Geodesic source code for Silverlight 4.0

Geodesics

The shortest path between two points on an arbitrary surface is called a Geodesic, and on a sphere, it is a Great Circle. Modelling the surface of the earth as a perfect sphere, the shortest distance between any two locations on the surface is then described by a section of a Great Circle, ie an arc that lies on the plane that is described by the vectors between its start and end points and the Earth’s centre ( see figure 1 ).

With this information, one way ( and the way I have adopted ) to plot such a curve is as follows:

1. Generate the points of the curve in two dimensions using the parametric equation of a circle.
2. Transform the plane of the 2d curve into 3D space such that it intersects the end points on the sphere, and the sphere’s centre.
3. Project the transformed points back into 2D space using the Mercator projection equations.

A Parametric Representation of a Great Circle

The Mercator projection gives the 2D rectilinear coordinates (x,y) as a function of the latitude and longitude of a point on a sphere. However, it is easier to draw the point using a typical drawing API, if we have a representation that gives each point of the curve in terms of a single parameter. To derive such a function, we observe that the parametric equation for a circle is given by

Since a great circle is a rigid transformation of a circle in 3D space, it can also be represented as a function of a single parameter

where is a 2×3 matrix that transforms the plane circle to a location on a sphere. Using spherical coordinates, the Mercator projection of the curve specified above is then

Now that we have a suitable parametric equation, we can draw the geodesic with a series of connected line segments by varying the parameter, t.

Mathematical note – the parameterization given by this expression is highly non-uniform, meaning that there are many more points generated in some parts of the curve than in others. The mathematics of generating uniform ( or natural ) parameterizations belongs to the field of differential curve geometry and is beyond the scope of this article ( and my brain ).

Implementation notes

Inevitably, the mathematics alone is not sufficient to produce an implementation of a reuseable class for the Bing Silverlight control. There are two main issues to resolve; firstly, it is not immediately obvious how to derive from the provided MapShapeBase class to create new shape overlays and secondly, how to handle drawing the curves when they ‘wrap’ beyond the map’s viewable area ( this is easier to illustrate than to describe – see the figure opposite ).

Inheriting from MapShapeBase

I must confess that I cheated slightly in implementing the MapGeodesicPath class, in that I used Reflector to peer into the implementation of the base class. The existing derivations of this class simply defer to MapShapeBase for most of the work, which they can do since for a MapPolygon and MapPolyline there is a one-to-one relationship between Locations, latitude and longitude points on the map and Points, the actual 2D cartesian coordinates used to draw the shape. For the Geodesic, this is not the case, because we only want to specify the start and ending points of the curve, not every point in between. A solution is to delegate the point generation code to a secondary class, one that can be independently tested.

Splitting the curves at the map boundary

When the curves wrap around the map projection, they need to be split at the boundary. This is done by finding the parameter t for the longitude value of the boundary. Where the longitude value is +/- 180, this is straightforward as the equation above reduces to,

One of Alan Turing’s many contributions to mathematics and science during the 20th century was his 1952 paper on “The Chemical Basis of Morphogenesis” in which he suggested that a simple model of coupled differential equations could account for pattern formation in natural processes such as those found on animal coats. Such models are known as Reaction-Diffusion models, and take the following general form

The equation describes how the concentration of each chemical components in q evolves as a function of the other components. I have chosen to illustrate the Gray-Scott model; the physical derivation of the reactant term is described in detail in [2], the addition of the diffusion term and the resultant behaviour in [3]. The model has two chemical components, U and V and is described as follows.

The model is what is knows as an activator-inhibitor model, in which one chemical acts to inhibit the growth of the other.

Simulation

The simulation is implemented with Silverlight 2.0. For full details of the implementation see the separate post. For guidance in parameter selection, see [3], or just select a preset and modify it. Note that the Png image generation feature currently requires the Firefox browser.

Gallery

References

1. Turing,A.,1952. The chemical basis of morphogenesis [pdf]. Philosophical Transactions of the Royal Society, 237 pp.37-72
2. Gray,P. and Scott,S.K. 1985. Sustained Oscillations and Other Exotic Patterns of Behavior in Isothermal Reactions. Journal of Physical Chemistry, 89 pp.22-32
3. Pearson,J.,1993. Complex patterns in a simple system. Science, 261(5118) pp.189-192

It also happens to be the model used by the Open University’s course on Mathematical Logic and Number Theory, and is the reason for this article since I studied the course in 2007.

Emulator

The implementation details of the emulator are quite interesting, using a mixture of Microsoft technologies including F# and Silverlight, however I will save that for another article for those interested in such things. To use the Emulator just hit Reset, enter the program and initial registers and click Step to trace through the program.