Lectures 3

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Lectures 3 4

• Geometry of the Ellipse

Need to understand ellipsoidal geometry becomes
obvious when computing distances, azimuths etc
on this surface. Useful for understanding shape
of the earth, error ellipses and satellite orbits
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The problem To precisely determine the shape of
the earth (geodesy) Complicated procedure
which requires geometry, geodesy, applied
mathematics. triangulation. astronomy, gravity
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Shape of the Earth

• Flat earth models are still used for plane
  surveying, over distances short enough so that
  earth curvature is insignificant (less than 10
  kms).
• Spherical earth models represent the shape of the
  earth with a sphere of a specified radius.
  Spherical earth models are often used for short
  range navigation (VOR-DME) and for global
  distance approximations. Spherical models fail to
  model the actual shape of the earth. The slight
  flattening of the earth at the poles results in
  about a twenty kilometer difference at the poles
  between an average spherical radius and the
  measured polar radius of the earth.
• Ellipsoidal earth models are required for
  accurate range and bearing calculations over long
  distances. Loran-C, and GPS navigation receivers
  use ellipsoidal earth models to compute position
  and waypoint information. Ellipsoidal models
  define an ellipsoid with an equatorial radius and
  a polar radius. The best of these models can
  represent the shape of the earth over the
  smoothed, averaged sea-surface to within about
  one-hundred meters.

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Shape of the Earth

• For precise work use spheroid of reference
• Spheroid is the figure described by the rotation
  of an ellipse about one of its axis oblate,
  prolate
• Earth has shape of oblate spheroid
• Local and global spheroids
• Geoid The equipotential surface that most
  closely approximates to mean sea level is known
  as the geoid.
• Spheroid Ellipsoid

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Geometry of the Ellipse
The Basic Ellipse
F1,F2 Foci of the ellipse O
Center of the ellipse a Semi major axis of the
ellipse b Semi minor axis of the ellipse P
An arbitrary point on the ellipse
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Definition of an Ellipse
An ellipse is a set of points in a plane whose
distances from two fixed points in a plane have
a constant sum
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Definition of a Rotation Ellipsoid
Reference ellipsoids are usually defined by
semi-major (equatorial radius) and flattening
(the relationship between equatorial and polar
radii). Other reference ellipsoid parameters
such as semi-minor axis (polar radius) and
eccentricity can be computed from these terms.
The polar flattening, f
The first eccentricity, e
The second eccentricity, e
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Ellipsoidal parameters of the Australian National
Spheroid
a semi major axis
6378160m b semi minor axis b a(1-f)
6356774.719m e eccentricity
0.081820180 e2 e2 (a2 - b2)/a2
0.00669438002290 e 2nd
eccentricity 0.082095437 f
flattening f (a - b)/a
0.003352892

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Comparison of Ellipsoidal Parameters
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Example
Given a 6378137 f
0.003353 Calculate b and e2 b a(1-f)
6356752.314 e2 1- b2/a2 0.006694
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Equation of Tangent
General form y mx c
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Equation of Normal
General form y mx c
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Recap from lecture 3
equation of ellipse
ellipse parameters
equation of the tangent equation of the normal

General form y mx c
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Position of a point on the spheroid
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The Meridian Ellipse
The ellipse which defines a spheroid is called
the meridian ellipse. The vertical is the
direction of gravity at that point, it is unique
and physically exists A normal at any point is
the direction of the perpendicular line to an
ellipsoid at that point
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The reduced latitude
Useful in determining the relationship between x
y, z coordinates and spheroidal f, l
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The Geocentric Latitude
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Relationships between various latitudes
What is the astronomic latitude fA?
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Parametric Equation of an Ellipse
x a cos b y b sin b
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Relationships Between the Different Latitudes
x a cos b y b sin b
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Other Geometrical Relationships
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Other Geometrical Relationships
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Coordinates of Intersection of Normal with the
Spin Axis
P
M
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Radius of Curvature
Figure on Pag173 Hosmer
On a spheroid the radius of curvature changes
from point to point and at a single point it
varies with direction. Radius of curvature is a
minimum in the plane of the meridian and a
maximum in the plane of the prime vertical.
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Radius of Curvature in the Meridian Plane
Pic Page 54 Jackson
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Radius of Curvature in the Prime Vertical
use diag from last year
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Radius of Curvature of Normal Section in any
Azimuth
Pic page 177 Hossman
Eulers theorem
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Mean Value of the Radius of Curvature
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Distances along a Meridian
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Length along a parallel