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12.215 Modern Navigation Thomas Herring tahmit.edu MW 11:00-12:30 Room 54-322http://geoweb.mit.edu/tah/12.215 Review of Wednesday Class Definition of heights –Ellipsoidal height geometric –Orthometric height potential field based Shape of equipotential surface: Geoid for Earth Methods for determining heights09/28/2009 12.215 Modern Naviation L04 2 Today’s Class Spherical Trigonometry –Review plane trigonometry –Concepts in Spherical Trigonometry Distance measures Azimuths and bearings –Basic formulas: Cosine rule Sine rule http://mathworld.wolfram.com/SphericalTrigonometry.html is a good explanatory site09/28/2009 12.215 Modern Naviation L04 3 Spherical Trigonometry As the name implies this is the style of trigonometry used to calculate angles and distances on a sphere The form of the equations is similar to plane trigonometry but there are some complications. Specifically in spherical triangles the angles do not add to 180o “Distances” are also angles but can be converted to distance units by multiplying the angles in radians by the radius of the sphere. For small sized triangles the spherical trigonometry formulas reduce to the plane form. 409/28/2009 12.215 Modern Naviation L04 Review of plane trigonometry Although there are many plane trigonometry formulas almost all quantities can be computed from two formulas: The cosine rule and sine rules. Angles A B and C Sides a b and c A Sum of angles ABC180 Cosine Rule: b c 2 a 2 b 2 2ab cos C c Sine Rule: a b c sin A sin B sin C B C a09/28/2009 12.215 Modern Naviation L04 5 Basic Rules discussed in following slides A B C are angles a b c are sides A all quanties are angles c b Sine Rule C sin a sin b sin c B a sin A sin B sin C O Cosine Rule sides 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 b cos a sin a sin b cos C Cosine Rule angles cos A cos B cos C sin B sin C cos a cos B cos Acos C sin Asin C cos b cos C cos Acos B sin Asin B cos c09/28/2009 12.215 Modern Naviation L04 6 Spherical Trigonometry Interpretation Interpretation of sides: –The spherical triangle is formed on a sphere of unit radius. –The vertices of the triangles are formed by 3 unit vectors OA OB OC. –Each pair of vectors forms a plane. The intersection of a plane with a sphere is a circle. If the plane contain the center of the sphere O it is called a great circle If center not contained called a small circle e.g. a line of latitude except the equator which is a great circle –The side of the spherical triangle are great circles between the vertices. The spherical trigonometry formulas are only valid for triangles formed with great09/28/2009 12.215 Modern Naviation L04 7 circles. Interpretation Interpretation of sides continued: –Arc distances along the great circle sides are the side angle in radians by the radius of the sphere. The side angles are the angles between the vectors. Interpretation of angles –The angles of the spherical triangles are the dihedral angles between the planes formed by the vectors to the vertic