Principal Directions with surf.ls

Suppose we plot a quadratic surface as in:

library(spatial)
example(surf.ls)

How can we plot lines on the graph to show the
principal directions/axes of the elliptical contours
and how do we get them?

In the example there are no closed ellipses, with an open, north-facing 
depression with flow from the west, south-west, south, south-east and 
east towards the north at the centre of the northern edge.

The contour lines can be retrieved with contourLines() on the same object
as contour() is used on. However, they are just a generalised
visualisation of the surface represented by the matrix. The underlying 
question might then be: what is the aspect and slope of the surface, in 
this case a smoothed surface, what are the plan and especially profile 
curvatures, and what are their length attributes (ie. patches of similar 
aspect along a profile curvature.

In GIS terms this would involve geomorphometrics, often computed using a 3 
x 3 filter, and then flow analysis to find the directions (on a digital 
elevation model, these would be similar to predicting drainage channel 
networks. Landslip and avalanche studies often need slope length too, 
which involves measuring patch characteristics in aspect and profile 
curvature - that is extracting patches with interesting pattern 
signatures, and checking whether the patch is "straight" in some sense, 
and longer than a minimum accumulation threshold to feed mass movement. 

These are general observations in the case that we don't have a smoothed 
surface. But here we have a model, and I'd guess that the underlying model 
could be used, rather than its predictions. 

I couldn't see a package with a vector field plot, there was a suggestion 
in:

http://finzi.psych.upenn.edu/R/Rhelp02a/archive/35933.html

but you'd still need the slope and aspect to get the length and direction 
of the arrows (for display only). A classic reference is:

Horn, B. K. P. (1981). Hill Shading and the Reflectance Map, Proceedings 
of the IEEE, 69(1): 14-47

and Horn's algorithm has stood the test of time, as has Zevenbergen and 
Thorne:

Zevenbergen, L. W., Thorne, C. R. 1987. Quantitative analysis of land
surface topography. Earth Surface Processes and Landforms, 12 (1), 4756.

It could well be that the surface you are considering is something other 
than elevation (temperature, etc.), but getting further would depend on 
the specific characteristics of the study, and whether the surface is a 
fitted trend surface or not. With a fitted trend surface model, it ought 
to be possible to use the model. If thinking in terms of geomorphometrics 
is unhelpful, please accept that geographers "see" things in those kinds 
of ways!

Hope this helps,

Roger

Thanks.

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Roger Bivand
Economic Geography Section, Department of Economics, Norwegian School of
Economics and Business Administration, Helleveien 30, N-5045 Bergen,
Norway. voice: +47 55 95 93 55; fax +47 55 95 95 43
e-mail: Roger.Bivand at nhh.no