(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 — a lineage that would culminate in the revolutionary Wright–Zeiler Projection.
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, software capable of displaying the true umbral shadow in 3-D. Together these works marked the transition from classical planar theory to physically realistic modeling.
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. John Irwin’s concurrent analyses confirmed that the umbral and anti-umbral cones do not touch but leave a small gap, further validating the new model.
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 what would become (with apologies to Luca and Bill) what I have dubbed (in all hopes that the name will be adopted by the eclipse community) the Wright–Zeiler Method.
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 effectively established the Wright–Zeiler Projection as the successor to classical Besselian methods.
After several years of refinement, Wright published his 2024 paper defining the Wright–Zeiler Projection (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 unified Zeiler’s cartography, Quaglia’s geometry, and Wright’s computational framework into one coherent system — closing the loop from discovery to definition.
Today the Wright–Zeiler Projection underpins Wright's NASA’s eclipse visualizations and Zeiler's Eclipseatlas.com and Eclipse-Maps.com. It essentially replaces Bessel’s 19th-century approximation with a true geometric solution. From Kramer’s first timing arrays in 2008 to Wright’s publication in 2024 and John Irwin's latest map of the path for the 2026 TSE, the journey spanned sixteen years and fused cartography, physics, and computer science into a single story of discovery beneath the Moon’s shadow. What began as an experiment with a handful of data points has become the global standard for eclipse prediction — a forward-looking algorithm that transformed the field for all time.