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Waldo,
My use of the word 'advanced' might be too strong a word for the mathematics...what I meant by it was really in the sense of software and graphics at this point. You are quite correct in fact that the integrated sum of squares yields equivalent results for single charts. I have actually tried it on a few in my study group.
I have been working with a group of mathematicians who are more concerned with morphometrics as applied to biological applications and imaging....much in the same way as you played with in the 1970's. The available statistical algorithms for analyzing non-Euclidean shape spaces is extremely large and I have had some really good luck using the thin-plate splines and being able to separate the linear and non-linear contributions to the distortion. I have looked at these on 22 charts so far for a wide date range and the use of the bi-harmonic equation and the eigenfunctions of the spline bending energy have allowed me to look not only at the variation from reality or modern forms but have also allowed intra sample variations to be calculated....
...any shape change can then be re-expressed as a sum of partial warps which are vector multiples of the principal warps just as in your vector field models which I agree need to be looked at more closely...the idea of a flow of distortion and the way it might change across an historic map intrigues me from both an historical and graphical standpoint.
The only real difference in my using the splines is the ease in extracting the intra sample statistics which have been quite useful in looking at the evolution of accuracy within the sample...not just applied to the modern form of the chart....so for me it is really the fact that I am simply more familiar and comfortable with programming the biharmonic equations....
The whole beginning of my talk at the conference will focus on your methods and their development....I often re-read your papers on Bi-dimensional regression and Medieval distortions and I am amazed at the geometric insight you brought to this work...I find new food for thought in them every time I re-read them....I have links to all of them on my research website....all of us who do any work in historical cartometry stand on your shoulders...
...the conference will be webcast a couple of weeks after it happens...I will let you know when it is available...also it seems like Adams was the first and last to use the Abelian functions...
Talk to you soon.
John
John W. Hessler
Fellow of the Royal Geographical Society Senior Cartographic Librarian
Geography and Map Division Library of Congress Washington DC 202-707-7223 What any picture, of whatever form, must have in common
with reality in order to depict it--correctly or incorrectly-- in any way at all, is logical form, i.e. the form of reality. --Wittgenstein
Tractatus Logico-Philosophicus This is a MapHist list message (when you hit 'reply' you're replying to the whole list) o + o + o + o + o + o + o + o + o + o + o + o + o + o + o + o + o + o + o + John, Thank you for sending this, with abstracts. I cannot come to the meeting, but hope to get the proceedings when published. But, I can not agree that radial basis functions and thin plate splines are more 'advanced' than my 'tuning' method since they give the same results but within a much more complicated mathematical structure. My notion of using the integrated sum of squares of the four partial derivatives as a measure of (local, then) total distortion has possible merit and my be simpler than integration Tissot's indicatrix. A more important and interesting direction is to recognize the vector fields can be decomposed into divergence and curl and potential fields can therefore be derived and applied to the charts. I too have recently gotten into Riemannian manifolds and laplacian eigenfunctions and their application to cartography. I assume that you are familiar with Mitchel's work for Hammond's Atlas with optimal conformal representations using Chebychev's conjecture. Waldo Tobler Geographer [email protected] http://www.geog.ucsb.edu/~tobler On May 12, 2010, at 6:24 AM, John W. Hessler wrote: > > Histers, > > Just a reminder that the LOC and Philip Lee Phillips Society > Conference on Portolan Charts is next week _______________________________________________ MapHist: E-mail discussion group on the history of cartography hosted by the Faculty of Geosciences, University of Utrecht. The statements and opinions expressed in this message are those of the author and do not necessarily reflect those of the University of Utrecht. The University of Utrecht does not take any responsibility for the views of the author. List Information: http://www.maphist.nl Maphist mailing list [email protected] http://mailman.geo.uu.nl/mailman/listinfo/maphist |
_______________________________________________ MapHist: E-mail discussion group on the history of cartography hosted by the Faculty of Geosciences, University of Utrecht. The statements and opinions expressed in this message are those of the author and do not necessarily reflect those of the University of Utrecht. The University of Utrecht does not take any responsibility for the views of the author. List Information: http://www.maphist.nl Maphist mailing list [email protected] http://mailman.geo.uu.nl/mailman/listinfo/maphist
