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

Slide 1

The Earths Shape, and How We Shape It.

OutlineGeoreferencingSpherical Coordinate SystemGeographic Coordinate SystemIs the Earth a Perfect Sphere?Oblate SpheroidGeoidReference EllipsoidMap Projections

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

Example: where are you located in this class room; 4th row 3rd column from the right.3When 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.

GeoreferencingSpherical Coordinate SystemDefined as (r, , ).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 0 180 ( 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 < 360 (2 rad). (For all r >0)

Draw out as explained in each bullet. (Figure 1)5 Plotting a point from its spherical coordinates (r, , )Move r units from the origin in the zenith direction.Rotate by about the origin towards the azimuth reference direction.Rotate by 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) = 45, and the yellow half-plane shows the points with azimuth = 300.

and lose their significance levels when r = 0, and loses significance if sin() = 0 (at = 0 and = 180.)6Spherical Coordinate SystemThe three spherical coordinates are converted to Cartesian coordinates by: x = r sin()cos() y = r sin()sin() z = r cos()Conversely, Cartesian coordinates are converted to spherical coordinates by:r = = =

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 SystemConsists 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).

Latitude is the angle in which a point is elevated above or below the equator plane. Measured from the center of the Earth.Longitude tells us how far east or west of the prime meridian a point is.

The equator is naturally defined by the Earths rotation, the prime meridian is arbitrary. Thats why we must agree on a prime meridian before we can use this system.

9Geographic Coordinate SystemIn 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 = 90- with a domain -90 90. Longitude is the azimuth angle shifted 180 from with a domain of -180 180.

Geographic Coordinate SystemLongitude 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. Example, a coordinate might be written (65 32' 15)Degrees as decimals: 65.5375Degrees and decimal minutes: 65 32.25' Degrees, minutes, and decimal seconds: 65 32' 15.275"11It 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.

A degree of latitude is approximately 69 miles, a minute is about 1.15 miles, and a second is about 0.02 miles (roughly 100 feet).Longitude:At the equator, it is about 111km (69 miles), the same size as a degree of latitude.At a latitude of 45 degrees, a degree of longitude is about 79km (49 miles).

12But..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 SpheroidThe diameter of the Earthat the equator (12,756km) is about 42km greater than the diameter through the poles (12,714km) .

If you were standing on the moon, looking at the Earth, it would be virtually impossible to see the bulge and the Earth would appear to be a perfect sphere. 14Oblate SpheroidAn 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 SpheroidSome basic spheroid equationsImplicit 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 So far, we have made the simplifying assumption that the Earth is a perfect sphere. Looking at the many bumps and hills we see, this is clearly not the case.16 ??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.

So we try to draw even out those curves and drops out with what we call a Geoid.17 GeoidGeoid takes a gravitational map, and then generates a mean value for numerous segments all around the earth.

18 Geoid

Here you can see the generated geoid compared to a normal sphere, and an ellipsoid.19 Geoid

Blue is roughly -80 meters below geoid, and red is roughly 80m above.

Now since this figure is still not a smooth shape, for mathimatical and computational reasons we carefully select a refenece ellipsoid that acts as a best fit line about the geoid. Reminds me of a best fit line in a scatter plot in R2.

20Reference EllipseIs 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 EllipseWe 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 a/b and none being less than b/a.

Draw picture23

Green covers the hole earth geoid vs reference ellipsoid.

24Some FactsThe 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 usedEverest (1830) 6,377,299.365 6,356,098.359300.80172554IndiaHayford (1910) 6,378,388 6,356,911.946297 USASouth American(1969) 6,378,160 6,356,774.719298.25 South AmericaWGS-72 (1972) 6,378,135 6,356,750.52298.26 USA/DoDGRS-80 (1979) 6,378,137 6,356,752.3141298.257222101Global ITRSWGS-84 (1984) 6,378,137 6,356,752.3142298.257223563Global GPS

The shape of the ellipsoid is given by the flattening, which indicates how much the ellipsoid departs from spherical. (the reciprocal of the flattening is set at 298.257223563 exactly for GPS.)

25Map ProjectionsMap 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 ProjectionsThere are three types of map projecting that are widely used:Cylindrical ProjectionsConical ProjectionAzimuthal/Plannar Projections

Cylindrical ProjectionMeridians 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 ProjectionA 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 ProjectionAzimuthally 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

Just for a clear recognition.34 -FYI-

FYI Topology has some very strong applications in map projection. 35Overview

Over all earth is shaped pretty funky, we always thought of it as a globe. But after estimations and calculations we come to our shape of the earth. And have found ways to put it on our walls : )36 So r? Noo, r ROUND!

Thanks CC!37Works Cited1. 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. .

3. Dana, Peter H. "Map Projections." University of Colorado Boulder. 2000. Web. 19 Apr. 2011. .

Works Cited4. Drbohlav, Zdenek. "Aquariu.NET Documentation." Aquarius.NET Main Page. 2002. Web. 19 Apr. 2011.