Thursday, October 05, 2006

The newest results on my studies of the Waldseemuller Map come from a technique known as a thin-plate spline. You can find more information on splines in the link section of the blog. The spline transformation technique allows for the isolation of scale and shape deformation at various scales and gives both local and global deformatiom information. This type of local information cannot be produced in more global techniques like Polynomial Warping. (see my paper in Cartographica, June 2006, Warping Waldseemüller: A Phenomenological and Computational Study of the 1507 World Map also see Coordinates article in links). A deformation grid can be generated for any area of the map. The regressions produced by these spline models show areas where different geographical sources may have been used.

The figure above shows the values of an eigenvector for the longitudinal displacement for the African sheet of the 1516 Carta Marina. These results have allowed us to compare the 1507 World map with the 1516 map.

The thin plate spline is the two-dimensional analog of the cubic spline in one dimension. It is the fundamental solution to the biharmonic equation.

Given a set of data points, a weighted combination of thin plate splines centered about each data point gives the interpolation function that passes through the points exactly while minimizing the so-called "bending energy." Bending energy is defined here as the integral over of the squares of the second derivatives. Regularization may be used to relax the requirement that the interpolant pass through the data points exactly. The name "thin plate spline" refers to a physical analogy involving the bending of a thin sheet of metal. In the physical setting, the deflection is in the direction, orthogonal to the plane. In order to apply this idea to the problem of coordinate transformation, one interprets the lifting of the plate as a displacement of the or coordinates within the plane. Thus, in general, two thin plate splines are needed to specify a two-dimensional coordinate transformation.

These thin-plate spline studies which will be highlighted in my November 29th talk at the Library of Congress show that both of these important maps are composites and that they come from very different geographical sources.