GE 161 – Geometric Geodesy

The Reference Ellipsoid and the Computation of the Geodetic Position:

Position:
Properties of the Ellipsoid

Fundamental
Fundamental Parameters
Parameters ofof the
the
Ellipsoid,
Ellipsoid, the
the Meridian
Meridian Ellipse,
Ellipse, and
and
Coordinate
Coordinate Conversion
Conversion

Lecture No. 7

Department of Geodetic Engineering

University of the Philippines

a.s. caparas/06

The Ellipse and its Fundamental Parameters

The fundamental
parameters of the ellipse
Formulas:
are:
a−b
1. Flattening or Polar f =
a
Flattening, f
a 2 -b 2 a 2 − b2
e= ; e2 =
2. First Eccentricity, e a a2
a 2 -b 2 a 2 − b2
3. Second Eccentricity, e’ e' =
b
; (e')2 =
b2
4. Angular Eccentricity, α cos α = 1 − f ; sin α = e ; tan α = e'

The Reference Ellipsoid and the

Lecture 7 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Properties of the Ellipsoid

1
 Latitudes on the Meridian Ellipse
There are three different
latitudes used to define the
position of the point on a
meridian ellipse: z
1.Geodetic Latitude (φ)- angle
between the line normal to the
point and the equatorial plane.
2.Geocentric Latitude (ψ)- angle P’
between the line connecting the p
center of the ellipse to the point P
and the equatorial plane. a r
b
3.Reduced Latitude(β)- obtained z
by projecting the ellipse on the β ψ φ
geocentric circle having a radius p
a
equal to the semi-major axis, a

The Reference Ellipsoid and the

Lecture 7 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Properties of the Ellipsoid

Parametric Representation of the

Meridian Ellipse
1.Using the geodetic latitude (φ):
a 2 cos ϕ b 2 sin ϕ
p= ,z=
a 2 cos 2 ϕ + b 2 sin 2 ϕ a 2 cos 2 ϕ + b 2 sin 2 ϕ

a cos ϕ a(1 − e 2 ) sin ϕ

p= ,z=
1 − e sin ϕ
2 2
1 − e 2 sin 2 ϕ

2.Using the geocentric latitude (ψ):

a(1 − e 2 )1 / 2 cos ψ a(1 − e 2 )1 / 2 sin ψ
p= ,z=
1 − e cos ψ
2 2
1 − e 2 cos 2 ψ

3.Using the reduced latitude (β):

p = a cos β , z = b sin β

The Reference Ellipsoid and the

Lecture 7 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Properties of the Ellipsoid

2
 Relationship Between the Various
Latitude
Comparing the parametric representations of
the meridian ellipse using the different latitudes,
we can find transformation between φ, β, and ψ:
• Geocentric to Geodetic:
2
b
tan ψ =   tan ϕ
a
• Reduced to Geodetic:
b
tan β = tan ϕ
a

The Reference Ellipsoid and the

Lecture 7 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Properties of the Ellipsoid

Differences Between the Various

Latitudes
We can find a series expansion that will give the
difference in the values of the different latitudes:
• Geodetic and Geocentric:
e2
ϕ − ψ = sin 2ϕ + ....
2
• Geodetic and Reduced:
( ϕ -ψ )
ϕ−β =
2
• The maximum difference φ- β is 5’50” and the maximum
difference φ-ψ is 11’40” in the case of Clarke Spheroid of
1866.

The Reference Ellipsoid and the

Lecture 7 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Properties of the Ellipsoid

3
 Example Problem
Problem: therefore:
A point on the ellipsoid has a a
2

geocentric latitude of 45°N if the ϕ = tan −1   tan ψ

flattening f of the ellipsoid is b
1/294.9786982. Compute the values ϕ = tan −1  
2
1  o
of the geodetic and reduced latitude.   tan 45 
 1 − 1/ 294.9786982  

Solution: ϕ = 45o11' 40.44"

Given: ψ= 45°N, f-1/294.9786982
Find: φ and β Using the relationship between φ and β :
Using the relationship between φ and b
ψ: tan β =   tan ϕ
2
b a
tan ψ =   tan ϕ therefore:
a

a−b β = tan −1  (1 − 1/ 294.9786982) tan 45o11' 40.44"

And knowing f = we have:
2 a
b
 a  = (1 − f)
2
β = 45o 05 '50.22"
 
The Reference Ellipsoid and the
Lecture 7 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Properties of the Ellipsoid

Geodetic Coordinates and the Space Rectangular

Coordinates

• We can determine the

space rectangular
(x,y,z) given the
geodetic coordinates
(φ, λ, h) and it is given
by:

x=(p+hcos φ)cos λ
y =(p+hcos φ)sin λ
z=(z+hsin φ)

The Reference Ellipsoid and the

Lecture 7 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Properties of the Ellipsoid

4
 Example Problem
Problem: Using the equations for converting
A point on the ellipsoid has a geodetic to cartesian:
geodetic coordinates φ=45°N, λ x=(p+hcos φ)cos λ
=121°E, and h=1500 m. If the y =(p+hcos φ)sin λ
flattening f of the ellipsoid is z=(z+hsin φ)
1/294.98 and the semi-major axis
a=6,378,206 m, compute the Solving for p and z:
space rectangular coordinates of a cos ϕ a(1 − e 2 ) sin ϕ
the points p= ,z=
1 − e 2 sin 2 ϕ 1 − e 2 sin 2 ϕ

Solution: Solving for e2 given a and f:

Given:
φ=45°N f=1/294.9786982 e2=0.00676865799760962
λ=121°E a=6,378,206.4 m Therefore:
h=1500 m p=
6378206.4 cos 45o
Find: (x, y, z) coordinates of the (1 − (0.00676865799760962) sin 2 45o )
point p = 4,517,724.209 m
The Reference Ellipsoid and the
Lecture 7 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Properties of the Ellipsoid

Example Problem
Solving for z: z=(4487145.279+1500sin 45)
z=4,488,205.939 m
6378206.4(1− 0.00676865799760962)sin 45o
z=
1− (0.00676865799760962)sin2 45o
The space rectangular
z = 4,487,145.279 m coordinates of the point are:

Substituting the values of p, z, φ,

λ and h, we get:
x=(4517724.209+1500cos 45)cos x= -2,327,346.260 m
121 y =3,873,354.629 m
x= -2,327,346.260 m z=4,488,205.939 m

y =(4517724.209+1500cos 45)sin
121
y =3,873,354.629 m

The Reference Ellipsoid and the

Lecture 7 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Properties of the Ellipsoid

5
 Geodetic Coordinates and the Space Rectangular
Coordinates
• We can get the geodeteic
coordinates (φ, λ, h) of a point
given its space rectangular
coordinates (x,y,z) using these
equations:
• However, most of the solution
in converting space
rectangular coordinates to
geodtic coordinates requires
iteration in the computation of
the geodetic latitude.
• There are several solutions
that can be used in this
conversion

The Reference Ellipsoid and the

Lecture 7 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Properties of the Ellipsoid

Geodetic Coordinates and the Space Rectangular

Coordinates
One solution is the following Then compute
iterative scheme: a
1.Calculate N=
(1 − e sin 2 ϕ )1/ 2
2
y
λ = tan−1 x2 + y 2
x h= −N
2.Iterate for φ; consequently cos ϕ
for h. the initial value for φ is giving
the spherical latitude,

   z  N  
−1

ϕo = tan−1 
z
 ϕ = tan  −1
1− e
2
 
 x2 + y 2   x2 + y2  N + h  
 

The Reference Ellipsoid and the

Lecture 7 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Properties of the Ellipsoid

6
 Geodetic Coordinates and the Space Rectangular
Coordinates
Another solution: 3.Then compute for h:
1.Calculate

y x2 + y 2
λ = tan−1 h= −N
x cos ϕ
2.Iterate for φ using as an
initial value for φ: where:

z  e2 N sin ϕ  a
ϕ = tan−1 1+  N=
x2 + y 2  z  (1 − e sin 2 ϕ )1/ 2
2

 z 
ϕinitial = tan−1  
 (1− e ) x + y 
2 2 2

The Reference Ellipsoid and the

Lecture 7 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Properties of the Ellipsoid

Geodetic Coordinates and the Space Rectangular

Coordinates
A non-iterative solution to this conversion was proposed by
Soler and Hothem (1988) which is based on the works of
Bowring:

 y
λ = tan−1   in which:
x  
p = x2 + y 2
 z + e a sin µ 
2 3
ϕ = tan−1  
 p − e a cos µ 
2 3 r = p2 + z 2
 a2  z(1− f )  ae2 
h = p cosϕ + z sin ϕ −   tan µ = 1+ 
p  r 
N

The Reference Ellipsoid and the

Lecture 7 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Properties of the Ellipsoid