112.215 Modern NavigationThomas Herring ([email protected]),MW 11:00-12:30 Room 54-322http://geoweb.mit.edu/~tah/12.21509/28/2009 12.215 Modern Naviation L04 2Review of Wednesday Class• Definition of heights– Ellipsoidal height (geometric)– Orthometric height (potential field based)• Shape of equipotential surface: Geoid for Earth• Methods for determining heights209/28/2009 12.215 Modern Naviation L04 3Todayʼs 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.htmlis a good explanatory site09/28/2009 12.215 Modern Naviation L04 4Spherical Trigonometry• As the name implies, this is the style of trigonometryused to calculate angles and distances on a sphere• The form of the equations is similar to planetrigonometry but there are some complications.Specifically, in spherical triangles, the angles do notadd to 180o• “Distances” are also angles but can be converted todistance units by multiplying the angles (in radians) bythe radius of the sphere.• For small sized triangles, the spherical trigonometryformulas reduce to the plane form.309/28/2009 12.215 Modern Naviation L04 5Review of plane trigonometry• Although there are many plane trigonometry formulas,almost all quantities can be computed from twoformulas: The cosine rule and sine rules.ABCabcAngles A, B and C; Sides a, b and cSum of angles A+B+C=180Cosine Rule:Sine Rule:€ c2= a2+ b2− 2ab cos C€ asin A=bsin B=csin C09/28/2009 12.215 Modern Naviation L04 6Basic Rules(discussed in following slides)ABCacbA B C are anglesa b c are sides(all quanties are angles)Sine Rule€ sin asin A=sin bsin B=sin csin CCosine Rule sidesCosine Rule angles€ cos a = cos b cos c + sin b sin c cos Acos b = cos c cos a + sin c sin a cos Bcos c = cos b cos a + sin a sin b cos C€ cos A = − cos B cosC + sin B sin C cos acos B = − cos A cosC + sin A sin C cosbcosC = − cos A cos B + sin A sin B cos cO409/28/2009 12.215 Modern Naviation L04 7Spherical 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 aplane with a sphere is a circle.• If the plane contain the center of the sphere (O), it is called agreat circle• If center not contained called a small circle (e.g., a line of latitudeexcept the equator which is a great circle)– The side of the spherical triangle are great circles between thevertices. The spherical trigonometry formulas are only validfor triangles formed with great circles.09/28/2009 12.215 Modern Naviation L04 8Interpretation• Interpretation of sides (continued):– Arc distances along the great circle sides are theside angle (in radians) by the radius of the sphere.The side angles are the angles between thevectors.• Interpretation of angles– The angles of the spherical triangles are thedihedral angles between the planes formed by thevectors to the vertices.– One example of angles is the longitude differencebetween points B and C if A is the North Pole.509/28/2009 12.215 Modern Naviation L04 9Interpretation• In navigation applications the angles and sides ofspherical triangles have specific meanings.• Sides: When multiplied by the radius of the Earth, arethe great circle distances between the points. On asphere, this is the short distance between two pointsand is called a geodesic. When one point is the Northpole, the two sides originating from that point are theco-latitudes of the other two points• Angles: When one of the points is the North pole, theangles at the other two points are the azimuth orbearing to the other point.09/28/2009 12.215 Modern Naviation L04 10Derivation of Cosine rule• The spherical trigonometry cosine rule can be derivedform the dot product rule of vectors fairly easily. Thesine rule can be also derived this way but it is moredifficult.• On the next page we show the derivation by carefullyselecting the coordinate axes for expressing thevector.• (Although we show A in the figures as at the Northpole this does not need to be case. However, in manynavigation application one point of a spherical triangleis the North pole.)609/28/2009 12.215 Modern Naviation L04 11Derivation of cosine ruleABCacbOVector OB has components:[sin c, 0, cos c]Vector OC has components:[sin b cos A, sin b sin A, cos b]XZYFrom dot product rule:€ cos a = O B •O C = sin c,0,cos c[ ]• sin b cos A, sin b sin A, cos b[ ]= sin c sin b cos A + cos c cos bTaking OA as the Z-axis, andOB projected into the plane perpendicular to OA as the X axis09/28/2009 12.215 Modern Naviation L04 12Area of spherical triangles• The area of a spherical triangle is related to the sum of theangles in the triangles (always >180o)• The amount the angles in a spherical triangle exceed 180o or πradians is called the spherical excess and for a unit radius sphereis the area of the spherical triangle– e.g. A spherical triangle can have all angles equal 90o and sothe spherical excess is π/2. Such a triangle covers 1/8 thearea of sphere. Since the area of a unit sphere is 4π ster-radians, the excess equals the area (consider the triangleformed by two points on the equator, separated by 90o oflongitude).709/28/2009 12.215 Modern Naviation L04 13Typical uses of SphericalTrigonometry• Spherical trigonometry is used for most calculations in navigationand astronomy. For the most accurate navigation and mapprojection calculation, ellipsoidal forms of the equations are usedbut this equations are much more complex and often not closedformed.• In navigation, one of the vertices is usually the pole and the sidesb and c are colatitudes.• The distance between points B and C can be computed knowingthe latitude and longitude of each point (Angle A is difference inlongitude) using the cosine rule. Quadrant ambiguity in the cos-1is not a problem because the shortest distance between thepoints is less than 180o• The bearing between the points is computed from the sine ruleonce the distance is known.09/28/2009 12.215 Modern Naviation L04 14Azimuth or Bearing calculation• In the azimuth or bearing calculation, quadrant ambiguity is aproblem since the sin-1 will return two possible angles (that yieldthe same value