Transcript Slide 1

Outline
 Georeferencing
 Spherical Coordinate System
 Geographic Coordinate System
 Is the Earth a Perfect Sphere?
 Oblate Spheroid
 Geoid
 Reference Ellipsoid
 Map Projections
Georeferencing
 Georeferencing defines an existence in physical space,
establishing its location in terms of map projections or
coordinate systems.
Georeferencing
 When describing our earth we reference sets of points,
lines, and/or surfaces.
So we must define a metric space, using the positions
of the points in the space.
Spherical Coordinate System
 Defined as (r, θ, λ).
 We set two orthogonal directions, the Zenith and the Azimuth. Such that
an origin point can be placed at the intersection.
 A reference plane is added and contains both the origin and Azimuth, and
is perpendicular to the zenith.
-Let P be a point on the sphere.
 The angle between the zenith and the point is called the inclination (or normal
angle) described in either degrees or radians with 0° ≤ θ ≤ 180° (π rad) .
 The angle between the azimuth direction and the reference point from the
right side is the azimuthal angle described in degrees or radians with
0° ≤ λ < 360° (2π rad).
(For all r >0)
Plotting a point from its spherical coordinates (r, θ, λ)
 Move r units from the origin in the zenith direction.
 Rotate by θ about the origin towards the azimuth reference direction.
 Rotate by λ about the zenith in the proper direction.
 (2,45°,300°)
The red sphere shows the points with r = 2,
the blue cone shows the points with inclination
(or elevation) θ = 45°, and the yellow half-plane
shows the points with azimuth λ = 300°.
Spherical Coordinate System
 The three spherical coordinates are converted to
Cartesian coordinates by:
 x = r sin(θ)cos(λ)
y = r sin(θ)sin(λ)
z = r cos(θ)
 Conversely, Cartesian coordinates are converted to
spherical coordinates by:
 r=
θ=
λ=
Geographic Coordinate System
The most common coordinate systems in use for geography is the
Geographic Coordinate System, and is used by mathematicians and
physicists for many earth related applications.
Geographic Coordinate System
Consists of:
 Lines of latitude running parallel to the equator and divide the
earth into 180 equal portions from north to south (or south to
north). The reference latitude is the equator and each
hemisphere is divided into 90 equal portions, each representing
one degree of latitude.
 Lines of longitude run perpendicular to the equator and
converge at the poles. The reference line of longitude is the
prime meridian, and runs from the north pole to the south pole
through Greenwich, England. Subsequent lines of longitude are
measured from zero to 180 degrees east or west of the prime
meridian (values west of the prime meridian are assigned
negative values).
Geographic Coordinate System
 In general r in the spherical system is simply dropped
due to a fixed value representing elevation or altitude.
 Latitude then becomes the complement of the zenith
λ= 90°- θ
with a domain -90° ≤ θ ≤ 90°.
 Longitude is the azimuth angle
shifted 180° from θ
with a domain of -180° ≤ λ≤ 180°.
Geographic Coordinate System
 Longitude 80 degree East and latitude 55 degree
North.
Degrees of latitude and longitude can also be subdivided into minutes and
seconds of a degree for more precision.
There are 60 minutes (') per degree, and 60 seconds (") per minute.
 It is difficult to determine the lengths of the latitude lines, because
they are concentric circles that converge to a single point at the poles
where the meridians begin.
 At the equator, one degree of longitude is approximately 111.321
kilometers, and at 60 degrees of latitude, one degree of longitude is
only 55.802 km.
 Therefore, there is no uniform length of degrees of latitude and
longitude. Hence, the distance between points cannot easily be
measured accurately using angular units of measure.
But……..
 Is the Earth a perfect sphere?
Is the Earth a Perfect Sphere?
 No! The rotation of the Earth causes a slight bulge toward
the equator, making it actually a bit wider than it is tall.
“Oblate Spheroid”
 The diameter of the Earth
at the equator (12,756km) is
about 42km greater than the
diameter through
the poles (12,714km) .
Oblate Spheroid
 An oblate spheroid is a rotationally symmetric
ellipsoid having a polar axis shorter than the diameter
of the equatorial circle whose plane bisects it. It is
shaped by spinning an ellipse about its minor axis,
making an equator with the end points of the major
axis.
Oblate Spheroid
 Some basic spheroid equations
 Implicit Equation: Centered at the "y" origin and rotated
about the z axis.
 Surface Area:
with
 Volume:
Where a is the horizontal, transverse radius at the equator, and b is the vertical.
??
 Now that we know that the earth is not a perfect
sphere, is it a perfect ellipsoid?
Looking at the many massive mountains and deep sea levels, this is clearly not the case.
Geoid
 Geoid takes a gravitational map, and then generates a
mean value for numerous segments all around the
earth.
Geoid
Geoid
Reference Ellipse
 Is a is a mathematically-defined oblate spheroid that is
a "best-fit" to the geoid.
Reference Ellipse
The difference between the sphere and the reference
ellipsoid is very small, only about one part in every
300.
 The flattening factor is generally computed using
grade measurements, but other surveying techniques
such as meridian arcs, satellite geodesy, and the
analysis and interconnection of continental geodetic
networks have been used such as the ellipsoid radii of
curvature.
Reference Ellipse
 We know that oblate ellipsoids have constant radius of
curvature along axes, but varying curvature in any
other direction therefore, oblate spheroids have limits
to their radii of curvature: With no radii being larger
than a²/b and none being less than b²/a.
Some Facts
 The GPS receivers use the reference ellipsoid, so the
number you see on the screen is the elevation above the
ellipsoid and not the real sea level.
 Some famous numbers for the reference ellipsoid:







Reference
ellipsoid name
Equatorial radius (m)
Everest (1830)
6,377,299.365
Hayford (1910)
6,378,388
South American(1969) 6,378,160
WGS-72 (1972)
6,378,135
GRS-80 (1979)
6,378,137
WGS-84 (1984)
6,378,137
Polar radius (m)
6,356,098.359
6,356,911.946
6,356,774.719
6,356,750.52
6,356,752.3141
6,356,752.3142
Inverse flattening
300.80172554
297
298.25
298.26
298.257222101
298.257223563
Where used
India
USA
South America
USA/DoD
Global ITRS
Global GPS
Map Projections
 Map projections are attempts to portray the surface of
the earth or a portion of the earth on a flat surface. Some
distortions of distance, direction, scale, and area always
result from these processes, but the different projections
minimize distortions of these properties at the expense of
maximizing errors in others.
Map Projections
 There are three types of map projecting that are widely
used:
 Cylindrical Projections
 Conical Projection
 Azimuthal/Plannar Projections
Cylindrical Projection
 Meridians are mapped to equally spaced vertical lines and
parallels are mapped to horizontal lines. The mapping of
meridians to vertical lines can be visualized by imagining
a cylinder wrapped around the Earth and then projecting
onto the cylinder, then un raveling that cylinder. This
projection stretches distances east-west. The amount of
stretch is the same at any chosen latitude on all cylindrical
projections.
Cylindrical Projection
Conic Projection
 A conic projection distorts the scale and distance
except along standard parallels. Areas are proportional
and directions are true in limited areas. Used in the
United States and other large countries with a larger
east-west than north-south extent.
Conic Projection
Planner Projection
 Azimuthally projections hold the strong property that
directions from a central point are always preserved.
Typically these projections have radial symmetry,
hence the distortions in map distances from the
central point can be computed by setting a function
£(c) of the true distance c. The mapping of radial lines
can be visualized by imagining a plane tangent to the
Earth, with the central point as tangent point.
Planner Projection
Projections
-FYI-
Overview
So πr²?
Noo, πr ROUND!
Works Cited
 1. Alpha, Tau Rho., and Daan Strebe. Map Projections. Menlo
Park, CA: U.S. Geological Survey, 1991. Print.
 2. Buckley, Aileen. "Mapping Center : Tissot's Indicatrix Helps
Illustrate Map Projection Distortion." Esri News | Esri Blogs
for the GIS Community. 2003. Web. 19 Apr. 2011.
<http://blogs.esri.com/Support/blogs/mappingcenter/arch
ive/2011/03/24/tissot-s-indicatrix-helps-illustrate-mapprojection-distortion.aspx>.
 3. Dana, Peter H. "Map Projections." University of Colorado
Boulder. 2000. Web. 19 Apr. 2011.
<http://www.colorado.edu/geography/gcraft/notes/mappr
oj/mapproj_f.html>.
Works Cited
 4. Drbohlav, Zdenek. "Aquariu.NET Documentation." Aquarius.NET Main
Page. 2002. Web. 19 Apr. 2011.
<http://www.mgaqua.net/AquaDoc/Projections/Projections_Conic.aspx
 5. Erickson, Jon. Making of the Earth: Geologic Forces That Shape Our Planet.
New York: Facts on File, 2000. Print.

 6. Grafarend, Erik W., and Friedrich W. Krumm. Map Projections:
Cartographic Information Systems. New York: Springer, 2006. Print.

 7. Grewal, Mohinder S. "Chapter 9.4.3: INERTIAL SYSTEMS TECHNOLOGIES:
Earth Models On GlobalSpec." GlobalSpec - Engineering Search &
Industrial Supplier Catalogs. 2007. Web. 19 Apr. 2011.
<http://www.globalspec.com/reference/14801/160210/chapter-9-4-3inertial-systems-technologies-earth-models>.
Works Cited
 8. Neutsch, Wolfram. Coordinates. Berlin: De Gruyter, 1996.
Print.
 9. Olds, Shelley. "Education and Outreach - Tutorial: The Geoid
and Receiver Measurements | UNAVCO." UNAVCO
Homepage | UNAVCO. 2011. Web. 19 Apr. 2011.
<http://www.unavco.org/edu_outreach/tutorial/geoidcorr.
html.
 10. Shuckman, Karen. "Geodesy, Datums, and Coordinate
Systems | GEOG 497L: LIDAR." Welcome to the EEducation Institute! | John A. Dutton E-Education Institute.
2007. Web. 19 Apr. 2011. <https://www.eeducation.psu.edu/lidar/l3_p4.html>.