New Spline Interpolation Function

Click on figures to enlarge

ESRI has finally undated their rubber-sheeting functions in ArcMap (9.2) to allow for Spline georectification and full rubber-sheeting. The new function also allows for the use of raster data making it much easier to compare the features of old and new maps and to display them in a pleasing and informative way. Below is a georectified sheet of the Waldseemüller 1516 Carta Marina that shows how it must be rotated in order to get the best spline fit.

For more information on splines see http://mathworld.wolfram.com/Spline.html

How to Map a Sandwich:
Potential Theory,Topological Existence Theorems, and the Changing History of the Ontology of Cartographic Objects

Abstract (full article coming soon)

In the 1960s and 1970s the most important work being accomplished in mathematical cartography had to do with the topological properties of surfaces and their relationship to geographical and spatial analysis. The Harvard Laboratory for Computer Graphics and Spatial Analysis was a hotbed of such work and was led into new areas by the ideas of the theoretician William Warntz. While most other researchers in the field where looking at the numerical properties of surfaces Warntz’s approach centered on understanding their topology. He recognized that the most important properties of surfaces from a mathematical point of view had nothing to do with numbers but rather their invariance under transformations. Warntz described the relationship of the topological properties of a surface to cartography in a number of important papers that adopted a terminology and methodology built on the work of the mathematician Arthur Cayley (1859). Warntz was particularly interested in mapping thematic surfaces and adopted a macrogeographical theoretical perspective that led not only to fundamental mathematical breakthroughs but also yielded philosophical insight into the nature of the objects described by the “science” of cartography. This paper focuses on one particular aspect of the work of Warntz and one of his students at the Harvard Laboratory; existence theorems. Existence theorems contain a statement of existential quantification such as “there is” and prove the existence of a particular set of mathematical objects. They do not however contain any directions of how such objects might actually be constructed algorithmically or numerically.

The researchers at the Lab published two very important works on existence theorems in the influential and now largely forgotten series the Harvard Papers in Theoretical Geography. We will provide a close reading of two of these papers, “The Sandwich Theorem: A Basic One for Geography”, and “Geography and an Existence Theorem: A Cartographic Solution to the Localization of Sets of Equal-Valued Antipodal Points”, in order to show how the Lab used a mathematical approach that was underexploited in cartography and in doing so changed the accepted notions of the nature of cartographic objects.

This shift in the nature of what constituted geographical and cartographic objects is discussed in this study within the framework of Thomas Kuhn's Structure of Scientific Revolutions. Kuhn provides an example in his analysis of the development of the theory of relativity in the beginning of the 20^th century of the type of profound conceptual shifts that took place in cartography in the 1960s and 70s. These shifts were not simply dramatic changes in beliefs about the world or even in scientific and geographic methodology, but rather in the very concepts that define the structure and formal properties (topological and transformationally invariant) of the objects of inquiry. In this way Kuhn’s framework and lexicon provides us with a solid philosophical and historical framework in which to discuss the same type of radical shifts that took place at the foundations of mathematical cartography. These changes in the conceptual framework of cartographic science redefined the nature of geographical objects (what is it that is mapped) and laid the foundations for the development of topological data structures and modern GIS.

Schöner's Cosmographical Miscellany and His Annotations in the Waldseemüller Sammelband

In 1656 the Emperor Ferdinand III of Austria purchased the Library of Georg Fugger for the Hofbibliothek in Vienna. The collection included the Library of Johannes Schöner and was handed down by Fugger to his son (Phillip Eduard, 1546-1618) and his great-grandson (Albert III, 1624-1682). Besides Schöner's Library Fugger's collection also contained the best mathematical and scientific literature then available.

Portrait of Schöner

The codex containing the 1507 and 1516 World Maps by Martin Waldseemüller was once part of this collection but how and when it became separated from Schöner's original Library and made its way to the Wolfegg Castle in Wurttemberg where it was discovered by Joseph Fischer in 1901 remains an historical mystery. The Waldseemüller Sammelband contained additional items besides the two famous world maps and originally included a set of Celestial Gores by Schöner and the star-chart of Stabius as rendered by Albrecht Dürer. Only one of the star-charts was bound into the codex and it shows the stars visible in the southern hemisphere. The figure below shows Schöner's annotations on the chart.

Contained in Schöner's Library that now resides in the Austrian National Library in Vienna are several volumes that resemble the Waldseemüller Sammelband in that they are bound in the same manner with heavy wooden covers connected with leather backs and also display Schöner's bookplate. These volumes include Schöner's copy of the 1482 Ulm edition of Ptolemy and his copy of Waldseemüller's 1513 edition of the same book. Both books are annotated with the same red-lines found on the 1507 and 1516 World maps and are held into the volumes with slices of printed vellum globe gores and pieces of the Elinger Map.

Schöner's Vellum Manuscript Drawing of a Sheet from the 1516 Carta Marina

Also included in of Schöner's Library is an interesting cosmographical miscellany that is unpublished but holds a great deal of interest for historians of cartography. The miscellany bares the title "Cosmographia" and contains the following items:

1. A short treatise with title "Regionum sive civitatum distantae."
2. Tables of latitudes and longitudes that are similar in content and structure to the so-called University Tables.
3. Notes on various units used to measure distance.
4. A table that displays the number of miles in a degree of longitude for each parallel similar to that found in Waldseemüller and Ringmann's Cosmographiae Introductio.
5. Instructions for measuring the distances between two cities on a map that has coordinates.
6. The University Coordinate Tables.
7. The Tabula Regionum of Regiomontanus. This is of course not the only Regiomontanus in Schöner's Library. Schöner inherited Regiomonanus' manuscripts and published his very important work On Triangles.
8. An outline for the chapter headings of an incomplete work on Cosmography.
9. Excerpts from the Geographiae.

The most interesting of all these works is the Regionum sive Civitatum. The treatise begins by describing at set of instructions for constructing a terrestrial globe. The initial part of the text describes the process by which one inscribes on a globe the locations of the cities and regions on the earth. The first set divides the earth into four equal areas by means of two arcs that intersect each other at ninety degree angles. Once these circles have been inscribed on the globe another great circle is drawn that bisects the other two and forms the equator. The next step divides that part of the equator that lies along the "habitable" part of the earth into 180 degrees of longitude numbering them in units of five. For marking the globe with latitude lines a strip of heavy vellum is used, equal in length to the distance from the pole to the equator. This type of construction continues in the various regions until the whole surface of the globe has coordinates. In order to transpose the points and locations of cities and regions from the globe to a plane the method is essentially that of an azimuthal projection from any point on the surface of the earth. At first the globe-maker selects the city or point that he wishes to make the center of the projected map. Then with a compass he inscribes a circle on the surface of the globe that is large enough in diameter to include the area to be reproduced. The smaller the circle, the larger the scale of the resulting map and the greater ease involved in measuring distances. The second method described in the book outlines what appears to be a conic projection. To do this two new vellum strips are used equal in length to the diameter of the circle drawn on the surface of the globe. The strips are divided into the same number of degrees as the strips used in the first method. They are them placed tangentially along the circle, running north to south. The text says that this method can be used either on a square (quadratam) or a circler (rotundam) map. All this is simply to suggest that Schöner was experimenting a great deal with different methods for measuring distances and for transferring coordinates from maps to globes and vise versa and obviously drew his annotations on the 1507 and 1516 World maps by Waldseemüller for this purpose.