(source: Eclipse-Maps.com).
Friedrich Wilhelm Bessel’s nineteenth-century formulation of eclipse geometry laid the foundation for all local-circumstance calculations. Building on that, Chester B. Watts created detailed lunar-limb profiles for occultation and eclipse prediction, later refined by Julena S. Duncombe of the U.S. Naval Observatory and by Dave Herald, who applied those profiles to improve eclipse-timing accuracy. In the 1970s and 1980s, Fred Espenak modernized the approach, integrating it into NASA’s computational pipeline through the early 2000s. By 2009, Japan’s SELENE (Kaguya) mission released high-definition lunar-altimetry data, which Herald reduced into Watts-format limb profiles. Using that data, Bill Kramer and Michael Zeiler applied advanced GIS methods to map eclipses with topographic realism — part of a broader transition toward modern topographic and three-dimensional eclipse projection techniques.
In 2008, cartographer Michael Zeiler sought to visualize eclipse paths using GIS tools. “Michael needed data points for his mapping experiment,” recalled veteran eclipse calculator Bill Kramer. “He contacted me about the possibility. So I hacked a little and generated arrays of data for a series of points inside and outside the path of totality with latitude, longitude, and C1–C4 times.” When Zeiler plotted those values, the umbral edge formed irregular polygons, each vertex corresponding to a feature of the lunar limb. “Mike started to play with the data points and discovered these polygon shapes,” Kramer said. “This was in 2008–2009. We had the first map ready to show off for the 2009 TSE.” The discovery was the first visible signature of the Moon’s topography imprinted directly on the shadow path.
By 2010, Zeiler had produced the first limb-corrected eclipse maps (2010 July 11 TSE). At the same time, Kramer built an online calculator to generate grids of limb-corrected circumstances. Zeiler noted that the central line was no longer identical to the line of longest duration — a subtle clue revealing the true geometry of the umbral cone. Meanwhile, in England, John Irwin developed UmbraView, a system capable of displaying and analyzing the true umbral shadow in three dimensions. His work represented one of the earliest known departures from purely planar eclipse modeling toward fully geometric eclipse projection.
In 2012, Zeiler published additional limb-corrected maps (2012 Nov 13–14 TSE). “Here are some C2/C3 maps from 2012,” he later wrote. “I believe this was the first time the polygonal boundaries were apparent.” Short-duration eclipses exposed the effect most clearly. Working with Xavier Jubier, Zeiler confirmed that contact timings varied with specific lunar valleys and peaks — proof that the polygon was real geometry, not a numerical artifact.
In 2013, Zeiler met Luca Quaglia, whose background in celestial geometry helped formalize the observations. “The first time I saw the true umbral outline plotted against actual terrain data, I realized the map was physically real, not a mathematical abstraction,” Quaglia recalled. Together they determined that observers along each polygonal edge experienced C2 simultaneously, each witnessing the final Baily’s bead through the same lunar valley. At the same time, John Irwin was independently developing geometric analyses and 3-D umbral visualizations that reinforced and extended the emerging understanding of polygonal umbral geometry and the transition toward fully geometric eclipse projection. His approach treated the eclipse shadow as a true three-dimensional geometric structure rather than as a purely planar projection, helping establish a fundamentally new framework for eclipse modeling.
At NASA Goddard’s Scientific Visualization Studio, Ernie Wright was known for precise lunar renderings but initially doubted the polygonal concept. “Mike Kentrianakis, Charles Fulco, and Shadia Habbal visited me at Goddard on October 9 2015 and asked what I thought about Zeiler’s polygonal umbra shapes,” Wright later recalled. “I expressed some skepticism.” That skepticism persisted into the following year, when several parallel investigations—Zeiler’s mapping, Kramer’s visual analyses, and Quaglia’s geometric modeling—were beginning to converge. Wright remained intrigued but unconvinced, waiting for a clearer geometric rationale that would justify what Zeiler was capturing.
During the June 2016 Solar Eclipse Conference in Carbondale, the author met Luca, and enjoyed a detailed conversation regarding his ongoing work on solar-radius calculations. Our conversation immediately convinced me that he was on the right track — his geometric reasoning was rigorous and deeply thought through. In the years since, as Luca and his team have refined their methods and incorporated additional datasets, that early impression has only strengthened. His results have been compelling enough that the author now computes his Besselian Elements using Luca's revised and expanded solar radius.
At this time, Luca was also collaborating with Michael Zeiler to understand the strange polygonal shapes appearing in modeled umbras. Luca had produced some powerful ideas, though these had not yet taken formal shape via a robust definition or explanation. A truly pivotal moment came as the conference ended: instead of taking a short (and bumpy!) commuter flight back to St. Louis, Luca accepted Ernie Wright’s offer of a ride to the airport. That drive became one of the defining moments in this entire story. As Luca attempted to outline his early geometric concepts, Ernie — still cautious but intrigued — began to envision how such a model could be tested computationally. As usual in science, the answers generally lie very close to our success in properly framing the questions.
This drive from Carbondale to St. Louis will be remembered as the moment when Ernie became convinced of the idea of experimenting with the model. His talk with Luca was the culmination of several independent lines of inquiry converging into one coherent framework — the birth of a new generation of topographic and three-dimensional eclipse projection methods.
In the days and weeks that followed, Wright began diligently experimenting with LRO topography and SPICE-based ray-tracing at Goddard, translating the ideas from the ride to STL into code. When he finally rendered the umbral shadow with true lunar relief, the result was unmistakable: the edges formed naturally from tangent rays. “The polygons just fell out,” he later wrote.
Meanwhile, Luca continued his collaboration with John Irwin, producing some of the first purely geometric eclipse-path maps derived from the true shape of the lunar umbra.
By late 2016, Wright had refined his raster-based model, and unveiled it at the AAS Eclipse Workshop in Columbia, South Carolina (video). He introduced the “pinhole effect” — explaining why observers along the same polygonal edge share identical final rays of sunlight. “A fair number of skeptical emails arrived about the shape of the path,” Wright said, “but once they saw the model projected live, the geometry spoke for itself.” The presentation earned wide acclaim and helped establish modern topographic and three-dimensional eclipse modeling as a successor to purely classical smooth-limb Besselian approximations.
After several years of refinement, Wright published his 2024 paper defining a complete raster-oriented three-dimensional eclipse model (with Alex Young) as a complete three-dimensional model of the umbral and anti-umbral cones derived directly from lunar topography, solar diameter, and geocentric vectors. “Publishing the visualization happened in 2024,” Wright wrote, summarizing sixteen years of progress. The paper brought together ideas and developments that had emerged over many years through the work of Zeiler, Quaglia, Irwin, Wright, Kramer, and others — helping formalize the transition from classical smooth-limb eclipse modeling to modern topographic and three-dimensional geometric projection.
Today these modern topographic and three-dimensional eclipse projection methods underpin NASA eclipse visualizations, EclipseAtlas.com, Eclipse-Maps.com, UmbraView, and other high-precision eclipse-mapping efforts. They replace purely smooth-limb approximations with physically realistic geometric models derived from lunar topography, solar geometry, and modern computational methods. The transition from classical eclipse prediction to modern topographic and three-dimensional projection emerged through the overlapping contributions of cartographers, eclipse calculators, geometric modelers, and software developers including Bill Kramer, Michael Zeiler, John Irwin, Luca Quaglia, Ernie Wright, Xavier Jubier, and others. From Kramer’s first timing arrays in 2008 to Wright’s 2024 publication and Irwin’s continuing geometric eclipse-modeling work, the evolution spanned more than fifteen years and fused cartography, physics, celestial geometry, and computer science into a new generation of eclipse prediction and visualization.