# 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  • GeoGridUnitDistance describes the local distorsion 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 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 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

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## Basic 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:

Use any other angular unit:

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:

Choose a spherical model of specific radius:

### UnitSystem(1)

Select the unit system used to return the distance scale:

They are the same value, but in different units:

## 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:

## Possible Issues(1)

If a geo location cannot be projected, then the geo grid unit distance cannot be computed either:

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

## Neat Examples(1)

Compare the geo grid unit distance in the Mercator projection at different latitudes:

The following diagram shows at the bottom a scale at lower latitudes and how it changes when it is projected at higher latitudes:

Sometimes this diagram is presented using the inverse ratio: