The Earth’s Shape, and How We Shape It.

1. The Earth’s Shape, and How We Shape It.

2. Outline • Georeferencing • Spherical Coordinate System • Geographic Coordinate System • Is the Earth a Perfect Sphere? • Oblate Spheroid • Geoid • Reference Ellipsoid • Map Projections

3. Georeferencing • Georeferencing defines an existence in physical space, establishing its location in terms of map projections or coordinate systems.

4. 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.

5. 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)

6. 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°.

7. 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 =

8. 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.

9. 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).

10. 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°.

11. 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.

12. 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.

13. But…….. • Is the Earth a perfect sphere?

14. 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) .

15. 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.

16. 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.

17. ?? • 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.

18. Geoid • Geoid takes a gravitational map, and then generates a mean value for numerous segments all around the earth.

19. Geoid

20. Geoid

21. Reference Ellipse • Is a is a mathematically-defined oblate spheroid that is a "best-fit" to the geoid.

22. 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.

23. 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.

25. 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) Polar radius (m) Inverse flattening Where used • Everest (1830) 6,377,299.365 6,356,098.359 300.80172554 India • Hayford (1910) 6,378,388 6,356,911.946 297 USA • South American(1969) 6,378,160 6,356,774.719 298.25 South America • WGS-72 (1972) 6,378,135 6,356,750.52 298.26 USA/DoD • GRS-80 (1979) 6,378,137 6,356,752.3141 298.257222101 Global ITRS • WGS-84 (1984) 6,378,137 6,356,752.3142 298.257223563 Global GPS

26. 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.

27. Map Projections • There are three types of map projecting that are widely used: • Cylindrical Projections • Conical Projection • Azimuthal/Plannar Projections

28. 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.

29. Cylindrical Projection

30. 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.

31. Conic Projection

32. 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.

33. Planner Projection

34. Projections

35. -FYI-

36. Overview

37. So πr²? Noo, πr ROUND!

38. Works Cited • 1. Alpha, Tau Rho., and DaanStrebe. Map Projections. Menlo Park, CA: U.S. Geological Survey, 1991. Print. • 2. Buckley, Aileen. "Mapping Center : Tissot'sIndicatrix 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-map- projection-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>.

39. 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-9-4-3- inertial-systems-technologies-earth-models>.

40. 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 E- Education Institute! | John A. Dutton E-Education Institute. 2007. Web. 19 Apr. 2011. <https://www.e- education.psu.edu/lidar/l3_p4.html>.