Surveying geodesy ajith sir
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- 1. The Shape of the Earth The Earth is not a perfect sphere Equatorial diameter slightly greater than polar diameter Earth is an oblate ellipsoidslightly flattened The geoid exaggerates small departures from spherical
2. The Earths Rotation Earth rotates on its axis: Counterclockwise at North Pole Left to right (eastward) at Equator One rotation is a solar day (24 hours) Axis: an imaginary straight line through the center of the Earth around which the Earth rotates Poles: the two points on the Earths surface where the axis of rotation emerges 3. The Earths Rotation Environmental Effects of the Earths Rotation: Day and night Fluctuating air temperature Tides 4. The Geographic Grid Parallels and Meridians Geographic grid: network of parallels and meridians used to fix location on the Earth Parallel: east-west circle on the Earths surface, lying on a plane parallel to the equator Meridian: north-south line on the Earths surface, connecting the poles 5. The Geographic Grid Parallels and Meridians Equator: Parallel of latitude lying midway between the Earths poles; it is designated latitude 0 Longest parallel of latitude Midway between poles Fundamental reference line for measuring position Latitude: arc of a meridian between the equator and a given point on the globe Longitude: arc of a parallel between the prime meridian and a given point on the globe 6. The Geographic Grid Latitude and Longitude Latitude is measured north and south of the equator, up to 90 Longitude is measured east and west of the Prime Meridianmeridian that passes through Greenwich, Englandup to 180 7. Visualizing Physical Geography Copyright 2008 John Wiley and Sons Publishers Inc. Earths Revolution Around the Sun Tilt of the Earths Axis Earth has seasons because of the tilt of the axis. Axis aims toward Polaris (North Star) Axis tilted at an angle of 23 from a right angle to plane of the ecliptic Plane of the Ecliptic: plane of the Earths orbit around the Sun 8. Earths Revolution Around the Sun The Four Seasons Earths axis tilted toward North Star throughout Earths orbit. December 22: N hemisphere tilted away from the sun at the maximum angle June 21: N hemisphere tilted toward the sun at the maximum angle 9. Earths Revolution Around the Sun The Four Seasons Summer solstice: solstice occurring on June 21 or 22, when the subsolar point is at 23 1/2 N; June Solstice Winter solstice: solstice occurring on December 21 or 22, when the subsolar point is at 23 1/2 S; December Solstice Equinox: time when subsolar point falls on equator and circle of illumination passes through both poles Circle of illumination: separates day hemisphere from night hemisphere 10. Earths Revolution Around the Sun Equinox Conditions Subsolar point: point on the Earths surface where the sun is directly overhead at noon Circle of illumination passes through both poles Subsolar point at equator Day and night of equal length everywhere on the globe Occurs twice per year Vernal Equinox: March 21 Autumnal Equinox: September 23 11. Earths Revolution Around the Sun Solstice Conditions Circle of illumination grazes Arctic and Antarctic Circles June Solstice: north pole has 24 hours of daylight; daylength increases from equator to north pole December Solstice: south pole has 24 hours of daylight; daylength increases from equator to south pole 12. Earths Revolution Around the Sun Earth revolves around the sun every 365.242 days Orbit is an ellipse Leap year corrects for the extra quarter day Orbit is counterclockwise Perihelion: point in orbit when Earth is closest to Sun Aphelion: point in orbit when Earth is farthest from Sun 13. A great-circle arc, on the sphere, is the analogue of a straight line, on the plane. Where two such arcs intersect, we can define the spherical angle either as angle between the tangents to the two arcs, at the point of intersection, or as the angle between the planes of the two great circles where they intersect at the centre of the sphere. (Spherical angle is only defined where arcs of great circles meet.) 14. A spherical triangle is made up of three arcs of great circles, all less than 180. The sum of the angles is not fixed, but will always be greater than 180. If any side of the triangle is exactly 90, the triangle is called quadrantal. 15. Set up a system of rectangular axes OXYZ: O is at the centre of the sphere; OZ passes through A; OX passes through arc AB (or the extension of it); OY is perpendicular to both. Find the coordinates of C in this system: x = sin(b) cos(A) y = sin(b) sin(A) z = cos(b) 16. sine rule: sin(a)/sin(A) = sin(b)/sin(B) = sin(c)/sin(C) cosine rule: cos(a) = cos(b) cos(c) + sin(b) sin(c) cos(A) cos(b) = cos(c) cos(a) + sin(c) sin(a) cos(B) cos(c) = cos(a) cos(b) + sin(a) sin(b) cos(C) 17. Napier's Rules for a spherical right triangle 1. The sine of an angle is equal to the product of cosines of the opposite two angles. 2. The sine of an angle is equal to the product of tangents of the two adjacent angles. 18. Nautical Mile It is the distance measured along the great circle joining the points which subtends one minute of arc at the centre of earth 19. Exercise: A point A, has longitude 2W, latitude 50N. And another place B, has longitude 97W, latitude 50N. How far apart are they, in nautical miles, along a great circle arc? 20. Use the cosine rule: cos AW = cos WP cos AP + sin WP sin AP cos P = cos240 + sin240 cos 95 = 0.5508 So AW = 56.58 = 3395 nautical miles 21. (This is 7% shorter than the route along a parallel of latitude). If you set off from Alderney on a great-circle route to Winnipeg, in what direction (towards what azimuth) would you head? Use the sine rule: sin A / sin WP = sin P / sin WA so sin x = sin 40 sin 95 / sin 56.58 = 0.77 so x = 50.1 or 129.9 . Common sense says 50.1 (or check using cosine rule to get PW). Azimuth is measured clockwise from north, so azimuth is 360 - 50.1 = 309.9 (Note that this is 40 north of the obvious due-west course.) Back to "Spherical trigonometry". 22. (Figure 4-2). In the continental United States, longitude is commonly reported as a west longitude. To convert easterly to westerly referenced longitudes, the easterly longitude must be subtracted from 360 deg. 23. I. Latitude and Longitude on Spherical Earth Latitude and longitude are the grid lines you see on globes. For a spherical earth these are angles seen from the center of the earth. The angle up from the equator is latitude. In the southern hemisphere is it negative in the convention used in geodesy. It has a range of 90 degrees to 90 degrees. The reference for latitude is set by the equator - effectively set by the spin axis of the earth. The angle in the equatorial plane is the longitude. There is no natural reference for longitude. The zero line, called the prime meridian, is taken, by convention, as the line through Greenwich England. (This was set by treaty in 1878. Before that each major nation had its own zero of longitude.) 24. astronomical latitude, . The point where the plumb-lines direction meets the equatorial plane is not, in general, the centre of the Earth. The angle between the line joining the observer to the Earths centre and the equatorial plane is the geocentric latitude, (see figure 7.3). There is yet a third definition of latitude. Geodetic measurements on the Earths surface show local irregularities in the direction of gravity due to variations in the density and shape of the Earths crust. The direction in which a plumb-line hangs is affected by such anomalies and these are referred to as station error. The geodetic or geographic latitude, , of the observer is the astronomical latitude corrected for station error. The geodetic latitude is, therefore, related to a reference spheroid whose surface is defined by the mean ocean level of the Earth. If a and b are the semi-major and semi-minor axes of the ellipse ofrevolution forming the geoid, the flattening or ellipticity, , is given by 25. The longitude used in geodesy is positive going east from the prime meridian. The values go from 0 to 360 degrees. A value in the middle United States is therefore about 260 degrees east longitude. This is the same as -100 degrees east. In order to make longitudes more convenient, often values in the western hemisphere are quoted in terms of angles west from the prime meridian. Thus the 2 longitude of -100 E (E for East) is also 100 W (W for West). Similarly latitudes south of the equator are often given as "S" (for south) values to avoid negative numbers. 26. Latitude and Longitude on Ellipsoidal Earth The earth is flattened by rotational effects. The cross-section of a meridian is no loner a circle, but an ellipse. The ellipse that best fits the earth is only slightly different from a circle. The flattening, defined in the figure below, is about 1/298.25 for the earth. Latitude and longitude are defined to be "intuitively the same as for a spherical earth". This loose definition has been made precise in geodesy. The longitude is the exactly the same as for a spherical earth. The way latitude is handled was defined by the French in the 17th century after Newton deduced that the world had an elliptical crosssection. 27. Before satellites latitude was measured by observing the stars. In particular observing the angle between the horizon and stars. The horizon was taken to be perpendicular to the vertical measured by a plumb bob or spirit level. The "vertical line" of the plumb bob was thought to be perpendicular to the sphere that formed the earth. The extension to an ellipsoidal earth is to use the line perpendicular to the ellipsoid to define the vertical. This is essentially the same as the plumb bob.1 28. The figures below show the key effects of rotation on the earth and coordinates. The latitude is defined in both the spherical and ellipsoidal cases from the line perpendicular to the world model. In the case of the spherical earth, this line hits the origin of the sphere - the center of the earth. For the ellipsoidal model the up-down line does not hit the center of the earth. It does hit the polar axis though 29. The length of the line to the center of the earth for a spherical model is the radius of the sphere. For the ellipsoidal model the length from the surface to the polar axis is one of three radii needed to work with angles and distance on the earth. (It is called the radius of curvature in the prime vertical, and denoted RN here. See the note on radii of the earth for details.) There are not two types of latitude that can easily be defined. The angle that the line makes from the center of the earth is called the geocentric latitude. Geocentric latitude is usually denoted as f, or fc . It is commonly used in satellite work. It does not strike the surface of the ellipsoid at a right angle. The line perpendicular to the ellipsoid makes an angle with the equatorial plane that is called the geodetic latitude. (Geodetic" in geodesy usually implies something taken with respect to the ellipsoid.) The latitude on maps is geodetic latitude. It is usually denoted as g 30. Geodetic Coordinates. Geodetic coordinate components consist of: latitude (f), longitude (l), ellipsoid height (h). Geodetic latitude, longitude, and ellipsoid height define the position of a point on the surface of the Earth with respect to the reference ellipsoid. 31. 1) Geodetic latitude (f). The geodetic latitude of a point is the acute angular distance between the equatorial plane and the normal through the point on the ellipsoid measured in the meridian plane Geodetic latitude is positive north of the equator and negative south of the equator. 32. (2) Geodetic longitude (l). The geodetic longitude is the angle measured counter-clockwise (east), in the equatorial plane, starting from the prime meridian (Greenwich meridian), to the meridian of the defined point 33. (3) Ellipsoid Height (h). The ellipsoid height is the linear distance above the reference ellipsoid measured along the ellipsoidal normal to the point in question. The ellipsoid height is positive if the reference ellipsoid is below the topographic surface and negative if the ellipsoid is above the topographic surface.