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.

[silverlight: Geodesic.xap,520,384,false]

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

$$!begin{pmatrix} x \ y end{pmatrix} = begin{pmatrix} r cos t \ r sin t end{pmatrix} $$

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

$$!begin{pmatrix} x \ y \ z end{pmatrix} = mathbf{R}cdotbegin{pmatrix} r cos t \ r sin t end{pmatrix} $$

where $$mathbf{R}$$ 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
$$!mathbf{C}(t) =begin{pmatrix} lambda \ tanh^{-1} left(sin phi right)end{pmatrix} = begin{pmatrix} tan^{-1}left(y/xright)\ tanh^{-1}z end{pmatrix}$$
Where $$lambda$$ and $$phi$$ are the longitude and latitude of the point to be projected, respectively. Writing the equation out in full gives,

$$!mathbf{C}(t) =begin{pmatrix}

tan^{-1}left(frac{R_{2,1}cos t + R_{2,2}sin t}{R_{1,1}cos t + R_{1,2}sin t}right)\
tanh^{-1}(R_{3,1}cos t +R_{3,2}sin t)
end{pmatrix}
$$

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,
$$!t = -tan^{-1}frac{R_{1,2}}{R_{2,2}} $$
For other longitude values, we simply offset the longitude values by the required amount, and calculate the value of t for the new matrix.

Last week myself and a group of friends took a winter camping trip up to the Lake District. An odd decision, it could be argued, seen as the region has recently seen both some of its worst flooding and coldest winters in recent years.  Nonethless, equipped with ice axe, crampons and a generous supply of Kendal Mint Cake we set out to climb Scafell.

Some of the things I learned

1. Elephants, though clearly signposted, are not common in the lake district. I didn’t see a single one all weekend. Sheep, however, are plentiful.
2. It may be effective as a deterrent against the Ravenous Bugblatter Beast of Traal, but a towel is no replacement for an insulating mattress.
3. Pop up tents, though useful on the beach, are less effective in torrential rain and gale force winds.
4. Retracing your steps back through the snow sounds easy, and it probably is for other people.
5. The Wasdale Head Inn serves an excellent selection of whiskeys, of which Lagavulin is the best. I should know, I tried all of them.
6. Swimming in Wastwater in January is almost exactly as cold as you’d expect.

The Route

The map below, which requires Microsoft Silverlight, was exported from my GPS wristwatch and shows the route we took up Scafell before visibility made further progress a bit risky, and so we retraced our steps back through the snow. Or at least tried to before our steps mysteriously disappeared. There is also a Photosynth from part of our route available.

[silverlight: osmap.xap,520,350,show]

Swimming in Wastwater

Although as an amateur triathlete I’ve done many open water swims, I’ve never done one in winter and never without a wetsuit. So, in an attempt to kill both birds with one stone I decided to go for a swim in Wastwater in January. I lasted about two minutes. I’m no Lewis Gordon Pugh. It was cold, but it really wasn’t as bad as you might think.  The water was beautifully clear but unfortunately sharp rocks do not a beach make.

Gallery

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