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

2. 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)

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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)

9. 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)

10. 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 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

11. 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 = metres. Mean radius of curvature in the area = metres. And the observed angles are :
A = 59° 34' 16.14"
B= 52° 13' 22.00"
C = 68° 12' “
Calculate the Spherical Excess and the triangular misclosure.

12. 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 = ̏
Then, the corrected angles are :
A' = A - ɛ /3
B' = 8 - ɛ /3
C' = C - ɛ /3

13. Coordinate Systems
2 D
Cartesian
Polar
3 D

14. 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

15. 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)

16. 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

17. 2 D Coordinate Transformation
From P (x, y) to P (x′, y′)

18. 2 D Coordinate Transformation continued
To transform point P from XY system to X’Y’ system
Translation
X′ = X – Tx
Y′ = Y – Ty

19. 2 D Coordinate Transformation continued
2. Rotation (β) X′ = X Cos β + Y sin β

20. 2 D Coordinate Transformation continued
2. Rotation (β)
Y′ = Y Cos β – X Sin β

21. 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

22. 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.

23. Thanks for your attention