Geodesy part1

INTRODUCTION TO GEODESY AND CARTOGRAPHY

• Course Overview: Introduction to Geodesy and Cartography, led by Philippe De Maeyer, specifically designed for UGent students.

REPRESENTING THE WORLD

• Description of the Earth's surface:

  • The "real Earth" refers to the topographical and bathymetric surface, which is complex and difficult to describe.

  • The need arises to represent both planimetric (horizontal plane) and altimetric (vertical/hieght) positions of surface points.

• Importance of a Cartographic System:

  • To transform points from the topographical surface to a mathematically describable reference body known as the "datum".

  • Cartographic System Functions:

    • The points on the datum are:

      • Scaled

      • Projected (or projected and scaled)

      • Defined within a coordinate system.

  • Definitions of Various Datums:

    • Horizontal Datums:

      • Serves to describe positions (latitude and longitude) relative to a defined reference body (datum).

    • Vertical Datums:

      • Used for describing land elevations and water depths relative to a reference height (vertical datum).

CARTOGRAPHIC SYSTEM AND GEODESY

• Definition of a Cartographic System:

  • A combination of a geodetic reference system, cartographic projection, and a coordinate system.

• Functionality of Coordinates:

  • The mapping equations are defined as:

    • x = f(φ, λ) (longitude)

    • y = f’(φ, λ) (latitude)

• Geodesy:

  • Defined by the International Association of Geodesy (IAG) as the science concerned with the Earth's shape, size, and gravity field.

COMPLEXITY OF GEODESY

• The "real Earth", or topographical surface, involves:

  • Macro-level forces:

    • Interaction of gravity and centrifugal forces impacting the viscous body of the Earth, resulting in an ellipsoidal shape.

  • Meso-level forces:

    • Induced by geological (endogenic) and external (exogenic) processes that model the Earth's surface.

GRAVITY AND EARTH'S SHAPE

• Explanation of the Earth's Shape:

  • The Earth assumes an ellipsoidal form due to:

    • Centrifugal force arising from its rotational speed.

    • Gravitational attraction that shapes the ellipsoid.

  • Gravitational acceleration varies:

    • At the poles: 9.832 ext{ m/s}^2

    • At the equator: 9.780 ext{ m/s}^2

    • The reduction in acceleration is quadratic concerning the distance from the center of the Earth.

• Discussion of Equatorial vs. Polar Acceleration:

  • The acceleration due to gravity at the poles is greater than that at the equator leading to a weight increase of approximately 0.5% at the poles versus the equator.

• Universal Law of Gravitation:

  • Newton's Law states that any two point masses attract each other with a force given by:

    • F = G rac{m1 m2}{r^2} ,

    • Where:

      • F = gravitational force

      • m1 and m2 = masses of the objects

      • r = distance from the center of each mass

      • G = gravitational constant.

ELLIPSOIDS AND ELLIPTICAL MATHEMATICS

• Definition of an Ellipsoid:

  • An ellipsoid of revolution (spheroid) is defined as a quadratic surface created by rotating an ellipse around its minor axis.

• Parameters of Ellipses:

  • Major axis (a), minor axis (b) and foci location.

  • Equation of an ellipse:

    • !f(x,y) = rac{x^2}{a^2} + rac{y^2}{b^2} = 1

  • The relationship of the ellipse centered at origin, foci coordinates, and definitions of vertices and co-vertices.

• Further Definitions:

  • Flattening (f) and eccentricity (e):

    • f = rac{a - b}{a} ,

    • e = rac{ ext{sqrt}(a^2 - b^2)}{a} .

TRANSITION FROM SPHERE TO ELLIPSOID

• Historical Perspective on Earth’s Shape:

  • Ancient perceptions characterized the Earth as a sphere.

  • Eratosthenes (3rd Century BC) calculated Earth's circumference using shadow lengths at different locations:

    • Estimated values ranging approximately between 39,375 km and 46,620 km depending on stadium length.

  • Discussions during the Renaissance examined the Earth's actual shape:

    • Prolate spheroid (Descartes-Cassini) and oblate spheroid (Newton-Huygens) distinctions made.

MULTIDIMENSIONAL SPHERE AND GEOID

• Discussion of Geoid:

  • The concept of the geoid is established by Gauss to represent the Earth's mathematical figure based on gravity.

  • Defined as an equipotential surface with gravitational characteristics aligned with gravity's action.

• Modern evolution in definition:
  

  • Shift from defining geoid relative to sea levels towards understanding its dynamic with gravitational measurements informed by satellite data.

GEOID VERSUS ELLIPSOID

• Variations in the geoidal surface relate primarily to density distributions inside Earth and are not accurately represented on the crust.

• New geoid models help address complex variances in gravitational measurements, evidenced by interpolated extensions of geoid measurements from satellite data.

GEODETIC REFERENCE SYSTEMS

• Components defining a cartographic reference system include:

  • A geodetic reference system based on a defined ellipsoid.

  • The need for local and global datum recognition depending on requirements of accuracy and application.

GEOGRAPHIC COORDINATES: LATITUDE AND LONGITUDE

• Latitude: Angles measured in the meridian plane

• Longitude: Angles measured in the equatorial plane relative to the prime meridian.

• Calculation of geographic positioning employs both geodetic and geocentric latitude references for precision in measurements.

CARTOGRAPHIC SYSTEM FUNCTIONALITY

• Geodesy and cartography interplay defined by understanding the geometry of the Earth through various applications including:

  • Geographic to Cartesian transformations.

  • Scale representation within maps affected by projection types and geometrical distortions.