This blog post explores the geometric relationship between the observer, the sun, and the horizon during sunset. It explains how the perceived "flattening" of the sun near the horizon is an optical illusion, and that the sun maintains its circular shape throughout its descent. The post utilizes basic geometry and trigonometry to demonstrate that the sun's lower edge touches the horizon before its upper edge, creating the illusion of a faster setting speed for the bottom half. This effect is independent of atmospheric refraction and is solely due to the relative positions of the observer, sun, and the tangential horizon line.
In a blog post titled "Sunset Geometry (2016)," published on the website shapeoperator.com, the author delves into a fascinating exploration of the geometric principles governing the visual phenomenon of a sunset. The central question posed is: disregarding atmospheric refraction, what is the precise shape of the Earth's shadow projected onto the sky during a sunset? The author meticulously dismantles the common misconception that this shadow is a semi-circle.
The investigation commences with a simplified two-dimensional model, representing the Earth and the observer as a circle and a point external to it, respectively. By constructing tangent lines from the observer's point to the circle, representing the lines of sight to the first and last points illuminated by the setting sun, the author demonstrates that the angle formed by these tangents is less than 180 degrees. This implies that the terminator, the line separating the illuminated and shadowed portions of the Earth, appears as a curved line from the observer's perspective, and consequently, the Earth's shadow projected onto the celestial sphere is not a semicircle.
The author then elevates the analysis to three dimensions, employing a more rigorous geometrical approach. The Earth is modeled as a sphere, and the sun's rays are considered parallel due to the sun's immense distance. The observer's horizon plane is defined, and the intersection of this plane with the cylindrical shadow cast by the Earth is meticulously analyzed. The author demonstrates through vector calculations and the derivation of parametric equations that the curve of the Earth's shadow on the observer's horizon plane is, in fact, a hyperbola.
This hyperbolic shape arises from the intersection of the observer's horizon plane with the Earth's cylindrical shadow, a geometric consequence of the relative positions of the observer, the Earth, and the infinitely distant sun. The post concludes with a visual representation of the hyperbolic shadow projected onto a plane, solidifying the counterintuitive conclusion that the Earth's shadow during sunset, when abstracted from atmospheric effects, is not a semicircle but a portion of a hyperbola. This elucidates a nuanced aspect of the geometry underlying everyday phenomena, highlighting how seemingly simple observations can possess intricate mathematical underpinnings.
Summary of Comments ( 6 )
https://news.ycombinator.com/item?id=43372431
HN users discuss the geometric explanation of why sunsets appear elliptical. Several commenters express appreciation for the clear and intuitive explanation provided by the article, with some sharing personal anecdotes about observing this phenomenon. A few question the assumption of a perfectly spherical sun, noting that atmospheric refraction and the sun's actual shape could influence the observed ellipticity. Others delve into the mathematical details, discussing projections, conic sections, and the role of perspective. The practicality of using this knowledge for estimating the sun's distance or diameter is also debated, with some suggesting alternative methods like timing sunset duration.
The Hacker News post "Sunset Geometry (2016)" has several comments discussing various aspects of the linked article about the geometry of sunsets.
Several commenters delve into the specifics of atmospheric refraction and its impact on the perceived position of the sun. One commenter highlights how the article simplifies the explanation by assuming a constant refractive index, which isn't entirely accurate due to variations in air density with altitude. This leads to a discussion of how temperature gradients and atmospheric conditions influence the refraction and thus the appearance of the sunset. Another commenter mentions how these variations can lead to phenomena like the "green flash," a rare and fleeting optical phenomenon occurring right at sunset.
Another thread focuses on the mathematical model presented in the article. A commenter points out the elegance of the model despite its simplifications, appreciating its ability to capture the essence of the sunset geometry. Another user questions the practicality of the provided calculations for predicting the exact time of sunset, emphasizing the complexities involved in real-world scenarios.
One commenter links the discussion to historical astronomical observations, mentioning how ancient civilizations might have perceived and interpreted these phenomena. This leads to a brief tangent about archaeoastronomy and how celestial events influenced ancient cultures.
Some users discuss the broader implications of understanding sunset geometry, relating it to navigation and timekeeping. They highlight the importance of such calculations in pre-modern societies.
Finally, several commenters express their appreciation for the article, praising its clear explanation and engaging presentation of a complex topic. They find the visual aids particularly helpful in understanding the underlying geometry. One commenter mentions the overall enjoyment of the article and its ability to spark curiosity about a seemingly mundane phenomenon.