GeoGridUnitDistance
GeoGridUnitDistance[proj,loc,α]
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
 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 ("largescale maps") correspond to large values of geo distance scale, and points of "smallscale 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)
Compute the unit geo distance induced by the Mercator projection at Copenhagen in the northeast direction:
Compute the final position of a geodesic of that length starting from Copenhagen in the northeast direction:
Compare the geo path joining those locations with the map units and the scale bar:
If map units are made to correspond to inches, the traditional scale notation is 1:2468243, as given by the following:
Use the Mollweide projection to construct a flat map based on a reference sphere of radius 6371:
Then a unit of projected distance at London corresponds to a geo distance between these two values:
Scope (9)
Compute the geo grid unit distance for a geo projection at your current geo location in the northward direction:
These are the default values of the parameters of the "Mollweide" projection:
Specify other values for the parameters of the projection:
Specify a location using a pair {lat,lon} in degrees:
Use locations with geo position heads:
Specify a location using a geo Entity object:
Compute the geo distance scale for a list of locations, all along the same direction:
Convert the QuantityArray output into its normal form:
Specify the azimuth as a number of degrees:
Specify the same azimuth as a Quantity angle:
Compute the geo grid unit distance for a list of different azimuths at the same location:
The input can also be given as a QuantityArray object:
Compute the range of possible values of geo grid unit distance at a given location:
Compare with the MinMax of values for each integer degree azimuth:
GeoGridUnitDistance can efficiently process values for large numbers of locations:
Select the same reference model and geo model to eliminate the effect of nominal scale:
The inverses of geo grid unit distance along meridians and parallels are traditionally denoted as h and k:
The behavior for these cylindrical projections is identical along parallels, but different along meridians:
Options (2)
GeoModel (1)
By default, GeoGridUnitDistance returns values for Earth:
Performing the same computation on the corresponding point of the Moon returns smaller scales:
Properties & Relations (11)
Take the Mercator projection on the default ellipsoidal model of Earth, a location and a direction:
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:
Compare with the computed value:
GeoGridUnitDistance is periodic in azimuth with a period of 180 degrees:
Find the positions of one minimum and one maximum:
Those correspond to the semiaxes of this Tissot ellipse:
Geo distance scale can vary strongly with azimuth at a given point:
These are the minimum and maximum values attained:
Geo grid unit distance can vary strongly from point to point for the same projection and azimuth:
The result varies by more than two orders of magnitude:
Geo distance scale is proportional to the geo model parameter:
Geo grid unit distance is inversely proportional to the reference model and central scale parameters:
For an ellipsoidal projection, geo grid unit distance depends only slightly on the choice of datum or ellipsoid:
Equidistant projections have constant geo grid unit distance along special paths on the map:
For conic and cylindrical projections, this usually happens along meridians, at any location:
For the azimuthal equidistant projection, this happens for all directions from its centering:
For short distances, GeoDistance can be approximated as a product of projected distance by geo grid unit distance:
Compute projected distance in a given projection:
Multiply by geo grid unit distance in the direction from p to q:
The difference with the true result is smaller than 1%:
Compute geo distance along a meridian using any projection:
Extract the projection selected by GeoGraphics and compute the projected points:
Here is the geo grid unit distance along the meridian, as a function of the projected y coordinate:
Compute the distance through a numerical integration:
Compare with the unprojected geo distance:
Compare intervals of geo grid unit distance for different projections at the same point:
The geo grid unit distance in conformal projections, like Mercator, is isotropic (does not depend on azimuth):
The actual value of the projected unit distance varies from point to point for any given projection:
Both isotropy and the dependence on latitude are clear in a map showing Tissot indicatrices: