# Reference Frames Map Projections And Their Applications Computer Science Essay

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When our civilization is raised on this planet around 5000 - 8000 years ago, the human already had curiosity about the world (for example, the shape and the area of it). Before Raise of Greek, most people just imagine how the shape of the world is. For example, some people thought the Earth to be flat and a Heave of dome spreads over it. Someone even thought Earth is a lunar shape, cone and rectangular. A Greek Philosopher - Anaximander (Borned -550BC) believed the Earth is on the flat and circular top with a short cylinder.

From the left of the model, Europe, Libya and Asia are surrounded by the ring of the ocean. Due to same distance, the top is very stable.

However, after 500BC, most philosophers especially after Pythagoras thought the earth is a spherical. Aristotle, the Great Greek Thinker, attempted to calculate the size of the Earth by measuring tis circumference.

Further, Eratosthenes from Egypt could also calculate the circumference without leaving the country!

Figure 1: Anaximander's Flat World Model

Since he knew the summer solstice at local noon, the sun will appear tropic of cancer in Syene during Summer Solstice. In this time, he used a stick to put the sand straight without shadow. And then using the well, the mid-day sun can shine to the bottom. The conclusion was the sun overhead. However his answer calculated is only 16.3% too large!

Because Eratosthenes is not able to go to centre of the earth himself or use his statellite to calculate the Earth size, he must use the angle of Sun rays then calculate the angle hence the whole size of Earth, Supposing the Earth is completely a Spherical (A Ball!)

In the left diagram, early co-ordinate system may be used. The model is Geocentre that can be referenced by an angle. The angle can be calculated between Alexandria and Syene. Because the Sun ray is perpendicular to the surface on tropic of cancer only plus Alexandria is exactly north From Syene. Although the error is 16% from now, this achievement is really huge without satellite, computer, travelling out from the country and atomic clock

Figure 2: Model used for Calculation by Eratosthenes

After Dark Age and Medieval age, the technology for Astronomy and Measurement are developing, Scientists and surveyors tried to use Triangulation that will be used widely. This method is used for determination of the position of a fixed point. The angles were measured from two other fixed points in a known distance. It is very common without the satellite around 16 - 19th centuries. The network is very wide.

Finally, usage of Satellite and computer are widely increasing after 1950s. The measure is very precise which can be only ten metres under sub-sealevels. One of the satellite calls NASTAR, a kind of Global positioning System Satellite which can also adjust to only a few centimetre.

## a Definition of Coordinate, Reference System

The Following term will be defined before further discuss about Reference System. This system was introduced by Rene Descartes based on orthogonal coordinates, which defined as a set of n co-ordinate, (x, y, z for 3 dimensional). Because x, y and z are planes that all have right angles for each other due to perpendicular. It is very common for reference system.

Firstly, co-ordinate means the relationship between origin and the primary plane. This is a definition for Two-Dimensional Coordinate Systems

Figure 3 and 4: Comparison between the differences between 2D and 3D (Cartesian) of Coordination System. Source: Coordinate Systems Overview, http://www.colorado.edu/geography/gcraft/notes/coordsys/coordsys.html

From these 2 figures, even if more and more dimensions are added with respect to the previous orthogonal planes, they have a right position to define a point. And they can even be conversed back to Cartesian Form from polar form whether it is 2D or 3D. The follow diagram in 3D (Figure 5) is showed that the polar coordinate which are presented in angles and radius only.

## Defining a Good System

In order to define a good co-ordination System, the following factors have to be considered. First is origin, Geocentre is the best origin because it is natural enough (Which doesn't change over billion of years) for heaps of measurement from satellite. Second is the fundamental plane. Horizon plane is the best example.

Note that: there is an angle between X and Y Co-ordinate called Greenwich sidereal time. However, a general relativity due to gravitational field may be account because Earth is orbiting the Sun.

Therefore, Geocentric system is the best way to describe the movement of Satellite orbiting the earth rather than Earth and Sun. Finally is the principal direction, the best example should be geodetic system, where X-axis for Greenwich Meridian intersecting equatorial plane and the point calls Vernal Equinox which is space-fixed system. And the rotation axis of the Earth becomes Z-axis.

Figure 6: A Typical equatorial coordinate system for our Planet earth.

Source: ArcGIS Explorer SDK, http://edndoc.esri.com/arcobjects/9.2/explorer/CustomDrawingWithOpenGL.html

## c)Co-ordinate Transformation and Conversion

So what is the difference between "transformation" and "Conversion"? How do they affect coordinate? Coordinate operation is the change of the system. Firstly, Transformation will be discussed. In this case, Datum will be changed based on different datum but the coordinate system must be same. The parameters could only be used under same coordinate system in empirical way. Therefore, the accuracy for transformation is higher in small area by using mathematical operation.

Datum 1 and Coordinate System A

Transformation

Datum 2 and Coordinate System A

Datum 1 and Coordinate System A

Coordinate Conversion

Datum 1 and Coordinate System B

Figure 7: Process between Coordinate Transformation and Conversion.

Shown from above graph, conversion preserves the same datum but different Coordinate System. Another mathematical rule is applied, for example: map projections. It is very easy to do it just changing the location and orientation for frame definition. This conversion can achieve very high accuracy. Additional information about high and gravity are required.

## a)Realization and Reference Frame

How do Conventional Terrestrial Reference System (CTRS) and Conventional Celestial Reference System (CCRS) use practically? A term calls "frame" is the best way to describe them. How does "frame" apply them?

## b) Concept of the frame with Rotational Matrices

The object of this is to transform a target coordinate from a reference frame to other with using same coordinate system. This can be done by rotational matrix that coordinates a unstable frame to a stable frame with rolling the pitch angles. It is also possible that transforms from a system to a frame or backward. However, fictional force may be considered for more precise results. In fact, Frame is the "realization" of both systems, CTRS and CCRS. Hence CCRS becomes CCRF (Conventional Celestial Reference Frame). Basically, if a set of coordinate is implanted to the system, Frame will be formed. In this process, a series of specific number was provided in a table of coordinates so that it is valid for reference. During realization, earth deform always deforms, for example, tectonic motion, tidal effect, glacial effect, man-made, war and Earthquake. As a result, the motion of the vertices can be up to around 30cm per year. Frame is also used to provide a monitor for systems because the vertices always move due to a series of Earth Deformation. Actually, the frame is very useful for precise Satellite Positioning. The Accuracy for realization is extremely high, HIPPAROCS satellite can achieve up to +/- 0".001.

Figure 8: HIPPARCOS, a satellite that can measure very accuate up to precision of 2 milliarcseconds

Source: From Hipparchus to Hippacos By Catherine Turon http://wwwhip.obspm.fr/heritage/hipparcos/SandT/hip-SandT.html

The following Equations and matrices will be used for transformation between Frames:

In one axis rotation, X"p = Rz (Î³)Xp

Xp = [ xp yp zp]â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.(i)

Rz = [cosÎ³ sinÎ³ 0; -sinÎ³ cosÎ³ 0; 0 0 1]â€¦â€¦â€¦â€¦â€¦â€¦â€¦..(ii)

For roation matrics for x - axis, y-axis and z-axis

Rx(Î±) = [1 0 0; 0 cosÎ± sinÎ±; 0 -sinÎ± cosÎ±]â€¦â€¦â€¦â€¦â€¦â€¦â€¦..(iii)

Ry(Î²) = [cosÎ² 0 -sinÎ²; 0 1 0; sinÎ² 0 cosÎ²]â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦(iv)

Rz(Î³) = [cosÎ³ sinÎ³ 0; -sinÎ³ cosÎ³ 0; 0 0 1]â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦(v)

Any rotation can be represented in 3 x 3 matrix

[x'1 x'2 x'3] = [a11 a12 a13; a21 a22 a23; a31 a32 a33] [x1 x2 x3]â€¦â€¦â€¦â€¦.(vi)

Note: big rotation/transformation and small rotation/transformation will also be considered.

## Properties of Celestial and Terrestrial Reference Frames/system

Terrestrial Reference Frames: This may include earth fixed - Earth Centre. There are so many versions of definition by countries and organisation in past few hundred years. However, the trend has to be unified by a system called World Geodetic System 84 (WGS84). This system was developed by US military system. From now, WGS84 became the Emperor of Terrestrial Reference Frames! Its system is CRS which spins with the Earth. Geocentric, Cartesian, Z - axis in rotation axis and X - axis with intersection of Greenwich and Equatorial Planes

Celestial Reference Frames: The latest celestial datum frame is Fundamental Katalog in German, and it is usually called in FK 1, 2, etcâ€¦There are two Factors calling nutation and precision that can adjust the coordinates so that how they will look like in mid 2010s. Also, brightest star will be used for z-axis for celestial reference frame and system. Proper motion may be gotten Since the stars which look small but they have measurable change. y-axis is the completion for right handed coordinate system and x-axis is Vernal Equinox.

## "Conventional Celestial/Terrestrial Reference System" (CCRS/CTRS) and Transformation

Conventional Terrestrial Reference System: its origin is based on geocentre, z-axis is the Conventional Earth Spinning axis. X-axis is the mean and intersection of Greenwich Meridan and equator. Finally the y-axis is the right handed Cartesian System. And Conventional Celestial Reference System has Geocentre as origin, North Celestial Pole as z-axis, x-axis in Vernal Equinox (first point of Aries) and y-axis is the right handed Cartesian System. These two systems can be realised by a set of coordinates. In CTRS, CTRF can be achieved by rotating the Earth, a number of observatories on the Earth Surface. A relative sense will be described in Geodesy. For CCRS, as described in last chapter, CCRF can be realised by using FK5, a fundamental star coordinates or quasar coordinates.

Figure 9 & 10: Distribution of quasar coordinates for CCRF and Coordination of CTRS/CCRF

Source: http://www.stsci.edu/stsci/meetings/shst2/jannuzib.html & http://civiliancomms.tripod.com/spacecomms/iss/iss_appendix_c.html

## c) International Terrestrial Reference System (ITRS) and International Terrestrial Reference Frame (ITRF) AND International Celestial Reference System (ICRS) and International Celestial Reference Frame (ICRF)

ITRS is defined as a geocentric non-rotating system and coordinate by a spatial rotation that leads to a quasi-Cartesian System using SI system (in metre). It is used to create reference frames that suits for use to measure on or next to the surface of the Earth and ensure the time evolution with no-net-rotation dealing with tectonic motion in the whole planet. ITRF is a realised version by using 3D position observation and GPS tracking station. Different version such as ITRF92, 94 and 2000 are available. And ICRS is based on the movement of the stars and any bodies in the universe by using standard constants, data and algorithms for interpretation for data of Astrometry. Finally, space-fixed coordination system is formed. ICRF can be realised by using extreme precise equatorial coordinates in angular position of radio wave source such as quasar observed

## A lists of Formula for Transformation

Euclidean Transformations will be used

First is Translation by using a set of points of plane, preserving the distance and direction between them. (X,Y) and (x,y)

(x', y') = (x + X, y + Y)â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.(vii)

Second is scaling, this method can make the figure larger or smalle in order to equalise by multiplying the Cartesian coordinate of every point by +ve m.

(x',y') = (mx, my)â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.(viii)

Third is rotation, and it was discussed in Chapter 3

Forth is Reflection. Just like a mirror, (x,y) is a coordinate of a point while (-x,y) are the reflection across the 2nd axis.

b) Example

In general, transformation can be achieved from Geographic to Cartesian with latitude, longitude and height to Cartesian coordinates (X, Y, Z)

Polar to Cartesian Coordinate Transformation

It is also possible to do this by:

(Ï•, Î¸,r) to (x,y,z)

x = r cos(Ï•) cos (Î¸) y = r cos(Ï•) sin (Î¸)

z = r sin (Ï•) X = (v + h) cosÏ• cosÎ»

Y = (v + h) cosÏ• sinÎ»

Z = [v( 1-e2) + h] sinÏ•

## Geodetic Reference System

There are 3 types of height systems: Ellipsoidal, Orthometric and Geoid heights. Ellipsodial height is purely mathematics and is only detected by the GPS. And it is not related to the gravity. And the height is very unique to Reference System. Orthometric height is very useful for Engineering due to independent of the RE. And the data can be obtained from height different observation. This height can be calculated by H = h -N (where H is Orthometric height, N is Geoid height and h is ellipsoidal eight.

Note: Geoid is a 3D surface that supposes all Gravity and Geopotential are all equal around the world and perpendicular to the surface of the Gravity vector. And it links Ellipsoidal and Orthometric height together and approximate Mean Sea Level

## How do the frames apply in Geodesy?

ITRFyy can be transformed to other realisation by using 7-parameter similarity transformation models which is used for accounting crust motion.

The most accurate ECEF is ITRS as GPS surveys connected both ITRF and the modern datums. It was Realised by very precise GPS detection.

Figure 11: The relationship between Three Heights Source: http://users.erols.com/dlwilson/gpsvert.htm

WGS84 (World Geodetic System) is the modern GPS surverying system and maintained/assigned by Cartesian Coordinates. The Master Control station in Colorado helps transfer the coordinate system to users during orbit determination process. It is redefined by using ITRF 2000 (previously is 94, 91 and 84.) It uses the motion of station to track the movement of site coordinate. Originally is was developed by US military datum. Here is the formula for using 7-parameter similarity transformation models:

XB = s Rz (Îº) Ry (Î¸) Rx(Ï‰) XA + T where

T = [Tx Ty Tz]'

## c)Applications in Australia

In Australia, Australian Geodetic Datum which has two versions, AGD 66 and 84 which are used for Pre-satellite saga, at which ANS (Australian National Speroid) is used. Their semi-major axis is 6378000 m and Inverse Flatten is exactly 298.25. When Australia has adapted GPS satellites in 1990s, GRS80 model (Geodetic Reference System) is used. GDA94 is based on this. Its semi-major axis is almost same as ANS'. Vertical datum is used to fix up, down and Z axis, while Horizontal one fix X and Y axis only and also for ellipsoid and the origin.

Without using Satellites, AGD is very useful for surveying by using non-geocentric datum with using thousands of trigonometric control permanent stations across Australia. AGD66 realised only part of Australia and AGD88 for the whole nation. Johnston Trigonometrical Station acted as origin of surveying with ellipsoidal coordinates.

When satellites arrived in 1990s, GDA was used with the accuracy is at least 1ppm or even higher. All Datums were refined to be geocentric and by using ITRF2000 and ITRS-based for more and better accuracy. 3D frame was adjusted for all very precise GPS baseline and fundamental plane and primary axis replaced AGD66 and 84. The height datum still keeps AHD. GDA 94 is very close to WGS84.

Nowadays, Australian National Network contains many nations spread across the nation with 500 km internal. It was Monitored by GNSS and ICSM (Intergovernmetal Committee on Surveying and Mapping) in 1993. And the accuracy is extremely high by using GDA94 and WGS84 together monitored by ITRF2000 and the system together.

Figure 12 and 13: ANN distribution and the model of 3D earth centred for GDA94

Source: http://www.ga.gov.au/earth-monitoring/geodesy/geodetic-datums/GDA/ANN.html

And http://www.ga.gov.au/servlet/BigObjFileManager?bigobjid=GA15778

## a) Concept

In reality, map projection is the way that tries to show the surface on the earth or part of the earth on a flat surface such as 2D map on Computer screen, iphone, GPS and even traditionally, on paper. Because of this, some distortions of area, direction, scale, distance may be discovered during the process. None of the projections is perfect as some projections attempt to minimise the error, but it will also maximise the errors in others too.

For example, if a map wants to preserve direction, azimuths have to be presented in all directions correctly. If the map wants to preserve the area, an equal-area map will be used. If the map in a scale wants to be same in any direction, the projection is conformal.

## Types of Projections

There are three main types of projections: Cylindrical projection, Conic Projection, Azimuthal projection

Cylindrical projection will project the spherical surface to a cylinder. Mercator projection is the most common choice for cylindrical projection because it used the equal area for projection. It also has straight meridians and parallels that intersect at right angles. Because Pure Mercator projection has very high distortion in polar region while there is almost distortion in lower altitude and equator. And cylinder is tangent to the sphere that contacts along a great circle. Therefore transverse Mercator will be used. The cylinder will "sleep" on the floor instead of standing. It Results from projection the sphere onto the cylinder tangent to a central meridian. But the distortion still occurs when it is away from central meridian.

Since then, UTM (Universal Transverse Mercator) is used to define horizon position around the world by dividing the whole world into 60 zones. Each zone has 6 degree. From east to west, the zone contains from one to sixty while the zone zero to 84 from Equator to the North Pole. However, only -80 is available in South Pole. Easting has 500km false while Northing has 10000km false.

Figure 14 and 15: Portrait of Geradus Mercator and his final modified product after him: UTM

Azimuthal projection is very useful for air-route distances. Any distances measuring from the centre are all true, although the distortion increases away from the centre point. And Lambert Azimuthal Equal Area contains larger ocean areas. Meridian is straight while others are curve. Generally they are almost same. It is useful for Australia which is a large country occupies the whole smallest continent in the world.

For Conic Projections, scale and distance will be distorted except the standard parallels. And the areas are proportional with exact true direction. In Australia, ACRESLC is based on WGS83, it has parallels at 18 and 36 degrees south and central meridian at 132 degree east. It also has 27 degree south of latitude. And GALCC (Geoscience Australia Lambert Conformal Conic Projection) is used based on GDA94 with 18 and 36 degree south and central meridian at 134 degrees east.

Figure 16: Australia under Lambert Conformal Conic Projection, source: http://www.icsm.gov.au/mapping/map_projections.html

## Better system and any future trends?

As technology is more advanced than the past, GIS has been advancing to the best evolution. The use of internet database can result from geo-spatial application more accurate and accelerate. Users can use Geo-database anytime, anywhere around the world. For Coordinate system, a 3D system will be commonly used that involves X, Y plus Z. This kind of model calls Digital Elevation Model (DEM) that expresses the topographic trends and distribution. Z is responsible for reference for Geode above sea level. The trend of Mapping is transferring from computer (display focus) to multimedia, that means using fast internet and bring full view of our world. 3D visualisation may make the mapping for flexible and dynamic. Newer and newer Geo-referencing framework may solve more complex problem of space. Finally, Clould-computing can use for resource virtualisation, more dynamic and acts as service.