Navigation Lecture: Geometric Shapes and Dimensions of the Earth PDF

LECTURE THEME :  Geometric Shapes and Dimensions of the EARTH Geoid - is the shape that the force of gravity acts everywhere perpendicular to the its surface. The surface of the oceans would take under the influence of Earth's gravity and rotation alone, in the absence of o...

LECTURE THEME :  Geometric Shapes and Dimensions of the EARTH Geoid - is the shape that the force of gravity acts everywhere perpendicular to the its surface. The surface of the oceans would take under the influence of Earth's gravity and rotation alone, in the absence of other influences such as winds and tides. This surface is extended through the continents (such as with very narrow hypothetical canals). All points on a geoid surface have the same effective potential (the sum of gravitational potential energy and centrifugal potential energy). Specifically, the geoid is the equipotential surface that would coincide with the mean ocean surface of Earth if the oceans and atmosphere were in equilibrium, at rest relative to the rotating Earth, and extended through the continents.  Geometric Shapes and Dimensions of the EARTH Geoid. According to Gauss, who first described it, it is the "mathematical figure of Earth", a smooth but highly irregular surface whose shape results from the uneven distribution of mass within and on the surface of Earth. It does not correspond to the actual surface of Earth's crust, but to a surface which can only be known through extensive gravitational measurements and calculations.  Geometric Shapes and Dimensions of the EARTH For normal navigational needs, it is assumed that the Earth is a sphere. With the development of radionavigation and satellite systems, there is a need to treat the Earth as a rotary ellipsoid for greater accuracy. An Earth ellipsoid is a mathematical figure approximating the shape of the Earth, used as a reference frame for computations in geodesy, astronomy and the geosciences. Various different ellipsoids have been used as approximations. EARTH elipsoid, whose short (polar) axis (connecting the two flattest spots called geographical north and south poles) is approximately aligned with the rotation axis of the Earth. The ellipsoid is defined by the equatorial axis a and the polar axis b; their difference is about 21 km or 0.335 percent. Geometric Shapes and Dimensions of the EARTH Elipsoid - a body formed by the rotation of the ellipse around the minor axis. On the ellipsoid: - Equator and parallels are circles, - The meridians are ellipses. - The value of the ellipse's radius varies according to the latitude. In fact, the Earth is not an ellipsoid, but very close to it. Geometric Shapes and Dimensions of the EARTH Elipsoida as a shape of the Earth a – large semimajor axis of the ellipsoid b – small semimajor axis of the ellipsoid The measurements of the elements of the earth ellipsoid were dealt with by Clarke, Bessel, Hayford, Krassoswski.  Geometric Shapes and Dimensions of the EARTH The parameters determined are usually the semi-major axis, a, and either the semi-minor axis,b , or the inverse flattening 1/ f , (where the flattening is f = (a-b)/a) The shape of an ellipsoid is determined by the shape parameters of that ellipse which generates the ellipsoid when it is rotated about its minor axis. The semi-major axis of the ellipse, a, is identified as the equatorial radius of the ellipsoid: the semi-minor axis of the ellipse, b, is identified with the polar distances (from the centre). These two lengths completely specify the shape of the ellipsoid but in practice geodesy publications classify reference ellipsoids by giving the semi-major axis and the inverse flattening, 1/f, The flattening, f, is simply a measure of how much the symmetry axis is compressed relative to the equatorial radius. For the Earth, f is around 1/300 corresponding to a difference of the major and minor semi-axes of approximately 21 km (13 miles). Some precise values are given in the table below and also in Figure of the Earth. For comparison, Earth's Moon is even less elliptical, with a flattening of less than 1/825.  Geometric Shapes and Dimensions of the EARTH The reference system includes: Reference surface (ellipsoid), Orientation of the ellipsoid it is: 1. point of application of the ellipsoid to the geoid, 2. orientation in the geoid body, which consists in the parallel arrangement of the small axis of the ellipsoid to the axis of rotation of the Earth and the simultaneous parallelism of the other two axes of the ellipsoid to the same geoid axis. Geometric Shapes and Dimensions of the EARTH GEODEZYJNE UKŁADY ODNIESIENIA WSPÓŁRZĘDNYCH Reference systems used in marine cartography: Local (eg ED 50, PUŁKOWO 42, NAD 1983, OSGB 1936, INT 24); Global (geocentric): WGS 72, WGS 84. Geometric Shapes and Dimensions of the EARTH GEODETIC COORDINATES - GLOBAL SYSTEMS Currently the basic global ellipsoid (for satellite systems as well as for newly developed maps) is the WGS-84 ellipsoid, which best approximates the geoid at every point of it, not just in the matching area.  Geometric Shapes and Dimensions of the EARTH EARTH AS A SPHERE By cutting the globe with the planes, we will obtain circles of different radiuses. We distinguish great circles and small circles.  Geometric Shapes and Dimensions of the EARTH EARTH AS A SPHERE The great circles - is created on the surface of the globe as a result of the intersection of its plane passing through the center of the globe. The radius of the great circle equals the radius of the globe. The shorter arc of a large circle passing through two points (A and B) is the shortest distance between these points and is called the orthodrome (Great Circle - GC). Great Circle  Geometric Shapes and Dimensions of the EARTH EARTH AS A SPHERE The small circles - a circle that is found on the surface of the globe as a result of cutting it with a plane that does not pass through the centre of the globe. The radiuses of the small circles (r) are smaller than the radius of the earth (R).  Geometric Shapes and Dimensions of the EARTH EARTH AS A SPHERE Equator - a large circle whose plane is perpendicular to the axis of the Earth and passes through the centre of the Earth. Parallel - a small circle whose plane is perpendicular to the axis of the Earth and parallel to the plane of the equator.  Geometric Shapes and Dimensions of the EARTH EARTH AS A SPHERE Earth meridian axis equator An infinite number of planes can be traversed by the Earth's axis, which intersect the globe along the great circles. Earth meridian - is half the great circle running from the north pole to the south pole. It is created on the surface of the globe as a result of the intersection of its plane passing through both poles of earth and through the centre of the Earth. Meridians on the globe cross the parallels and the equator at right angles.  COORDINATES Latitude φ - the middle angle between the plane of the equator and the line joining the point (A) with the centre of the earth - the angular distance north or south from the equator of a point on the earth's surface, measured on the meridian of the point. Longitude λ - the angle between the plane of the Prime (Greenwich) meridian and the plane of the meridian passing through the point (the angular distance of the meridian of a given point from the Prime meridian measured at the Equator) COORDINATES The Plane of the Prime Meridian (Greenwich) divides the globe into two hemispheres: East (E) of the angle between 000º to 180º east from the meridian zero. The eastern sign in algebraic operations accepts the sign (+); West (W) of the angle 000º to 180º West from the meridian zero. The western length in algebraic operations accepts the sign (-);  COORDINATES The Prime meridian is a conventionally accepted meridian passing through Greenwich, London  The difference in latitude The difference in latitude Δφ - is the difference in the angular distance of the two parallel points from the equator measured per meridian - the angular distance between the parallel of the destination to the parallel of the point of departure. The difference in latitude between two points on the globe is calculated as the algebraic difference between the destination lattitude φB and the lattitude of the departure point φA Δφ = (±) φB -(±) φA Lattitude difference can reach the highest value of 180º N (North) or S (South) between poles; Δφ is (+) N if the destination is north of the departure point; Δφ is (-) S if the destination is south of the departure point; The sign of the latitude difference Δφ is derived from the position of the destination relative to the departure point.  The difference in latitude Example 1: φA= 30º48,4’ N φB = 52º25,7’ N Δφ =(±)φB -(±)φA equator φB = + 51º 85,7’ -φA= + 30º 48,4’ Δφ = + 21º 37,3’ Δφ =21º 37,3’ N 1º = 60’ 0,1º = 6’ 1’ = 60” 0,1’ = 6”  The difference in longitude Longitude difference Δλ - this is the difference in the angular distance between the meridians of two points from the Prime meridian (the dihedral angle between the plane of the meridian of the departure (A) and the plane of the meridian of the destination (B), which tells the length of the equator in the angular distance between the meridian of the departure and meridian of destination). We calculate it as an algebraic difference between the length of the destination λB and the length of the departure point λA. Δλ = (±) λB – (±) λA Δλ is (+) E, if the destination is east (East) from the departure point; Δλ is (-) W if the destination is west (West) from the departure point; Δλ reaches the highest value of 180º E (East) or W (West); If in the algebraic operation we get Δλ larger than 180º, it must be completed to 360º and the resulting difference in length is given the opposite sign.  The difference in longitude Example: 1 Known: λA = 044º 48’30” E. Greenwich equator λB = 115º 35’14”E Δλ = (±) λB – (±) λA λB =+115º 35’14” 1º = 60’ - λA =+044º 48’30” 0,1º = 6’ Δλ= +070º 46’44” 1’ = 60” Δλ= 070º 46’44”E 0,1’ = 6”  The difference in longitude Example 2: Known: λA = 154º 28,7’ W λB = 136º 55’36” E Greenwich Δλ = (±) λB – (±) λA λB = +136º 55,6’ 359º 60,0’ - λA = - 154º 28,7’ - Δλ= 291º 24,3’ Δλ= +290º 84,3’ Δλwł.= 068º 35,7’ Δλ= 291º 24,3’E > 180º Δλwł.= 068º 35,7’ W Opposite sign of the Δλ