History of Geodesy

  • Geodesy is the science that deals with the measurement and representation of Earth, including its gravitational field, in a three-dimensional time-varying space.
  • The understanding of Earth’s shape has evolved significantly over time.

Early Beliefs about Earth’s Shape

  • Historically, there was a widespread belief in a flat Earth.
  • This belief was prevalent in many ancient cultures and persisted for many centuries.

Modern Understanding of Earth’s Shape

  • The ideas of Eratosthenes demonstrate an early understanding of the Earth's spherical shape. He famously calculated the Earth's circumference.
  • Modern geodesy has replaced outdated beliefs with scientific evidence demonstrating that Earth is an oblate spheroid.

What is Geodesy?

  • Geodesy is the scientific study of Earth's geometric shape, orientation in space, and gravitational field.

Types of Geodesy:

  • Physical Geodesy: Examines the Earth's gravity field and how it affects the shape of the geoid, which is the theoretical shape of the Earth under the influence of Earth's gravity.
  • Satellite Geodesy: Involves measurements from satellites to determine the Earth’s shape, orientation, and gravitational field.
  • Geometric Geodesy: Focuses on measuring distances, angles, and coordinates on the Earth's surface.

Geometric Geodesy

Shape of the Earth

  • While it is impractical to mathematically describe the Earth's entire topological surface, geodesy aims to develop mathematical models that approximate Earth's mean sea-level surface.
  • The most commonly used model is the ellipsoid, which is formed by rotating an ellipse about its minor axis.
  • The properties of the ellipsoid include:
      - Major Axis: In the equatorial plane.
      - Minor Axis: Coincides with the Earth’s spin axis.

Ellipse Definition

  • An effective geometrical representation used in geodesy is the ellipse, defined as:
      - A conic section in which the total distance from two fixed points (foci) remains constant as a point moves along the curve.
Parameters of the Ellipse:
  • Semi-Major Axis (a): The longest radius, aligned with the ellipse's major axis.
  • Semi-Minor Axis (b): The shortest radius, aligned with the ellipse's minor axis.
  • Foci (F1, F2): Two fixed points equidistant from the center, where the total distance to any point on the ellipse is constant.

Properties of the Ellipse

Flattening

  • Flattening measures how much a sphere is compressed to form an ellipse:
      - First Flattening: f=a−baf = \frac{a - b}{a}
      - Second Flattening: f′=b−abf' = \frac{b - a}{b}
      - Third Flattening: f′′=1ff'' = \frac{1}{f}

Eccentricity

  • Eccentricity (e or ε) quantifies how much an ellipse deviates from being circular:
      - First Eccentricity: e=a2−b2ae = \frac{a^2 - b^2}{a}
      - Second Eccentricity: e′=a2−b2be' = \frac{a^2 - b^2}{b}
      - Linear Eccentricity: E=aimeseE = a imes e
      - Angular Eccentricity: extαext{α} (to be defined in context).

Definitions of Latitude

Geodetic Latitude ($ ext{φ}$)

  • The angle between the normal to the ellipsoid at a point and the equatorial plane.

Geocentric Latitude ($ ext{ψ}$)

  • The angle at the center of the ellipse between the plane of the equator and a line to a point on the surface.

Parametric/Reduced Latitude ($ ext{β}$)

  • The angle at the center of the sphere tangent to the ellipsoid along the equator. Defined between the plane of the equator and the radius to the point along the sphere intersected by a line perpendicular to the equator.

Mathematical Relationships Between Different Latitudes

Geodetic Latitude Relationships

  • Relationship Equations:
      - Geodetic Latitude vs. Reduced Latitude: b=aimesextsinφextcosφb = \frac{a imes ext{sinφ}}{ ext{cosφ}}

Geocentric Latitude Relationships

  • Geocentric Latitude Formulation:
      - General formulations can be derived based on previous equations for tight coupling of geodetic coordinates.

Special Latitude Considerations

Significant Latitudes

  • Arctic Circle: 66° 34′ N (66.57°)
  • Tropic of Cancer: 23° 26′ N (23.43°)
  • Tropic of Capricorn: 23° 26′ S (23.43°)
  • Antarctic Circle: 66° 34′ S (66.57°)

Problems for Practice

  1. Given values: a=6378206.4a = 6378206.4, find bb.
  2. Determine the minor axis length.
  3. Find polar flattening using values.
  4. Calculate first eccentricity.
  5. Calculate second eccentricity.
  6. Determine sine, cosine, and tangent of angle a.
  7. Find the linear eccentricity.
  8. **Express first eccentricity as a function of flattening.
  9. **Compute equivalent reduced latitude from geocentric latitude of a station.
  10. **Calculate equivalent geocentric and reduced latitudes from given station data.