Lecture 3 Geodesy 1-A 1. Plane and Geodetic Triangles 2. Two Dimensional Coordinate Transformation

Contents Plane surveying and Geodesy Plane Triangle and its solution Spherical Triangle with its formulas 2 D coordinate System 2 D Coordinate Transformation Quiz (1)

Plane Vs Geodetic Surveying Plane Surveying Geodetic Surveying Shape of Earth Plane Sphere, Spheroid, or Ellipsoid Distances Straight Lines (≤ 10 km ) curved line ( > 10 km) Triangles Plane triangles Spherical or Ellipsoidal Triangles Coverage area Small Areas Large portions of earth’s surface Required accuracy Low High

Plane Triangle 6 Elements - Three Angles (A, B, C) Three Sides (a, b, c) Sine Rule Cosine Rule Area or Where s is the semi-perimeter

Geodetic (Spherical)Triangle A triangle drawn on the surface of a sphere is only a spherical triangle if : The three sides are all arcs of great circles Any two sides are together longer than the third side The sum of the three angles is greater than 180º Each individual spherical angle is less than 180º The 3 vertex angles (A, B, and C) and 3 arclengths are related by : 1. Sine Rule a b c

Geodetic (Spherical)Triangle continued 2. 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 cos A = ـــ cos B cos C + sin B sin C cos a Right Angled Spherical Triangle Napier’s Rule a b c

Geodetic (Spherical)Triangle continued The rule specifies that: Sin (90-c) = Tan (90-A) * Tan (90-B) or = Cos (a) * Cos (b) Sine (Middle Part) Product of Tangents of Adjacent Parts Product of Cosines of Opposite Parts

Geodetic (Spherical)Triangle continued Area of Spherical Triangle A = 0.5 * a * b * Sin C = 0.5 * b * c * Sin A = 0.5 * a * c * Sin B = 0.5 * c^2 *(Sin A* Sin B) / Sin (A+B)

Geodetic (Spherical)Triangle continued Spherical (Ellipsoidal)Excess Excess in sum of interior angles of geodetic triangle over 180º because of spherical ( ellipsoidal) geometry. where, ɛ : The Spherical Excess F : Area of spherical triangle R : Mean earth’s Radius S : Semi-perimeter (2S = a+b+c)

Geodetic (Spherical)Triangle continued Cases of spherical excess It is neglected for triangles with all sides less than 10 km or of area less than 25000 km2 For triangles with sides length greater than 160 km, ellipsoidal excess is computed as follows : Where, Ɛ : The intial spherical excess M : The radius of curvature in meridian direction N : The radius of curvature in prime vertical direction

Geodetic (Spherical)Triangle continued Solved Example From two known stations A and B, a third station C to the east of them was fixed by observation of all the angles. The following information is available. Spheroidal Distance AB = 28866.149 metres. Mean radius of curvature in the area = 6 369750 metres. And the observed angles are : A = 59° 34' 16.14" B= 52° 13' 22.00" C = 68° 12' 25 .04“ Calculate the Spherical Excess and the triangular misclosure.

Geodetic (Spherical)Triangle continued Area of plane triangle = 0.5*c^2 (sin A sin B) / sin C Spherical excess = Area / (R^2* Sin 1̏ ) By substituting ɛ = 1.55 ̏ Triangle mis closure = Sum (angles) - ɛ - 180º = 1.69 ̏ Then, the corrected angles are : A' = A - ɛ /3 B' = 8 - ɛ /3 C' = C - ɛ /3

Coordinate Systems 2 D Cartesian Polar 3 D

Coordinate Systems continued 2 D Cartesian Coordinate System Point is defined by its perpendicular offsets from the system axes For point p in figure X = 244 Y = 249

Coordinate Systems continued 2 D Polar Coordinate System Any point is defined by two components Radial distance from system origin The angle (Ɵ) subtended at the origin between this radial distance (r) and one of the system axes e.g. P (r, Ɵ) = (235, 50º)

Conversion Between Cartesian and Polar From Cartesian to Polar Ɵ = arctan (Y/X) From Polar to Cartesian X = r * Sin Ɵ Y = r * Cos Ɵ

2 D Coordinate Transformation From P (x, y) to P (x′, y′)

2 D Coordinate Transformation continued To transform point P from XY system to X’Y’ system Translation X′ = X – Tx Y′ = Y – Ty

2 D Coordinate Transformation continued 2. Rotation (β) X′ = X Cos β + Y sin β

2 D Coordinate Transformation continued 2. Rotation (β) Y′ = Y Cos β – X Sin β

2 D Coordinate Transformation continued Now , the final formula is : X′ = X Cos β + Y sin β - Tx Y′ = Y Cos β – X Sin β - Ty In Matrix form

Quiz (1) How could spherical geometry of the earth affect the surveying observations? Explain how can you merge your Hypothetical coordinates to an existing map coordinates. The mean values of the angles A, B and C of a triangle as measured in a major triangulation were as follows A =50°22'32.55"; B = 65°40'47.53" ,C= 63°56' 46.56". The length of the side BC was 37.5 km and the radius of the Earth 6267 km. Calculate: (a) The spherical excess; (b) The probable values of the spherical angles.

Thanks for your attention