# Notes on Map Geometry and Projections

This is a back-up of the WIKI.
We're working on a new wiki.

Main Page | Recent changes | View source | Page history | Log in / create account |

MapWindow Users are not required to be experts in Geodesy, Mapping science or Coordinate computation. However, once in a while, information from these fields can be helpful.

Here an attempt to start wiki pages covering these topics in a simple way.

Geodesy, the Science of measuring the Earth

Map Projections

Coordinates and Coordinate Transformations

## Problem Overview

When a view of Earth or part of it is required which reaches beyond the horizon, truthful reduced models of it will be needed, as for example provided by maps.

Mapping the Earth is not a trivial process.

Consider that Earth's outer surface is in rough approximation spherical, like a ball. Creating a truthful model of Earth will only be possible in creating a reduced spherical model of it. But, who wants to carry a globe in his/her hip pocket? Flat paper or computer screen maps are much easier to handle.

And here lies the problem: How to flatten out a curved surface and still be truthful?

This will not be possible at all. All we can expect is to create a flat map with geometric properties we need and with distorted properties we don't need or to keep any distortion small enough to be neglegible. (For the latter, the shear size of Earth is helpful).

In order to start this flattening process (Map Projection) we need to know two things:

1. the SIZE of Earth and

2. the SHAPE of Earth as exact as possible

Geodesy is the science which delivers these two aspects through measurement and computation.

Once the size and shape of Earth is known to acceptable accuracy, this information can be used in the flattening out process: Map Projection.

Flat maps have been in use for several thausends of years. That means Map Projections have been used with and without knowledge of Earth's shape and size for various reasons:

• Overview and rough orientiering ==> Atlas type maps
• Rough positioning and compass orientireing ==> Mercator type maps
• Exact engineering and military stype maps ==> Universal Mercator Maps (UTM)

Today, all these map types are still in use and all are more or less based on the knowledge of Earth's size and shape empoying in their creation more or less rigid and accurate Map Projection mathematics based on positional data, e.g. Coordinates.

Since Earth does not carry an inherent natural positioning reference system (coordinates like latitude and longitude), assumptions have to fixed by common agreement as to how and from where to reference positions:

• Geographic Latitude, Logitude Reference: Meridian through Greenwich is Zero longitude with logitude angle measured easteward and westward both to 180 degree at the date line. Equator is ZERO latitude and latutudes measured (positive) northward as an angle in or neare Earth's center and (negative) southward. Brisbane at the east coast of Australia for example is at logitude 153 degree East and 30 degree South, stated as long=153 lat=-30.
This kind of position reference is agreed (fixed) to as distortion free spherical positioning system when obtained by accurate astronomical measurements.
• 3D geocentric coordinates X,Y,Z: obtained from longitude, latitude and the knowledge of the shape and size of Earth. The latter also is obtained from measurments and depends on the quality and accurcay of measurement techniques employed. Considering advances in technology, we must expect that resulting X,Y,Z coordinates will change (become more accurate) with progressing measurement technology.
• Map surface coordinates (x,y) plus height above sea level: These depend on two things, the Map Projection type and the knowledge and accuracy of size and shape of Earth (see above).
These x,y coordinates are usually rectangular coordinates defined on the plane mapping surface. This mapping surface is either a map 'paper' sheet of suitable size and scale or an imaginary plane surface at a scale resembling 1 in 1 reality (e.g. UTM coordinates).

In any form, map coordinates can be re-mapped by Transformations into different position systems (map and/or earth bound).
• Image coordinates: Most images used in GIS applications are created following some form of projective geometry. (Aerial and some satellite imagery ==> central projection; Radar and Sonar images ==> combined distance and central projection, etc.) In order to use these images in GIS applications, Georeferencing, (in principle Coordinate Transformations) is employed.
• Using projected maps and georeferenced images in a GIS system also requires Inverse Coordinate Transformations so that any feature's real position can be displayed when pointing with the mouse cursor.

This page may contain inconsistencies and errors. Please correct or complain to --Gngdowid 04:53, 25 December 2009 (UTC)