These notes have been prepared by the Department of Geomatics
at the University of Melbourne to facilitate the understanding
of some of the concepts being presented during the three Horizons
programs featuring geomatics.
The fundamental basis of much of the material is the planar triangle,
and the relationship between the angles and sides. This geometry
is used to map and position objects and features on and below
the earth's surface, and some basic applications are presented
in the program. Detailed concepts are avoided, although some are
alluded to as the science of geomatics involves considerable high
order mathematics.
The Horizons program is supported by a World Wide Web site, located
at http://www.sli.unimelb.edu.au/Horizons These notes are available
as hypertext documents at the site, as well as additional information
on the program content and other programs being undertaken in
the Department. The site is also connected to the Department's
Home Page, which gives further details on the academic programs
offered, the staff, the research and so on. Further information
on careers in the geomatics industries is available from the Department
on (03) 9344 6806.
These notes are intended to give an overview of the concepts that
students will need to understand some of the concepts being presented
in the programs. They have been kept deliberately simple so they
can be used as a handout if necessary. It is accepted that teachers
will already understand many of these concepts.
In addition to these teachers' notes some of the relevant notes
that accompany the lectures series in Geomatics Science 1 have
also been made available. They can be found either as WWW documents
or Word 6 documents at http://www.sli.unimelb.edu.au/Horizons/Documents,
or by following this link. They are NOT stand alone publications,
they have been produced to accompany a lecture series which goes
into more detail on some topics and less on others. If they are
copied or used due credit to their origin would be appreciated.
Further background information of the history of Geomatics can
be found here at intro.html.
Angle measurement
The concept of angle:
The concept of angle is one of the most important concepts in
geometry. The subject of trigonometry is based partially on the
measurement of angles, partially on the measurement of distances.
There are two commonly used units of measurement for angles. The
most common unit of measurement is that of degrees. A circle is
divided into 360 equal degrees, so that a right angle is 90 degrees.
Degrees may be further divided into minutes and seconds. Each
degree is divided into 60 equal parts called minutes. So five
and a half degrees can be called 5 degrees and 30 minutes. Each
minute is further divided into 60 equal parts called seconds.
The division of degrees into minutes and seconds of angle is the
same as the division of hours into minutes and seconds of time.
A circle therefore has 359°59'60" (360°)
The other common measurement for angles is radians. For this exercise
we will ignore them as they are only rarely present in calculations,
but it is the unit needed by computers to perform trigonometric
calculations. There are 2xPI radians per 360°.
Trigonometry as computational geometry:
Trigonometry began as the computational component of geometry.
For instance, a statement of plane geometry states that a triangle
is determined by a side and two angles. In other words, given
one side of a triangle and two angles in the triangle, then the
other two sides and the remaining angle are determined. Trigonometry
includes the methods for computing those other two sides. (The
remaining angle is easy to find since the sum of the three angles
equals 180 degrees.)
Trigonometric functions such as sine, cosine and tangent are used
in computations in trigonometry. These functions relate measurements
of angles to measurements of associated straight lines.
Right Angled Triangles
Basic Sines, Cosines and Tangents
sin Ø = length of opposite side/length of hypotenuse
cos Ø = length of adjacent side/length of hypotenuse
tan Ø = length of opposite side/length of adjacent side
(S=OH, C=AH, T=OA, OR Some Old Hags Can't Always Hide Their Old
Age)
Let's have a convention for labelling the parts of a right triangle. Let the right angle be labelled C and the hypotenuse c. Let A and B denote the other two angles, and a and b the sides opposite them respectively.
|
cos A = b/c (adj/hyp)
tan A = a/b (opp/adj) |
 |
Solving right angled triangles
We can use the Pythagorean theorem and properties of sines, cosines,
and tangents to find unknown parts in terms of known parts.
Sines: sin A = a/c, sin B = b/c. Cosines: cos A = b/c, cos B = a/c. | Sine Rule: a/sin A = b/sin B = c/sin C Cosine Rule: a2 = b2 + c2 - 2bc cos A |
Let's first look at some cases where we don't know all the sides. Suppose we don't know the hypotenuse but we do know the other two sides. The Pythagorean theorem will give us the hypotenuse.
|
For instance, if a = 10 and b = 24, then c2 = a2 + b2 = 102 + 242 = 100 + 576 = 676. The square root of 676 is 26, so c = 26.
|
Now suppose we know the hypotenuse and one side, but have to find the other.
For example, if b = 119 and c = 169,
then a2 = c2 - b2 = 1692 - 1192 = 28561 - 14161 = 14400,
and the square root of 14400 is 120, so a = 120.
We might only know one side but we also know an angle.
For example, if the side a = 15 and the angle A = 41 degrees, we can use a sine and a tangent to find the hypotenuse and the other side.
|
Since sin A = a/c, we know c = a / sin A = 15 / sin 41.
Using a calculator, this is 15 / 0.6561 = 22.864.
Also, tan A = a/b, so b = a / tan A = 15 / tan 41 = 15 / 0.8693 = 17.256.
|  |
Whether you use a sine, cosine, or tangent depends on which side and angle you know.
For solutions to non right angle triangles use the sine and cosine rule, and see examples on page 6 of how to map a baseline map.
For more information see the lecture note series funda.html (or intro.doc).
Why use maps ?
General maps display the spatial relationship of the geographical
features of a landscape, for example the roads, rivers and buildings.
They provide a visual representation of the information describing
the features on the landscape, which is easier to understand than
facts and figures about each area. There are many types of maps
which are produced for different purposes. Maps are required for
navigation, mining, town planning and engineering projects such
as the construction of tunnels and bridges.
Specialised maps like Melways maps display roads, shopping centres,
schools and sporting grounds to provide the means to locate facilities
and navigate around Melbourne's network of roads.
A thematic map displays the spatial variation of a particular theme, such as the population density, rainfall, vegetation types or climate.
A chart is a type of map used by pilots of ships and planes for
navigation, it includes features like the depth of the water in
a port or the appropriate symbols to allow navigation in controlled
airspace.
Map features
When reading a map the following features will help to interpret what the objects are; their size, distance, and height above sea level. A north point provides a reference direction in which to orientate the map. The legend or key describes what object each symbol represents. The scale allows the determination of the size of the object and their distance from other objects, (scale = map distance / true distance, in the ratio 1:x).
The scale can be represented in two forms, as a numeric or a bar
scale. A numerical scale is a ratio, for example the atlas scale
is 1:1,000,000. A bar scale is a graphical representation of a
ground distance plotted to scale, with the advantage that the
map scale remains true when the map itself is enlarged or reduced
in photocopying.
How maps are made
The data for map production is collected by many different ways,
depending on the required accuracy and scale of the map. Field
survey, aerial photogrammetry and satellite imagery are some of
the methods employed to gather information of the spatial relationship
of the features
A topographic map is generally produced from a series of overlapping
aerial photographs, which when viewed as pairs can be seen in
three dimensions. Think of our eyes which view the world in three
dimensions, our right eye sees more of the right hand side of
the view, and the left eye sees more of the left hand side of
the view. The difference between the two 'views' is known as the
parallax. The brain combines the two images which produces our
perception of depth. If we were to replace the 'eyes' with cameras,
and we were to look at each photograph from the right hand position
with our right eye (and so on), then we can see three dimensions.
If one pair of the overlapping aerial photographs are placed in
a stereoplotter then the human brain combines them to recreate
the image in the three dimensions.
The cartographer, using the stereoplotter, can follow and trace
the course of rivers, roads, and surface heights and also identify
structures such as bridges and buildings. This is used to produce
the features on the map such as contour lines.
For more information see the lecture note series pandrs.html (or
pandrs.doc)
The concept of a baseline map has been used through the program
to indicate the use of the fundamental triangle for measurement
of large features on the earth's surface. Geomatic engineers do
not actually make 'baseline' maps, more sophisticated instruments
allow the measurement of three dimensional vectors which are then used to derive coordinates for the features.
In base line maps the sine and cosine rules are used to calculate
missing values in the system. For more information please consult
plane.html.
For information on angle measurement please see angle.html.
Examples For a Baseline Map:
|
This leaves you with enough information to calculate the location of your target ( and also gives a triangular shape)
Example 1:
Step 1:
The easiest calculation to make is that of the last interior angle of the triangle, C.
C = 180o - 67 o - 37 o = 80 oNow that we know all the interior angles we can start calculating the distances from the points A and B to the target.
Step 2:
Using the sine rule:
20/sin 80o = a/sin 63o = b/sin 37oAs more targets are measured to, the diagrams become more complex to the eye but the mathematics stays the same.20.3085 = a/0.891
20.3085*0.891 = a
a = 18.0948m
20.3085 = b/0.6018
20.3085*0.6018 = b
b = 12.2216m
Even the most complicated baseline observation diagrams are no
harder to do than the original single triangle, as you only use
one triangle at a time to calculate the location of the target.
For further information see planemap.html or planemap.doc
Distance Measurement
The basis of modern distance measurement uses the propagation
of electromagnetic waves in order to measure distances. The fundamental
basis is simple, if you know how fast something travels and for
how long it travels then you can work out the distance:
|
|
Surveyors often use either light waves (laser) or radio waves
(used with GPS) to measure distances, and generally the speed
of light is known (and denoted by 'c'). So if we bounce a light
wave off a reflector and measure the small amount of time taken
for the light to travel to the reflector and back again, the equation
becomes:
|
|
The actual process is a little more complicated as phase measurement
is used to determine the time component, but basically that is
it.
GPS Global Positioning System) -
NAVSTAR GPS (NAVgation System with Time And Ranging - Global Positioning
System) was designed for American military applications, to provide
positioning of troops, air and sea craft and accurate missile
firing. The constellation of satellites is made up of 24 satellites,
four satellites travel in each of the six orbital planes which
provides global coverage 24 hours a day (see diagram). The system
is available 24 hours per day in all weather, and is made available
in a less accurate mode for civilian use. This has had a major
impact on the modern science of geomatics.
Each GPS satellite transmits a signal to Earth containing information
about the satellite's location in space. When the location of
at least four satellites is known, the position on Earth can be
calculated (a satellite for each of the unknown dimensions, X,
Y, and Z plus an additional satellite for the error in the satellite
message). The position calculation is very complex, but is based
on the determination of the distances from the receivers to the
satellites (using knowledge on the propagation of electromagnetic
waves).
Surveyors use GPS to determine precise locations for the construction
of buildings, tunnels, and bridges, the excavation of mines, calculating
the depth of the seabed for dredging of ports or oil and gas exploration.
GPS also has many other applications, such as navigation of ships
and aeroplanes, tracking and monitoring of fleet services and
recreational uses like hiking and boating.
See gps.html for more details
Geographic Information Systems -
Geographic Information Systems (GIS) are a database system which
is spatially referenced to a map or plan. Information such as
vegetation types, rainfall, population, income or industry are
overlayed and manipulated to produce visual, geographically related
information to support decision making for business and government.
GIS has endless applications, for example, GIS can predict where bushfire threats are greatest, using information about the vegetation and the history of rainfall in the area. This enables bushfire awareness and preventative programs to be established in areas of greatest danger.
GIS can determine areas where there is demand for new businesses, such as a McDonalds restaurant. This is done by assessing trends in the population, such as age, employment, transport, income and shopping habits as well factors such as the location of competing McDonalds and other take away fast food restaurants. This will determine whether a new McDonalds restaurant will be profitable and therefore a viable option.
Photogrammetry and Remote Sensing -
Photogrammetry is the process of obtaining reliable information from photographs. This process dates back to 1839 when the first attempt was made to make a topographic map from a photograph.
Surveyors use photogrammetry to determine distances, elevations,
areas and volumes of the landscape. There are two main types of
photogrammetry, aerial and terrestrial. Aerial photogrammetry
is used for mapping and determination of the location of control
points. Terrestrial photogrammetry is Earth based, some examples
are the monitoring of a dam wall or volcano for movement, or the
survey and recording of historically important features such as
Rock Art paintings.
Remote sensing is similar to aerial photogrammetry in that it is taken above the Earth's surface. However remotely sensed images are captured by satellites, and the images are digital in nature and are capable of recording wavelengths of light to visible to the human eye.
Remote sensing is used to identify and monitor changes on the Earth, such as pollution, surface temperature, vegetation and weather patterns.
Information from photogrammetry and remote sensing techniques can be overlayed into a GIS.
Global Coordinate System
Latitude and longitude form a geographical coordinate system used
for locating places on the surface of the earth. They are angular
measurements, expressed as degrees of a circle measured from the
center of the earth. The earth's axis, which intersects the surface
at the north and south poles, is the origin for the spherical
grid of latitude and longitude.
Latitude
At the point of equal distance to the poles lies the equator. Latitude is expressed in degrees north or south of the equator. Latitude runs from 0 at the equator to 90N or 90S at the poles. Lines of latitude run in an east-west direction. They are called parallels because they run parallel to each other.
Longitude
Lines of longitude are called meridians. They run in a north-south
direction from pole to pole. Longitude is the angular measurement
of a place east or west of the prime meridian. The Prime meridian
is also known as the Greenwich Meridian because it runs through
the original site of the Royal Observatory Greenwich just outside
London, England. Longitude runs from 0° at the prime meridian
to 180° east or west, halfway around the globe.
Latitude and Longitude are measured in degrees, minutes and seconds.
One degree of latitude equals approximately 111 km. One minute
is just over a mile, and one second is around 35 metres. Because
meridians are not parallel (they converge at the poles) the length
of a degree of longitude varies, from 111km at the equator to
0 at the poles (longitude becomes a point at the poles).
The north and south poles are the earth's geographic poles, located
at each end of its axis of rotation. All meridians of longitude
meet at these poles. A compass needle points to either of the
earth's two magnetic poles, not to the geographic poles. The north
magnetic pole is located in the Queen Elizabeth Islands group,
in the Canadian Northwest Territories. The south magnetic pole
lies near the edge of the continent of Antarctica, off the Adelie
Coast. The magnetic poles are constantly moving.
 Go back to the Horizons Home Page
|
 Go on to Further Background Information
|
