GE 161 – Geomet Geometric ric Geodesy Geodesy The Reference Ellipsoid and the Computation of the Geodetic Posi Posi tion: tion: Properties of the Ellipsoid
Fundamental Fundamental Parameters Parameters of of 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 Paramete Parameterrss The fundamental parameters of the ellipse Formulas: are: 1. Fla Flatte ttening ning or or Polar Polar Flattening, f
f
a
b a 2
2. Fir First st Eccentr Eccentricit icity, y, e
e
a -b
e'
4. Ang Angular ular Eccentr Eccentrici icity, ty, α
cos α
Lecture 7
GE 161 161 – – Geometric Geodesy
a -b b 1
a
; e2
a 2
3. Seco Second nd Eccentr Eccentricit icity, y, e’
2
2
b a
2
; (e')
2
f ; sin α
2
2
a
2
b
2
2
b e ; tan α
e'
The Reference Ellipsoid and the Computation of the Geodetic Position: Properties of the Ellipsoid
Latitudes on the Meridian Ellipse There are three different latitudes used to define the position of the point on a meridian ellipse: 1.Geodetic Latitude (φ)- angle between the line normal to the point and the equatorial plane. 2.Geocentric Latitude (ψ)- angle between the line connecting the center of the ellipse to the point and the equatorial plane. 3.Reduced Latitude(β)- obtained by projecting the ellipse on the geocentric circle having a radius equal to the semi-major axis, a
Lecture 7
z
P’ p b
P
a r z β
ψ
φ
p
a
The Reference Ellipsoid and the Computation of the Geodetic Position: Properties of the Ellipsoid
GE 161 161 – – Geometric Geodesy
Parametric Representation of the Meridian Ellipse 1.Using the geodetic latitude ( φ): a 2 co s
p
a 2 co s 2
b 2 sin 2 a co s
p
1 e 2 sin 2
,z
,z
b 2 sin a 2 co s 2
b 2 si n 2
a(1 e 2 ) si n 1 e 2 sin 2
2.Using the geocentric latitude ( ψ): p
a(1
e2 )1 / 2 c o s
1 e 2 cos 2
, z
a(1
e 2 ) 1 / 2 sin
1 e 2 cos 2
3.Using the reduced latitude ( β): p
Lecture 7
a co cos , z
b si sin
GE 161 161 – – Geometric Geodesy
The Reference Ellipsoid and the Computation of the Geodetic Position: Properties of the Ellipsoid
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: • Ge Geoc ocen entr tric ic to to Geod Geodet etic ic:: b
tan
a
2
ta n
• Re Redu duce ced d to to Geo Geode deti tic: c: ta n
Lecture 7
b a
t an
The Reference Ellipsoid and the Computation of the Geodetic Position: Properties of the Ellipsoid
GE 161 161 – – Geometric Geodesy
Differences Between the Various Latitudes We can find a series expansion that will give the difference in the values of the different latitudes: • Ge Geod odet etic ic an and d Ge Geoc ocen entr tric: ic:
e
2
2
sin 2
.. . .
• Ge Geod odet etic ic an and d Red Reduc uced ed::
( -
)
2 • The The maxi maximu mum m diff differ eren ence ce φ- β is 5’50” 5’50” and the the maximum maximum difference φ-ψ is 11’40” in the case of Clarke Clarke Spheroid Spheroid of 1866. Lecture 7
GE 161 161 – – Geometric Geodesy
The Reference Ellipsoid and the Computation of the Geodetic Position: Properties of the Ellipsoid
Example Problem Problem: A point on the ellipsoid has a geocentric latitude of 45 °N if the flattening f of the ellipsoid is 1/294.9786982. Compute the values values of the geodetic and reduced latitude. Solution: Given: ψ= 45°N, f -1/294.9786982 -1/294.9786982 Find: φ and β
therefore: ta n
ta n
a
And knowing f b a
2
b a
(1
we have:
ta n
ϕ = 45o11'40.44" Using the relationship between φ and β : ta n
b a
ta n
o 1 β = tan− (1 − 1/ 294. 294.9786 9786982)tan 982)tan 45 11'40.44"
β = 45o 05'50.22"
f)2
Lecture 7
b
2
2 tan 4 5o 294.97869 9786982 82 1 − 1/ 294.
therefore:
tan
a
a
1
ϕ = tan −1
Using the relationship between φ and ψ: 2 b
1
GE 161 161 – – Geometric Geodesy
The Reference Ellipsoid and the Computation of the Geodetic Position: Properties of the Ellipsoid
Geodetic Coordinates and the Space Rectangular Coordinates • We ca can n det deter ermi mine ne th the e 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) Lecture 7
GE 161 161 – – Geometric Geodesy
The Reference Ellipsoid and the Computation of the Geodetic Position: Properties of the Ellipsoid
Example Problem Problem: A point on the ellipsoid has a geodetic coordinates φ=45°N, λ =121°E, and h=150 h=1500 0 m. If the the flattening f of the ellipsoid is 1/294.98 and the semi-major axis a=6,378,206 m, compute the space rectangular coordinates of the points Solution: Given: f=1/294.9786982 φ=45°N =121°E a=6,378,206.4 m λ =121 h=1500 m Find: (x, y, z) coordinates of the point Lecture 7
Using the equations for converting geodetic to cartesian: x=(p+hcos φ)cos λ y =(p+hcos φ)sin λ z=(z+hsin φ) Solving for p and z: p
a cos 1 e 2 sin 2
,z
a (1 e 2 ) s i n 1 e 2 sin 2
Solving for e2 given a and f : e2=0.00676865799760962 Therefore: p =
6378206.4cos 63782 06.4cos 45o (1 − (0.00 (0.0067686 6768657997 5799760962 60962)) sin 2 45o )
p = 4,517,724.209 m
GE 161 161 – – Geometric Geodesy
The Reference Ellipsoid and the Computation of the Geodetic Position: Properties of the Ellipsoid
Example Problem Solving for z: z =
6378206.4( 63782 06.4(1 1 − 0.00676 0.0067686579 8657997609 9760962)sin 62)sin 45o
z=(4487145.279+1500sin 45) z=4,488,205.939 m
1− (0.00676 (0.0067686579 8657997609 9760962)sin 62)sin 2 45o
The space rectangular coordinates of the point are:
z = 4,487,145.279 m Substituting the values of p, z, φ, λ and h, we get: x=(4517724.209+1500cos 45)cos 121 x= -2,327,346.260 m
x= -2,327,346.26 2,327,346.260 0m y =3,873,354.629 m z=4,488,205.939 m z=4,488,205.939
y =(4517724.209+1 =(4517724.209+1500cos 500cos 45)sin 121 y =3,873,354.629 m Lecture 7
GE 161 161 – – Geometric Geodesy
The Reference Ellipsoid and the Computation of the Geodetic Position: Properties of the Ellipsoid
Geodetic Coordinates and the Space Rectangular Coordinates •
We ca can n get get th the e geo geode dete teic ic coordinates (φ, λ , h) of a poi point nt given its space rectangular coordinates (x,y,z) using these equations:
•
Howeve Howe ver, r, mo most st of th the e sol solut utio ion n in converting space rectangular coordinates to geodtic geodt ic coord coordinate inates s requ requires ires iteration in the computation of the geodetic latitude.
•
There Ther e ar are e se seve vera rall sol solut utio ions ns that can be used in this conversion
Lecture 7
The Reference Ellipsoid and the Computation of the Geodetic Position: Properties of the Ellipsoid
GE 161 161 – – Geometric Geodesy
Geodetic Coordinates and the Space Rectangular Coordinates One solution is the following iterative scheme: 1.Calculate λ = ta tan n−1
N =
y x
h=
2.Iterate for φ; consequently for h. the initial value for φ is the spherical latitude, z ϕ o = ta tan n−1 x2 + y2
Lecture 7
Then compute a (1 − e 2 sin 2 ϕ )1/ 2
x 2 + y 2 cos ϕ
− N
giving −1 z N 2 ϕ = tan 1− e 2 2 N h + x + y −1
GE 161 161 – – Geometric Geodesy
The Reference Ellipsoid and the Computation of the Geodetic Position: Properties of the Ellipsoid
Geodetic Coordinates and the Space Rectangular Coordinates Another solution: 1.Calculate λ = tan−1
3.Then compute for h:
y
h=
2.Iterate for φ using as an initial value for φ: ϕ = tan−1
e2 Nsin ϕ 1+ 2 2 z x + y z
x 2 + y 2
− N
cos ϕ
where: N =
a (1 − e2 sin 2 ϕ )1/ 2
2 2 2 (1− e ) x + y
ϕ initial = tan−1
z
Lecture 7
The Reference Ellipsoid and the Computation of the Geodetic Position: Properties of the Ellipsoid
GE 161 161 – – Geometric Geodesy
Geodetic Coordinates and the Space Rectangular Coordinates A non-iterative solution to this conversion was proposed by Soler and Hothem Hothem (1988 (1988)) which is based based on the the works works of Bowring:
y λ = ta tan n−1 x
in which:
z + e asin 2 coss3 p − e aco a2 h = p cosϕ + z sin ϕ − N
ϕ = ta tan n−1
Lecture 7
2
3
p=
x2 + y2
r = p2 + z 2 (z1 − f) ae2 tan µ = 1 + p r
GE 161 161 – – Geometric Geodesy
The Reference Ellipsoid and the Computation of the Geodetic Position: Properties of the Ellipsoid
