
Hans Havlicek: Mathematical Cartography
This is a series of lectures and exercises for students of Mathematics or Descriptive Geometry. We present the mathematical background that is used when mapping the (round) surface of the earth to a (flat) plane.

The earth: Nov. 11th, 2000 at 16:00 UTC

The Earth and Mathematics
For the purposes of geography it is usually sufficient to assume that the earth is a sphere; this point of view is also adopted in our lectures.
In geodesy, however, the oblateness of the earth at the poles has to be taken into account. So an ellipsoid of revolution is taken as a mathematical model of the earth.
Neither a sphere nor an ellipsoid can be mapped in a plane without distortion. The theory of distortion is a central part of these lectures on mathematical cartography.
It seems that most people are not aware of the fact that the maps in their atlases are based upon a discipline of higher mathematics, called differential geometry.


Course
1. Introduction
 Mathematical models of the earth
 Earth and sun: tropics and polar circles
 Spherical coordinates (longitudinal and latitudinal circles)
 Basic differential geometry of curves and surfaces

Equidistant projection on a cylinder

2. Conical Projections
 Conical, azimuthal, and cylindrical projections
 Equidistant projections
 The theory of distortion I: Distances (Tissot's indicatrix)
 Conformal projections in general
 Conformal conical projections
 Stereographic projection
 Mercator's conformal cylindrical projection
 The theory of distortion II: Angles
 Equalarea projections in general
 Lambert's equalarea conical, azimuthal and cylindrical projections
 The theory of distortion III: Areas
 Perspective projections in general
 Gnomonic projection


3. Pseudoconical Projections
 Pseudoconical and pseudocylindrical projections
 Bonne, StabWerner, and Sanson
 Mollweide's projection
 Modifying a map projection
 Affine linear combination of maps: Eckert V (pseudocylindrical)


4. The Aspect
 Normal aspect
 Transverse aspect
 Oblique aspect
 Transformation formulas.
Lecture Notes
Lecture notes (Formelsammlung Kartenentwürfe, mit zahlreichen Illustrationen / Formulas for Map Projections, in German, with many illustrations) can be downloaded from my page on Publications.


Illustrations
All map projections that are exhibited in these lectures (and many more) can be found in our Picture Gallery of Map Projections.


The following material has been prepared in part by one of my students, Harald Veronik, with the programme 'Maple V'.
From North to South. This sequence of images starts with the equidistant azimuthal projection onto the tangent plane at the north pole. This is gradually changed to
equidistant conical projections with the standard parallel moving towards the equator. So we arrive at the equidistant cylindrical projection. As the standard parallel moves further to the south, we finally obtain the equidistant azimuthal projection onto the tangent plane in the south pole.


Tissot's Indicatrix. An easy way of visualising the distortion of a map at a given point P is Tissot's indicatrix, which is an ellipse centred at P. If the indicatrix is a circle then the map is conformal (angle preserving) at P, otherwise the main axes of the ellipse give the directions of maximal and minimal distortion.
In our example you can see indicatrices for Sanson's projection. Here the image is undistorted (up to the scale) along the equator and the meridian 0° (Greenwich). It is worth noting that all indicatrices, though they of different shape, have the same area. This illustrates the fact that Sanson's projection is area preserving.
