lecture 8

1

a.s. caparas/06a.s. caparas/06

GE 161 – Geometric Geodesy

Lecture No. 8

Department of Geodetic EngineeringUniversity of the Philippines

The Reference Ellipsoid and the Computation of the Geodetic PosiThe Reference Ellipsoid and the Computation of the Geodetic Position: tion: Properties of the EllipsoidProperties of the Ellipsoid

Radii of Curvature on the Ellipsoid and Radii of Spherical

Approximation of the Earth

Radii of Curvature on the Ellipsoid and Radii of Spherical

Approximation of the Earth

GE 161 GE 161 –– Geometric GeodesyGeometric GeodesyThe Reference Ellipsoid and the The Reference Ellipsoid and the

Computation of the Geodetic Position: Computation of the Geodetic Position: Properties of the EllipsoidProperties of the Ellipsoid

Lecture 8Lecture 8

Normal Sections on the Ellipsoid Normal Sections on the Ellipsoid • Consider first a normal to the

surface of the ellipsoid at some point.

• A particular plane will cut the surface of the ellipsoid forming a curve which is known as the normal section.

• At each point there exist an infinite number of normal section as there exist an infinite number planes that that contain the normal line.

• However, at each point, there exist two mutually perpendicular normal sections whose curvature will be maximum and minimum.

• These normal sections is called the principal normal sections.

2



Lecture 8Lecture 8

Principal Normal SectionsPrincipal Normal Sections

On the ellipsoid the two principal normal sections are:

1.The Meridian or Meridional Normal Section – a plane passing through the point and the two poles.

2.The Prime Vertical Normal Section – a plane passing through the point and perpendicular to the meridian at that point.

MeridionalMeridional Normal Normal SectionSection

Prime Vertical Prime Vertical Normal SectionNormal Section



Lecture 8Lecture 8

Radii of Curvature of a Normal SectionsRadii of Curvature of a Normal Sections

In order to find the radius of curvature of any normal section at any arbitrary direction, we may utilize the Euler’s formula:

where:ρ= is the radius of curvature of the section (any arbitrary section)θ= is the angle measured from the meridian of the pointρ1=is the radius of curvature of the principal normal section with the

maximum curvatureρ2=is the radius of curvature of the principal normal section with the

manimum curvature

21 ρθ

ρθ

ρ

22 sincos1+=

3



Lecture 8Lecture 8

Radius of Curvature of the Principal Normal SectionsRadius of Curvature of the Principal Normal Sections• Meridional Radius of Curvature, M:

at the equator:

at the poles:

)f1(a

)e1(

a

)e1(

)e1(aM21

223

2

2

90 −=

−=

−

−==ϕ

23

22

2

)sine1(

)e1(aMϕ−

−=

220 )f1(a)e1(aM −=−==ϕ



Lecture 8Lecture 8

Radius of Curvature of the Principal Normal SectionsRadius of Curvature of the Principal Normal Sections• If the values of the M

were tabulated, they could be plotted with respect to an origin at the surface of the reference ellipsoid.

• The endpoints of the various M values would fall on a curve known as the locus of the centers of the meridional radius of curvature.

locus of the centers locus of the centers of the curvature of of the curvature of the meridianthe meridian

∆2∆1

4



Lecture 8Lecture 8

Radius of Curvature of the Principal Normal SectionsRadius of Curvature of the Principal Normal Sections• Prime Vertical

Radius of Curvaturep=Ncosφ

At the equator:Nφ=0=a

At the poles:

21

22 )sine1(

aNϕ−

=φ

p

)f1(aN 90 −

==ϕ



Lecture 8Lecture 8

Comparing M and N…Comparing M and N…

• We can see that M and N are minimum at points on the equator.

• At the poles M and N are equal with value equal to a/(1-f).

• If we take the ration of M and N, we will find that:

• Thus, N≥M where equality holds at the poles.

)e1()sine1(

MN

2

22

−ϕ−

=

5



Lecture 8Lecture 8

Radius of Curvature of the normal section at any given azimuthRadius of Curvature of the normal section at any given azimuth• using Euler’s formula we can determine

the radius of curvature letting θ=α=azimuth of the normal section from the north, ρ1=N and ρ2=M by:

Mcos

Nsin

R1 22 α

+α

=α

α+α=α 22 sinMcosN

MNR



Lecture 8Lecture 8

Gaussian Mean Radius of Curvature of Normal Sections at a pointGaussian Mean Radius of Curvature of Normal Sections at a point• The Gaussian Mean Radius R of all the

radii of curvature of all the normal section containing the normal line is given by:

• The value of R is helpful when a radius of a sphere that is to approximate the ellipsoid is required.

MNR =

6



Lecture 8Lecture 8

Example ProblemExample ProblemProblem:Compute for the radii of curvature of the two principal normal sections and the Gaussian Mean radius of curvature at the point whose geodetic latitude is 45°N on the Clarke Spheroid of 1866.

Solution:Given:φ=45°Nf=1/294.98 a=6,378,206 m e2=0.006768628177Find: N, M, and R

Solving for N:

Solving for M:

Solving for R:

21

22 )sine1(

aNϕ−

=

21

o2 )45sin770067686281.01(

6378206N−

=

m399.026,389,6N =

23

22

2

)sine1(

)e1(aMϕ−

−=

23

o2 )45sin70067862817.01(

)770067686281.01(6378206M−

−=

m501.330,367,6M =

399.6389026)(501.6367330(MNR ==

m225.169,378,6R =



Lecture 8Lecture 8

Earth as a SphereEarth as a Sphere

• Since the computation of some quantities on the surface of the ellipsoid is sometimes too complex to handle, geodesists uses the sphere as a model.

• This reduces the complexity of deriving formulas and evaluating quantities.

• In order for us to use a sphere as a reference model, we need to find a sphere which is equivalent to the reference ellipsoid that we are using

7



Lecture 8Lecture 8

Earth as a SphereEarth as a Sphere

• There are several way of finding a sphere equivalent to the reference ellipsoid:1. Equal surface area2. Equal volume3. Ellipsoid’s mean radius

- Gaussian- Mean of the three semi-axes



Lecture 8Lecture 8

Radii Approximation to the Earth or Mean Radius of the Earth as a SphereRadii Approximation to the Earth or Mean Radius of the Earth as a Sphere

A suitable radius may be defined by equating the expressions of the quantities being compared:

1. Spherical radius having the same area s the ellipsoid

2. Spherical radius having the same Volume as the ellipsoid

−−−= ....e

302467e

36017e

611aR 642

A

3 2v baR =

8



Lecture 8Lecture 8

Radii Approximation to the Earth or Mean Radius of the Earth as a SphereRadii Approximation to the Earth or Mean Radius of the Earth as a Sphere

3. Spherical radius having the mean radius of the three semi-axes of the ellipsoid

4. Gaussian mean radius as the radius of the sphere

3)baa(Rm

++=

R= MN



Lecture 8Lecture 8

Example ProblemExample Problem

Problem:What are the radii of the equivalent spheres of the Clarke Spheroid of 1866.Solution:Given:f=1/294.98a=6,378,206 me2=0.006768628177Find: Rm, RA, and RV

Solving for Rm:

Solving for RA:

Solving for Rv:

3)baa(Rm

++=

3)497.635658363782066378206(R m

++=

m499.998,370,6R m =

−−−= ....e

302467e

36017e

611aR 642

A

m873.996,370,6R A =

3 23 2v )497.6356583()6378206(baR ==

m339.990,370,6R v =

lecture 8

Documents