GISC3325 Class 2 - Latitude and Longitude on Sphere 28 January 2013
Geodesy Defined The science describing the size, shape and gravity field of the Earth in its threedimensional time-varying states. Branch of applied mathematics concerned with the determination of the size and shape of the earth (geoid). Direct measurements determine the exact location of points on the earth s surface and its external gravity field.
Latitude and Longitude
Latitude and Longitude Latitude can be defined by physical measurements to the stars. Measured with respect to the equator along the meridian. Longitude is equivalent to time. Choice of reference longitude is by convention or treaty.
Definitions Earth s rotation axis Diameter on which earth rotates. Geographic poles Points at which rotation axis cuts the earth s surface. Great circle Circle on the surface of a sphere formed by the intersection of a plane that passes through the center of the sphere.
Example of Geocentric Latitude and Longitude
Meridians Great circles pass through the north and south geographic poles True north is the direction found by standing on a meridian and facing the north geographic pole Meridians are great circles on a sphere but ellipses on an ellipsoid of revolution.
Small Circles and Parallels Small circle - circle on the surface of a sphere formed by the intersection of a plane NOT passing through the center of the sphere. Parallels are small circles on the sphere whose planes are parallel to the plane of the equator. Equator is the only parallel that is a great circle.
Longitude Longitude of a place is its angular distance east or west of the Prime Meridian.
Prime Meridian and my GPS The Prime Meridian is zero degrees longitude but my receiver shows 0deg 0min 5.4sec W. Why is this? How far off is it?
Longitude and Time Time difference between a reference location and your location (determined by astronomic observation e.g local apparent noon) can be used to compute longitude with respect to a reference location. How accurately must time be measured?
The Problem Knowledge of longitude is crucial for navigation in treacherous waters. A fleet returned to England (1707)in foggy conditions. Consensus of navigators was that they were safe. They weren t. Almost 2,000 troops died and 4 of 5 ships sank due to the error. BTW, a sailor who kept his own reckoning of the fleet s location risked his life to warn them. The Admiral had the man hanged for mutiny.
The longitude prize The Parliament promised a prize of 20,000 for the solution to the longitude problem. John Harrison came up with a technology solution in the form of a precise chronometer.
The longitude problem Harrison s H1 and H4 clocks/chronometers H4 is 5.25 inches in diameter. During a two-month voyage from England to Jamaica it lost 5 seconds.
Relation of Time to Angle Time Angle Time Angle 1 hour 15 deg 1 deg 4 min 1 min 15 min 1 min 4 sec 1 sec 15 sec 1 sec 0.0667 sec
Distance on a sphere We can calculate the length of an arc segment (S) on a sphere given a spherical radius (R) and the interior angle (θ). S = R θ
Some computation on a sphere Given the latitude above (in decimal degrees) solve for the distance to the equator along the meridian. We need a spherical radius (R). Let s use R = 6,380,000 meters We convert our latitude to radians (θ) by multiplying it by /180. S = R θ =???
Distance between points on a parallel not the Equator? Let us recollect slide 11. My GPS shows a longitude of 00deg 00min 05.4sec. It should be 0deg 0min 0sec. How far off is my position? The radius of the small circle is NOT equivalent to the equatorial/spherical radius used above (R = 6,380,000 m) Use latitude: N51deg 28min 40.7sec
Solving (Part 1) We need to compute the radius of the small circle (Rp) using simple plane trigonometry. θ = latitude in radians multiply decimal degrees by /180 Rp = R cos(θ)
Solving (Part II) Having solved for the radius at the specified latitude, we use our arc length formula substituting Rp for R λ = difference in longitude (in radians) Sp = Rp λ =???
Geometry of the Sphere A sphere is a three-dimensional figure. We can determine coordinates of points with respect to a Cartesian [XYZ] system with origin at center.
Geocentric XYZ www-gpsg.mit.edu/~tah/12.215/12.215.lec02.ppt - Excellent presentation by Thomas Herring of MIT