**Getting into Shape:**

What Do Butterfly Wings and Renaissance Maps Have in Common?

What Do Butterfly Wings and Renaissance Maps Have in Common?

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**A Note on Method**

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**There are spaces in which the determination of position requires not a finite number, but either an endless series or a continuous manifold of determinations of quantity. Such manifolds are, for example, the possible determinations of a function for a given region, the possible shapes of a figure, and so on.**

**--Bernhard Riemann**

**Wing Patterns or Coastlines?**

*On Growth and Form*. Although Tobler’s work was graphically somewhat primitive, the numerical results from his regressions gave the first look at what types of information it might be possible to add to historical studies of early maps through the use analytical comparisons. Tobler’s methods were formalized to shape spaces by Small in several important papers such as “Techniques of Shape Analysis on Sets of Points. (5) ”

Through some very interesting algebra Bookstein derives functions that effectively separate the global and local error into affine and nonlinear components. The functions are vector valued and if the pairing of the points that are being transformed correspond to the homologies on our two maps we have effectively modeled the difference between the two point sets as a deformation.

*is the distance from the origin. The*

**r***function satisfies,*

**U(r)**.

Thus far I have utilized this thin-plate spline method on many maps and in a study that compares the longitudinal error in the Mediterranean basin on the 1507 and 1516 World Maps by Martin Waldseemüller. The method does have significant advantages over methods such as Polynomial Warping that I used in my study, “Warping Waldseemüller: A Phenomenological and Computational Study of the 1507 World Map” (9).

I have since utilized these methods on Roman and some Medieval cartography, like Portolan charts. Using these methods one can calculate distortion grids and scale and rotation isolines.

For the historian of cartography comparing the accuracy on a modern and early map has a number of formal and logical difficulties that must be considered before the application of any cartometric process.

1. The identification of tie points on the two maps to be compared is sometimes difficult and once selected are seldom distributed evenly across the surface of the two images. The problem is essentially one of homology. Finding homologous points on two maps may seem trivial but differences in landmark or coastline shape, an incorrect assignment of place names and changing scales require insight into the target image. The selection of points is especially important when using the simplest linear models without M-estimators where results can be error distribution sensitive.

2. The accuracy and error to be tested and compared may not be a relevant concept across the whole surface of the map. Accuracy is seldom evenly distributed especially in the case of maps of small scale that may be composed from a variety of sources. Discontinuities in error from scale changes are not only scalar but are vector quantities whose direction is important to determine. Rapid changes in accuracy across map surfaces may be difficult to handle with simple linear models and are better approached using local non-linear radial basis functions whose correlations may be statistically more relevant, but mathematically more complex to program. Decisions regarding the use of local or global methods cannot usually be made ahead of time and require experimentation in order to characterize the accuracy and decide on a methodology.

3. The substrate that the map is draw or printed on may have undergone distortion through shrinking, folding, or stretching. Distortion of this sort is especially important to consider in the case of environmentally sensitive materials such as vellum. The distortion of the medium not only effects the accuracy of statistical transformations that we are trying to perform but can also obscure the intent of the cartographer.

4. Spatial association does not necessarily imply causality. The warps and correlations that we calculate using cartometric techniques give real mathematical results but these may be extremely difficult to link to any historical meaning, event or cause. Care must be taken not to over interpret the results of these calculations and not to override historical and documentary context in the rush towards accuracy measures.

5. Historical cartometry is a problem in equifinality or process convergence. Similar types of distortion may arise from different causes making it difficult to derive exact causes for particular distortion patterns.

6. Experimentation is necessary and we must be prepared to use a variety of techniques to characterize the accuracy on a single map.

In the end the researcher who would employ these methods needs to critically consider each map to be studied on a case-by-case basis and would be wise to consider the words of R.A. Skelton, which although written in a different context, can be used as model for the cautions we must recognize in historical cartometry.

(3) Brian Harley, “Concepts in the History of Cartography: A Review and Perspective,” Cartographica 17 (1980): 54.

(5) C.G. Small, “Techniques of Shape Analysis on Sets of Points,” International Statistical Review 56 (1988): 243-257.

(6) Tomoki Nakaya, “Statistical Inferences in Bi-dimensional Regression Models,” Geographical Analysis 29 (1997): 169-185.

(7)P.J. Huber, “Robust Estimation of a Location Parameter,” Annals of Mathematical Statistics 35 (1964): 73-101 and Robust Statistics (New York: John Wiley, 1981) see also F. Hampel, Contributions to the Theory of Robust Estimation, PhD Thesis (Berkeley: University of California, 1968).

(8) Fred Bookstein, “Principal Warps: Thin-Plate Splines and the Decomposition of Deformations”, IEEE Transactions on Pattern Analysis and Machine Intelligence 11 (1989): 567-585 and Christopher G. Small, The Statistical Theory of Shape (Berlin: Springer-Verlag, 1996) 110

(10) Timothy R. Wallace and Charles van der Hanvel, “Truth and Accountability in Geographic and Historical Visualizations”, Cartographic Journal 42 (2005): 173-181.

(11) A. Braddeley, “Stochastic Geometry: an introduction and reading list,” International Statistical Review 50 (1982): 179-193.