lecture 9

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

GE 161 – Geometric Geodesy

Lecture No. 9

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: Curves on the Surface of the EllipsoidCurves on the Surface of the Ellipsoid

Normal Sections, Unique Normal Sections, and Reciprocal Normal

Sections

Normal Sections, Unique Normal Sections, and Reciprocal Normal

Sections

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: Curves on the Surface of the EllipsoidCurves on the Surface of the Ellipsoid

Lecture 9Lecture 9

Normal SectionsNormal Sections• Recall that we have

defined a normal section as a curve formed by the intersection of the plane that contains the normal at a given point to the surface of the ellipsoid

• Physically, the normal section can be viewed when an optical instrument such as a theodeolite or total station is set-up above a point

• A normal plane is the plane swept out by the moving the telescope in the vertical direction

Normal PlaneNormal Plane

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Lecture 9Lecture 9

Normal SectionsNormal Sections

• By sighting on a distant point, we define a plane that contains the normal at the observation site, and passes through the observed site

• The intersection of this plane with the ellipsoid forms the normal section from the observation to the observed point

A B



Lecture 9Lecture 9

Normal SectionsNormal Sections• Consider the normal line

to point B• This normal line will

intersect the minor axis at some point

• Now consider the normal line at point B

• The normal line at point B will intersect the minor axis at a point different from the point of intersection of the normal line at point A and the minor axis

A B

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Lecture 9Lecture 9

Normal SectionsNormal Sections

• Consider the two normal plane to the two points

• We can see that two normal planes will nor coincide in any way

• Thus, the two normal planes will create two different normal sections

• And if we have two normal planes, we have two normal sections

A

B



Lecture 9Lecture 9

Reciprocal Normal SectionsReciprocal Normal Sections• In general, if we have two

points on the ellipsoid whose latitudes and longitudes are different, there exist two different normal section that contain both points

• The normal section from point A to point B and the normal section from point B to point A

• These two normal sections is known as the reciprocal normal sections

A

B

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Lecture 9Lecture 9

Complication of having Reciprocal Normal SectionsComplication of having Reciprocal Normal Sections• The presence of

reciprocal normal sections creates a problem in when observation are used in the computations

• We can see that with the observed interior angles of the triangle, we cannot have a closed figure

• Therefore, theoretically, no matter how good our observations are, we still cannot have a closed observed polygon on the surface of the ellipsoid



Lecture 9Lecture 9

Unique Normal SectionUnique Normal Section• However, there are certain

cases in which the normal section between two points is unique

• There are two cases in which there exist a unique normal section between two points:

1.When the two points are on the same meridian

2.When the two points are on the same parallel

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Lecture 9Lecture 9

Separation Between RNSSeparation Between RNS

• We can express the differences between the RNS in terms of the quantities that separates them

• There are two principal separations between the RNS.

• However, a third quantity is needed to consider to at the two principal separation

• The separations between RNS are:

1. Angle in between2. Linear Separation3. Azimuth Separation

Azimuth Azimuth SeparationSeparation

Linear Linear SeparationSeparation

Angle between the Angle between the normal section normal section planesplanes

A

B



Lecture 9Lecture 9

Angle between the RNSAngle between the RNS

• The angle between the intersecting normal section planes denoted by f is given by:

2 212 12cos cos sinmf e A Aσ ϕ=

2 212

1 cos sin22 mf e Aσ ϕ=

2 2m 12

1

1 sf e cos sin2A2 N

ϕ=

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Lecture 9Lecture 9

Linear Separation between RNSLinear Separation between RNS

• The linear separation between the reciprocal normal section denoted by d is generally given by:

• The maximum linear separation occur when θ=σ/2, the equation becomes:

22 2

m 12ed s( - ) cos sin2A4

σ θ ϕ=

22 2

m 12ed s cos sin2A16

σ ϕ=

2 22m 122

1

ed s cos sin2A16

sN

ϕ=



Lecture 9Lecture 9

Linear Separation between RNSLinear Separation between RNS

As a numerical example:

• For a line whose φm=45°N and A12=45°:

s 200 km 100 km 50 kmdmax 0.050 m 0.006 m 0.0008 m

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Lecture 9Lecture 9

Azimuth Separation between RNSAzimuth Separation between RNS

• The azimuth separation between the reciprocal normal section denoted by ∆ is given by:

22 2 2 22m 12m 12

1

e cos sin2A e cos sin2A4 4

sN

σ ϕ ϕ

∆ = =



Lecture 9Lecture 9

Azimuth Separation between RNSAzimuth Separation between RNS

As a numerical example:

• For a line whose φm=45°N and A12=45°:

s 200 km 100 km 50 km∆” 0.36” 0.09”

0.023”

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Lecture 9Lecture 9

Reference:Reference:

• Rapp, Richard R., Geometric Geodesy, Ohio State University, Ohio State USA.

lecture 9

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