Flat card

Square flat map: Blue Marble : Land Surface, Ocean Color and Sea Ice , NASA

Plan map with Tissot indicatrix . The circles of identical size on the surface of the sphere show how much the respective areas of the projection are distorted.

As a planar map ( Plate-Carrée projection to French plate carrée "flat rectangle"), even Rektangularprojektion or rectangular bank or rektangulare projection ( English rectangular projection ) is referred to in the mathematical mapping a distance fidelity (faithfulness length) cylindrical projection . Its special form is the square flat map (equirectangular projection, simple cylinder projection , English equirectangular projection , French also carte parallélogrammatique ). Despite its old age, it was rarely used in the past, but is now one of the more important types of projection.

Flat map is specifically the designation of the main position of the rectangular projection, i.e. with the equator as the standard parallel. Their special square shape is equidistant in east-west as well as north-south direction, so it reflects the geographic coordinates directly, and is therefore also called geographic projection .

Basics

The longitudes are mapped true to the route. The circles of latitude are distorted except for the circles of contact or intersection. The poles are reproduced with the same length as the equator, the surface distortion increases towards the pole.

The basic shape of the flat map is created when geographical longitude and latitude are used directly as Cartesian coordinates : ${\ displaystyle \ lambda}$ ${\ displaystyle \ phi}$

{\ displaystyle {\ begin {aligned} x & = \ lambda \\ y & = \ phi \ end {aligned}}}

It is the simplest of the possible map network designs .

A transformation from spherical coordinates into plane coordinates or a reverse transformation from plane coordinates into spherical coordinates can take place.

The following is defined:

λ longitude on the surface of the sphere

φ latitude of the spherical surface

φ ₁ parallel to the equator

λ ₀ main meridian

x horizontal coordinate of the projected point on the map (illustration)

y vertical coordinate of the projected point on the map (illustration)

transformation

{\ displaystyle {\ begin {aligned} x & = (\ lambda - \ lambda _ {0}) \ cos \ varphi _ {1} \\ y & = (\ varphi - \ varphi _ {1}) \ end {aligned} }}

This formula only applies to a sphere (not an ellipsoid).

Inverse transformation

{\ displaystyle {\ begin {aligned} \ lambda & = {\ frac {x} {\ cos \ varphi _ {1}}} + \ lambda _ {0} \\\ varphi & = y + \ varphi _ {1} \ end {aligned}}}

Square flat card

The square flat map (equirectangular projection) is an equidistant cylinder projection in a normal position with a contact circle on the equator . The equator itself is true to length, areas near the equator are also shown as being relatively true to angle and area .

The main design principle is that at every point on the map the distance between two neighboring latitudes and two neighboring longitudes is identical. In the middle latitudes this leads to relatively strong distortions in the east-west direction, while the meridians are shown as long as half the equator - which is correct if the earth is assumed to be a sphere. This means that the map is true to length in north-south direction. The distortion of the latitudes increases with the factor towards the poles . ${\ displaystyle 1 / \ cos ({\ text {Width}})}$

Rectangular flat card

In this variant, the cylinder cuts the globe or ellipsoid. The intersection circles are true to length next to the longitudinal circles. This results in a rectangular map network in proportion . The circles of latitude and longitude have constant distances, but in the ratio that applies to the mean geographical latitude of the map ( intersection ). The widths between the intersection circles are shown shortened, outside they are extended towards the poles. The distortion is . ${\ displaystyle 1: \ cos ({\ text {Cutting width}})}$ ${\ displaystyle \ cos ({\ text {Cutting width}}) / \ cos ({\ text {Width}})}$

The maps show the middle latitudes better.

use

Show star map
as spherical panorama

According to Ptolemy, the map network design of the square flat map goes back to Marinos of Tire (around 100 AD), making it one of the oldest map projections .

Because of the distortions, this projection was rarely used in geography; it was found in early nautical charts in nautical science . The Portuguese astronomer and mathematician Pedro Nunes (1502–1578) was probably the first to deal with the resulting distortions and their influence on navigation . Early star maps , such as Bayer's Uranometria from 1603, are also drawn true to coordinates .

However, it is of particular importance in modern geographic information systems because the geographic coordinates can be entered directly on the map or read from it.

It is used for applications that handle worldwide data sets or that rely on satellite data, such as NASA World Wind . The best-known equirectangular data set is the Blue Marble : Cloudless Earth from NASA. It is particularly suitable for raster data sets ("image files"), while vector-based applications such as Google Maps or Openstreetmap have been using the conformal Mercator projection (Web Mercator) since the mid-2000s , because the server-side computing power is sufficient.
In addition, it is naturally found for very small-scale maps in which the curvature of the earth no longer plays a role; local maps or construction plans are defacto equirectangular projections. Therefore, direct conversion of the geodetic survey data into the plan position is also standard in the digitized cadastre .

The same applies to star maps; astronomy programs also use this representation, in particular for the view close to the horizon , while they usually switch to perspective (point-related) images for zenith-like images, such as stereographic projection that is true to the angle and circle or the azimuthal projection that is true to zenith (equidistant) , and for overall views into the orthographic projection . The distance fidelity is particularly well suited for astronomical distance estimates such as "fist's width" (10 °).

Web links

Commons : Plattkarte - collection of images, videos, and audio files

The Blue Marble: Land Surface, Ocean Color and Sea Ice (Engl.)
Equirectangular Projection , MathWorld (Eng.)
Image: Big ben equirectangular.jpg , on PanoTools - example of a 360 ° flat map of an interior

Individual evidence

↑ Andreas Weber: Rectangular projections (360 ° photography) , weber-fotografie-kassel.de, accessed on February 28, 2018

^ Hermann Haack : Cartographic monthly report , in: Petermanns Mitteilungen , v. 1-4, 1906-1911, p. 163

↑ For the spelling, see rektangular , duden.de, accessed on February 27, 2018.

↑ ^a ^b ^c Equidistant cylinder projection. desktop.arcgis.com, accessed June 1, 2019.

↑ The term "unprojected" is wrong, the earth figure is projected onto the plane.

↑ Map projections - cylinder designs: spaced cylinder design with equator true to length. In: Rolf Böhm: Map network drafts, on boehmwanderkarten.de, accessed June 1, 2019.

↑ John P. Snyder: Flattening the Earth: Two Thousand Years of Map Projections. 1993, ISBN 0-226-76747-7 , pp. 5-8.

[1] Andreas Weber: Rectangular projections (360 ° photography) , weber-fotografie-kassel.de, accessed on February 28, 2018