GE 161 – Geometric Geodesy
The Reference Ellipsoid and the Computation of the Geodetic Position:
Position:
Curves on the Surface of the Ellipsoid
The Elliptic Arc, Azimuth,
and Chord of a Normal
Section
Lecture No. 11
Department of Geodetic Engineering
University of the Philippines
a.s. caparas/06
The Elliptic Arc of the Normal Sections
• Recall that the equations giving the linear and
azimuth separations of the normal sections, we
have used the relationship s=Nσ in order for us
to express σ in terms of s and N.
• We assumed that the normal section is an arc
length of a circle.
• However, the normal section is in the surface of
the reference ellipsoid, therefore the normal
section must be an elliptic arc.
• Having this, we should find if the relationship
that we used would be valid to use in evaluating
the separations of the normal sections.
The Reference Ellipsoid and the
Lecture 10 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Curves on the Surface of the Ellipsoid
1
The Elliptic Arc of the Normal Sections
• In order to find
whether the our
assumption is correct,
we need to find the B
fB
of
no
di an
true relationship eri
ia
rid
M
Me
B
between s, N and σ. s
A12
S2
• Consider the plane A
containing the normal N1
σ
section from A to B.
The Reference Ellipsoid and the
Lecture 10 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Curves on the Surface of the Ellipsoid
The Elliptic Arc of the Normal Sections
• After some manipulation, we can find:
S2 1 1
= 1 − σ 2η12 cos 2 A12 + σ 3η12t1 cos 2 A12 + ....
N1 2 2
where: η12 = (e ') 2 cos 2 ϕ1
t1 = tan ϕ1
The Reference Ellipsoid and the
Lecture 10 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Curves on the Surface of the Ellipsoid
2
The Elliptic Arc of the Normal Sections
• To derive the relationship between s, N
and σ, we must consider the differential
curve ds:
(ds ) 2 = ( S2 dσ ) 2 + (dS 2 ) 2
• Since dS2 is negligible compared to S2dσ:
(ds ) 2 = ( S 2 dσ ) 2
The Reference Ellipsoid and the
Lecture 10 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Curves on the Surface of the Ellipsoid
The Elliptic Arc of the Normal Sections
• Substituting the derived equation for S2
and integrating ds:
1 1
s = N1σ 1 − σ 2η12 cos 2 A12 + σ 3η12t1 cos 2 A12
6 8
• This formula can be inverted to find σ as a
function of s:
s 1 s 2
2 3
1 s 2
σ= 1 + η1 cos A12 + η1 t1 cos A12
2 2
N1 6 N1 8 N1
The Reference Ellipsoid and the
Lecture 10 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Curves on the Surface of the Ellipsoid
3
The Elliptic Arc of the Normal Sections
• If we will evaluate the value of the terms
inside the bracket of the equation for any
normal section, we will see that the value
will be approximately equal to 1.
• With this we can conclude that the use of
the relationship s=Nσ for the computations
with normal sections separations is
justified to some degree of accuracy.
The Reference Ellipsoid and the
Lecture 10 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Curves on the Surface of the Ellipsoid
Curves or Arcs on the Ellipsoid
• A curve or an arc on the
surface of the ellipsoid Parallel Arc
generally connects two Meridian Arc
points on the ellipsoids
surface.
• We can classify this
basically into two
categories:
1. “Special” Curves/Arcs
- Arc along the Meridian
- Arc along the Parallel Normal Section
2. “General” Curves
- Normal Section/Curve
The Reference Ellipsoid and the
Lecture 10 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Curves on the Surface of the Ellipsoid
4
Length of Meridian Arcs
• In order to find the length
between two points with
latitudes φ1 and φ2, the ϕ2
differential arc length ds=Mdφ
ds=Mdφ must be ϕ1
integrated:
ϕ2 ϕ2
1
s = ∫ Mdϕ = a (1 − e 2 ) ∫ dϕ
ϕ1 ϕ1 (1 − e 2
sin 2
ϕ )3/ 2
• But this represents an
elliptical integral, which
cannot be integrated
using elementary integral
functions.
The Reference Ellipsoid and the
Lecture 10 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Curves on the Surface of the Ellipsoid
Length of Meridian Arcs
• Instead of elementary integral functions,
the use of series expansion is necessary
to evaluate the length of meridian arc.
• The McLaurin Series is used to expand
the term inside the integral.
• The length of the meridian arc after
expanding the term is given by:
B C D E F
Sϕ = a (1 − e 2 ) Aϕ − sin 2ϕ + sin 4ϕ − sin 6ϕ + sin 8ϕ − sin10ϕ + ....
2 4 6 8 10
The Reference Ellipsoid and the
Lecture 10 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Curves on the Surface of the Ellipsoid
5
Length of Meridian Arcs
• In which:
3 45 175 6 11025 8 43659 10
A = 1 + e2 + e4 + e + e + e + ....
4 64 256 16384 65556
3 2 15 4 525 6 2205 8 72765 10
B= e + e + e + e + e + ....
4 16 512 2048 65536
15 4 105 6 2205 8 10395 10
C= e + e + e + e + ....
64 256 4096 16384
35 6 315 8 31185 10
D= e + e + e + ....
512 2048 131072
315 8 3465 10
E= e + e + ....
16384 65536
693 10
F= e + ....
131072
The Reference Ellipsoid and the
Lecture 10 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Curves on the Surface of the Ellipsoid
Length of Meridian Arcs
• The length of the meridian arc from equator to the pole
from the derived equations is:
π
Sϕ =90 = a (1 − e 2 ) A
2
• For lines up to 400 km, the equation may be modified as
1
s = M m ∆ϕ 1 + e 2 (∆ϕ ) 2 cos 2ϕm
8
• For even shorter lines that reaches only 45 km, we may
dropped the term in the bracket:
s = M m ∆ϕ
The Reference Ellipsoid and the
Lecture 10 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Lecture 7 GE 161- Geometric Geodesy
Curves on the Surface of the Ellipsoid
6
Length of Parallel Arc
• Parallel arcs are arcs of
circle so the length of this
arc can be computed
using the arclength
formula for circular arcs. p λ2
λ1
• The length of arc or the
φ
distance between two
points on the same
parallel having longitudes
λ1 and λ2 is given by:
L=p∆λ=Ncosφ∆λ
The Reference Ellipsoid and the
Lecture 10 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Lecture 7 GE 161- Geometric Geodesy
Curves on the Surface of the Ellipsoid
The Azimuth, Chord and Length of the Normal
Section
• The normal section azimuth can be computed if we
know the latitudes and longitudes of two points on the
surface of the ellipsoid.
• The whole process of computing of the normal section
azimuth, chord or distance involves two steps
(1) Converting the geodetic coordinates to
cartesian coordinates.
(2) Substitution of the cartesian coordinates to the
equations giving the normal section azimuth, chord
and length.
• Note that this procedure is just one of the several
possible procedures for computing normal section
azimuth, chord and length.
The Reference Ellipsoid and the
Lecture 10 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Lecture 10 GE 161- Geometric Geodesy
Curves on the Surface of the Ellipsoid
7
The Normal Section Azimuth
• Recall the coordinate conversion from
geodetic to cartesian…
• If we have two points with latitudes φ1 and
φ2 and with longitudes λ1 and λ2, the
cartesian coordinates (assuming points
are on the surface of the ellipsoid) are:
x1 = N1 cosϕ1 cos λ1 x2 = N2 cosϕ2 cos λ2
y1 = N 1 cosϕ1 sin λ1 y2 = N 2 cosϕ2 sin λ2
z1 = N1(1− e )sin ϕ1
2
z2 = N 2 (1− e2 )sin ϕ2
The Reference Ellipsoid and the
Lecture 10 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Lecture 10 GE 161- Geometric Geodesy
Curves on the Surface of the Ellipsoid
The Normal Section Azimuth
• However, in computing the normal section, we
may assume that the first point is on the
meridian of origin such that:
x1 = N1 cos ϕ1 y’
y1 = 0
z1 = N 1 (1 − e 2 ) sin ϕ1
y
x 2 = N 2 cos ϕ2 cos ∆λ
y 2 = N 2 sin ϕ2 cos ∆λ ∆λ
z 2 = N 2 (1 − e2 ) sin ϕ2
x’; λ=0
x
The Reference Ellipsoid and the
Lecture 10 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Lecture 10 GE 161- Geometric Geodesy
Curves on the Surface of the Ellipsoid
8
The Normal Section Azimuth
The azimuth of the normal section can be derived
following this figure:
[(x2-x1)sinφ1+(z2-z1)cosφ1]
y2
A12 (z2-z1) Chord AB
A12
Normal Line at A
A
φ1
y2 (x2-x1)
The Reference Ellipsoid and the
Lecture 10 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Lecture 10 GE 161- Geometric Geodesy
Curves on the Surface of the Ellipsoid
The Normal Section Azimuth
• It follows that the normal section azimuth
is:
y2
tan A12 =
x2 − x 1 sin ϕ1 + z2 − z1 cos ϕ1
• The chord between the two points is
simply computed as:
s = y22 + ( x2 − x1 )2 + ( z2 −z1 )2
The Reference Ellipsoid and the
Lecture 10 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Lecture 10 GE 161- Geometric Geodesy
Curves on the Surface of the Ellipsoid
9
The Normal Section Distance
• We are not actually interested in the chord distance, but in the actual
normal section distance s. However this can be computed using the
chord distance:
1 s 2 3 s 4 5 s 6 µ1 s 3 3µ2 s 4
s = s 1 + + + + + + ....
6 2r
40 2r
112 2r
2 2
r 5 2
r
where: r = x12 + y12 + z12
e '2 sin 2ϕ1 cos A12
µ1 =
1+η12 cos2 A12
sin 2ϕ1 − cos2 ϕ1 cos2 A12
µ2 = e '2
1+η12 cos2 A12
• For lines up to 100 km, this equation reduces (with an accuracy of 1
cm) to:
1 s 2
s = s 1 + + ....
6 2r
The Reference Ellipsoid and the
Lecture 10 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Lecture 10 GE 161- Geometric Geodesy
Curves on the Surface of the Ellipsoid
Reference:
• Rapp, Richard R., Geometric Geodesy,
Ohio State University, Ohio State USA.
The Reference Ellipsoid and the
Lecture 10 GE 161 – Geometric Geodesy Computation of the Geodetic Position:
Lecture 10 GE 161- Geometric Geodesy
Curves on the Surface of the Ellipsoid
10
