/xid/0ixhk5yjlv9ebyq-v1h3yc


https://wolfram.com/xid/0ixhk5yjlv9ebyq-q0elxf

The actual value of the projected unit distance varies from point to point for any given projection:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-t75gdy


https://wolfram.com/xid/0ixhk5yjlv9ebyq-b4y1g5


https://wolfram.com/xid/0ixhk5yjlv9ebyq-wynyrq

Both isotropy and the dependence on latitude are clear in a map showing Tissot indicatrices:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-6abh65

Possible Issues (1)Common pitfalls and unexpected behavior
If a geo location cannot be projected, then the geo grid unit distance cannot be computed either:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-fjgf9


https://wolfram.com/xid/0ixhk5yjlv9ebyq-uf0hbg

This location is not on the half-Earth covered by the "Orthographic" projection with default center:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-bqunkz

Neat Examples (1)Surprising or curious use cases
Compare the geo grid unit distance in the Mercator projection at different latitudes:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-hun1ck

https://wolfram.com/xid/0ixhk5yjlv9ebyq-kbw07l

https://wolfram.com/xid/0ixhk5yjlv9ebyq-wnmbtu
The following diagram shows at the bottom a scale at lower latitudes and how it changes when it is projected at higher latitudes:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-4y26d4

Sometimes this diagram is presented using the inverse ratio:

https://wolfram.com/xid/0ixhk5yjlv9ebyq-o20k2x

https://wolfram.com/xid/0ixhk5yjlv9ebyq-59ccil

https://wolfram.com/xid/0ixhk5yjlv9ebyq-wxduzl

Wolfram Research (2019), GeoGridUnitDistance, Wolfram Language function, https://reference.wolfram.com/language/ref/GeoGridUnitDistance.html.
Text
Wolfram Research (2019), GeoGridUnitDistance, Wolfram Language function, https://reference.wolfram.com/language/ref/GeoGridUnitDistance.html.
Wolfram Research (2019), GeoGridUnitDistance, Wolfram Language function, https://reference.wolfram.com/language/ref/GeoGridUnitDistance.html.
CMS
Wolfram Language. 2019. "GeoGridUnitDistance." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/GeoGridUnitDistance.html.
Wolfram Language. 2019. "GeoGridUnitDistance." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/GeoGridUnitDistance.html.
APA
Wolfram Language. (2019). GeoGridUnitDistance. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GeoGridUnitDistance.html
Wolfram Language. (2019). GeoGridUnitDistance. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GeoGridUnitDistance.html
BibTeX
@misc{reference.wolfram_2025_geogridunitdistance, author="Wolfram Research", title="{GeoGridUnitDistance}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/GeoGridUnitDistance.html}", note=[Accessed: 19-June-2025
]}
BibLaTeX
@online{reference.wolfram_2025_geogridunitdistance, organization={Wolfram Research}, title={GeoGridUnitDistance}, year={2019}, url={https://reference.wolfram.com/language/ref/GeoGridUnitDistance.html}, note=[Accessed: 19-June-2025
]}