Map projections Flashcards
geographic coordinate system
- set up to identify every locations on earth by a set of number or letters (georeferencing)
- uses a system of meridians and parallels:Meridians or great circles:
- define the longitude
- They have all the diameter of the Earth.
- The half of a great circle on the earth’s surface, extending from the South to the North Pole, connects points of equal longitude.
*One of the meridians sticks out as it has been defined as the Prime meridian (0° longitude). It passes through London-Greenwich. - angle: Position of a point eastward and westward from the prime meridian (Greenwich). 0° - 180° East or West
Parallels:
* determine the latitude.
* They have different sizes, and they get smaller as the distance from equator increases. The only great circle is at the equator.
* Paralleles connect all locations with a given latitude.
* 2. angle: Position of a given point in terms of its angular distance from equator. 0°-90° North or South.
–> combine two angles which gives us the position (EASY)
Explain the difference between local and earth-centered geographic coordinate systems
- Problem starts when producing paper maps –> Before: Find reference body for earth because it not a sphere
–> result in datum, which is a reference coordinate system (not geographic coordinate system –> longitude, altitude) - Datum defines both:
- reference ellipsoid, representing the earth (because earth is not a sphere)
and 2. geodetic (determines, fix the position of the ellipsoid and hence that of the coordinate system) which is the actual datum
- reference ellipsoid, representing the earth (because earth is not a sphere)
- Because the earth is not an ellipsoid but a geoid, we want to define a datum in a way that it fits best for all purposes. –> A coordinate system can be earth-centered ((WGS84 datum–> used with GPS) the earth is described as an ellipse with the earths center of gravity as its midpoint/ cross section coincides with the center of gravity)
or local ((NAD27 datum, ED1950) the calculated ellipse is moved so that its outline matches best with the real earths surface at the point of observation) –> meaning, that the position of the ellipsoid is selected as to match the earth’s surface - Everywhere best (locally not best, because earth is not an ellipsoid)
- Or locally best
Map projection
-Problem
- explaining Map projection
- 2 main classes of projections
The problem is, however, to transform the shape of the earth’s surface in a two-dimensional (often, but not necessarily, Cartesian) coordinate system.
–> Leonard Euler proved as early as in 1777 that there is no way of projecting the three-dimensional surface of a sphere or ellipsoid to a two-dimensional map without distorting it. This process distorts at least one of the shape, area, distance or direction. –> The surface of a sphere is not developable.
Map Projection:
- are mathematical expressions that convert data from geographical
location (latitude – longitude) on a sphere or spheroid to a representative location on a flat surface.
- This process distorts at least one of shape, area, distance or direction.
- For small areas, distortions affect assessments to a smaller extent, but for bigger areas, a projection needs to be chosen that minimizes error.
Equal area projection
- preserves area, but not shape (circle to ellipse)
- circle with radius r in the original surface is deformed to an ellipse with 𝑎 𝑥 𝑏 = 𝑟2 but a ≠ b
conformal projection:
- Preserves the shape, but not the area (circle)
- Good for navigation
- Meridians and parallels intersect at right angles
- a = b ≠ r
- Projection preserves the length in one direction: 𝑎 = 𝑟 and 𝑏 ≠ 𝑟 or 𝑎 ≠ 𝑟 and 𝑏 = 𝑟
classification of map projections
(“How is it done?”)
mathematical operation, transformation of coordinates
Zenithal projections:
- Directions from a central point are preserved, and therefore, great
circles through the central point are represented by straight lines on the map.
- A plane tangent to one of the earth’s poles is the basis for polar azimuthal projection.
cylindrical projections:
- Projection in which meridians are mapped to equally spaced vertical lines and circles of latitude (parallels) are mapped to horizontal lines.
- In the first case (Mercator), the East – West scale always equals North – South scale
- Mercator conformal cylindrical projection
Conical projections:
- Any projection in which meridians are mapped to equally spaced
lines radiating out from the apex and circles of latitude (parallels) are mapped to circular arcs centered on the apex.
- The mapmaker arbitrarily picks two standard parallels, which are visualized as secant lines, where the core intersects the globe.
- Lambert conformal conical projection
Explain the terms orthodrome and loxodrome taking the Mercator projection as an example.
Orthodrome or great circle:
*Line of shortest distance between two points on the surface of a sphere, measured along the great circle connecting the two points.(the angles formed with longitude and line will change)
Rhumb line or loxodrome:
*Arc crossing all meridians of longitude at the same angle. (came with invention of mercator)
*Easiest (but not shortest) way to go from one point to another on a sphere’s surface.
Explain the issues of map projection and why the Mercator projection is useful?
- The problem with map projection is that transforming a 3-dimensional to a 2-dimensional map is not possible without distortion. This process distorts at least one of the shape, area, distance or direction.
*Mercator Projection is a conformal cylindrical projection which is considered a great achievement for nautics as it allows them to sail along lines of constant compass bearing. These intersect meridians always at the same angles.
* Mercator (of course) distorts the size of objects
as the latitude is increasingly shifted polewards. Greenland and Antarctica appear much
larger than the actually size
–> first conformal projection ever
- circle stays a circle (same shape) but not qual-area because circle gets bigger to south and north
What are the types of projections of earth
Equal area projection
- preserves area, but not shape (circle to ellipse)
- circle with radius r in the original surface is deformed to an ellipse with 𝑎 𝑥 𝑏 = 𝑟2 but a ≠ b
conformal projection:
- Preserves the shape, but not the area (circle)
- Good for navigation
- Meridians and parallels intersect at right angles
- a = b ≠ r
- Projection preserves the length in one direction: 𝑎 = 𝑟 and 𝑏 ≠ 𝑟 or 𝑎 ≠ 𝑟 and 𝑏 = 𝑟
Other famous projections:
**1. Lambert conformal conical projection: Conic map projection used for aeronautical navigation. Useful between 30° - 60° latitude. Used in the US because the deformation is the smallest possible. The scale increases poleward, and rapidly southward. All parallel – meridian inter sections are at right angles.
2. German Transverse Mercator Projection (Gauss-Krüger-System)
- Projected coordinate system that rotates a transverse cylinder around the earth to divide Germany into zones of 3° longitude width.
-The Y (Hochwert, Northing) coordinate is the distance from the
equator in meter.
-The X (Rechtswert, Easting) coordinate is computed by dividing the central meridian in each zone by three, multiplying this value by 1000 km and adding 500 km to it.
-Example: The 9° East meridian has been assigned the Y coordinate
3500 km. The coordinate of point A (or B) is then found by adding (or subtracting) the distance of the point from the meridian.
3. Universal Transverse Mercator system (NATO): This system is closely related to the Gauss-Krüger system but developed for worldwide use. The earth is divided into 60
zones, each of width of 6° longitude. The X value (easting) is assigned to a value of 500,000 m (see above) at the central meridian of each zone. The Y value (northing) is the distance to the equator, which has a Y value of 0 m for the northern hemisphere,
but a Y value of 10,000,000 m for the southern hemisphere. UTM is limited to the area between 84ºN and 80ºS. It is not valid in the areas around North and South Pole. Each
zone has been further divided into 20 latitude bands of 8° height, starting from “C” at 80ºS, increasing along the English alphabet up to X omitting I and O. Distortion inside each zone is below 1 per mille.
Which classes of map projections do you know? Please explain (Georeferencing and Map Projection).
a. Many possible projections, and why they are used
b. Describe and compare pros and cons of different projections
6. Please explain the differences between exact and inexact interpolation methods (Spatial data analysis and interpolation).
