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 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)
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:
Possible Issues (1)
Text
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.
APA
Wolfram Language. (2019). GeoGridUnitDistance. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GeoGridUnitDistance.html