Stories with Tag projection

• Sunset Geometry (2016)

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  Posted: 2025-03-15 13:32:53

  Verbose tl;dr

  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.

  • sunset
  • geometry
  • mathematics
  • sun
  • earth
  • horizon
  • spherical geometry
  • projection
  • perspective
  • visual perception
  • astronomy
  • Simulation
  • computer graphics
  • Blog Post
  • shapeoperator

  Summary of Comments ( 6 )
  https://news.ycombinator.com/item?id=43372431

  Verbose tl;dr

  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.

• Slicing the Fourth

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  Posted: 2025-01-28 21:09:40

  Verbose tl;dr

  "Slicing the Fourth" explores the counterintuitive nature of higher-dimensional rotations. Focusing on the 4D case, the post visually demonstrates how rotating a 4D cube (a hypercube or tesseract) can produce unexpected 3D cross-sections, seemingly violating our intuition about how rotations work. By animating the rotation and showing slices at various angles, the author reveals that these seemingly paradoxical shapes, like nested cubes and octahedra, arise naturally from the higher-dimensional rotation and are consistent with the underlying geometry, even though they appear strange from our limited 3D perspective. The post ultimately aims to provide a more intuitive understanding of 4D rotations and their effects on lower-dimensional slices.

  Within the realm of computer graphics and spatial reasoning, the process of constructing three-dimensional objects from two-dimensional cross-sections, akin to assembling a loaf of bread from its individual slices, presents a fascinating challenge. The blog post entitled "Slicing the Fourth" delves into this very subject, exploring the intricacies of visualizing four-dimensional shapes by examining their three-dimensional cross-sections. The author elucidates this complex concept by drawing parallels to more readily comprehensible lower-dimensional analogies. Just as a two-dimensional being might only perceive a three-dimensional sphere as a series of changing circular slices as it passes through their plane of existence, we, as three-dimensional beings, can attempt to grasp the nature of a four-dimensional hypersphere by observing its three-dimensional cross-sections as it theoretically intersects our three-dimensional space.

  The post meticulously details the mathematical underpinnings of this visualization process, utilizing the equation of a hypersphere to demonstrate how varying the fourth spatial coordinate, commonly denoted as 'w', results in a sequence of evolving three-dimensional spheres. These spheres initially appear as a single point, then grow in radius, reaching a maximum size when the hypersphere's center coincides with our three-dimensional space, and subsequently shrink back down to a point before vanishing entirely. This dynamic transformation of the three-dimensional cross-sections provides a tangible, albeit incomplete, representation of the four-dimensional hypersphere.

  Furthermore, the author extends this concept beyond the hypersphere, discussing how other four-dimensional shapes, such as the hypercube (also known as a tesseract), can be visualized through the same slicing technique. By meticulously calculating and rendering the three-dimensional cross-sections of a hypercube at different values of 'w', the author demonstrates how these slices morph from a point into a series of increasingly complex polyhedra, ultimately revealing the intricate structure of the four-dimensional object. The post concludes by emphasizing the power of this slicing method as a valuable tool for understanding and visualizing higher-dimensional objects, offering a glimpse into realms beyond our immediate spatial perception. This method, while inherently limited by our three-dimensional perspective, nonetheless provides a valuable framework for conceptualizing the complexities of higher-dimensional geometry.

  • dimensionality
  • geometry
  • Visualization
  • 4D
  • four-dimensional
  • hypercube
  • tesseract
  • projection
  • slicing
  • interactive
  • mathematics
  • polytopes
  • computer graphics

  Summary of Comments ( 0 )
  https://news.ycombinator.com/item?id=42858051

  Verbose tl;dr

  HN users largely praised the article for its clear explanations and visualizations of 4D geometry, particularly the interactive slicing tool. Several commenters discussed the challenges of visualizing higher dimensions and shared their own experiences and preferred methods for grasping such concepts. Some users pointed out the connection to quaternion rotations, while others suggested improvements to the interactive tool, such as adding controls for rotation. A few commenters also mentioned other resources and tools for exploring 4D geometry, including specific books and software. Some debate arose around terminology and the best way to analogize 4D to lower dimensions.

  The Hacker News post titled "Slicing the Fourth" has generated a moderate discussion with several insightful comments. Many of the commenters engage with the geometric concepts presented in the linked article about visualizing four-dimensional objects.

  One compelling thread discusses the challenges and rewards of visualizing higher dimensions. A commenter points out the inherent difficulty in truly visualizing 4D, as our minds are trained in 3D. They suggest that the article's approach, using slicing, offers a practical way to grasp some aspects of 4D objects. This leads to another comment suggesting alternative methods like projection and immersion, each with its own strengths and weaknesses for understanding different facets of higher-dimensional geometry.

  Another commenter draws a parallel between visualizing 4D shapes and understanding complex systems. They argue that just as slicing a 4D object reveals different 3D cross-sections, analyzing complex systems requires examining various aspects to build a comprehensive understanding. This analogy highlights the broader applicability of the slicing concept beyond pure geometry.

  The practical applications of 4D visualizations are also discussed. One commenter mentions the use of similar techniques in medical imaging, where 3D scans are essentially "slices" of a 4D spacetime representation of the body. This example grounds the abstract mathematical concepts in a real-world context, making them more relatable.

  Several comments focus on specific aspects of the article's visualizations. Some commenters praise the interactive elements, allowing readers to manipulate the slices and gain a more intuitive understanding. Others point out the limitations of representing complex 4D shapes, emphasizing that these visualizations are merely projections or cross-sections and not a complete representation of the true object.

  Some commenters share personal anecdotes about their experiences trying to comprehend higher dimensions, highlighting the individual challenges and "aha" moments in this process. Others suggest additional resources, such as books and software, for those interested in exploring the topic further.

  Overall, the comments on the Hacker News post reflect a general appreciation for the article's attempt to make 4D visualization accessible. They acknowledge the inherent difficulties of the subject matter while emphasizing the value of different approaches and the potential for real-world applications. The comments also reveal the diverse perspectives and levels of understanding within the Hacker News community, ranging from casual interest to expert knowledge in the field.