A Lénárt sphere is a teaching and educational research model for non-Euclidean geometry, especially spherical geometry, spherical trigonometry, and projective geometry. The Lénárt sphere has been called a "spherical blackboard" for use in visualizing polygons (especially triangles) on the sphere and the relationships between the sides and the angles. The sphere is used like visualization software programs such as The Geometer's Sketchpad, GeoGebra and Spherical Easel (see External Links for Spherical Easel information, and visit List of interactive geometry software for Non Euclid, and many other interactive Projective geometry apps and programs). Example educational applications of the Lénárt sphere include Geodesy, GIS, astronomy, geometry, ray tracing (graphics), perspective (graphical), trigonometry and celestial navigation.
The Lénárt sphere was invented by István Lénárt in Hungary in the early 1990s and its use is described in his 2003 book comparing planar and spherical geometry.
Spherical trigonometry used to be an important mathematics topic from antiquity through the end of World War II, and has been replaced in modern education and (in navigation) with more algorithmic methods as well as GPS, including the Haversine formula, linear algebraic matrix multiplication, and Napier's pentagon. The Lénárt sphere is still widely used throughout Europe in non-Euclidean geometry as well as GIS courses.
Following Glen Van Brummelen (Reference 1 below, p. 129, stereographic projection), spherical trigonometry, though certainly no longer relevant to the older scientific needs of navigation, astronomy, geography, etc., other than as historical mathematics, has nevertheless seen a "rebirth" today due to simulation, game programming, Autodesk Maya, kinematics, physics engines, and many other new fields as diverse as optics, photography, art and medicine.
Going (spherically) from 2D to 3D and vice versa was an old map making technique that has now found a new home in computer graphics, game engines and even GPU architecture (although looking at spherical trigonometry in "reverse" by mapping spherical data onto planes – see Stereographic projection for details). These developments affect the continuing relevance of the Lénárt sphere and analogous tools, spherical trigonometry, and numerous other geometric modeling tools beyond their initial and historic value in astronomy, navigation, and geography. Mathematically, many of these newer developments fall under the older, more general family of perspective. Perspective itself, along with its many applications to spherical projection, currently falls under the rubrics of descriptive and projective geometry.
The historic applications of perspective geometry have also been reprised in several modern astronomical sphere-projection arenas such as photometric surveying (specifically, inverse problems of perspective and spherical vanishing points in photometric systems). Many Lénárt spheres come with both GIS and astronomy as well as trigonometric overlays.
In layman's terms, when you project a sphere onto a plane and vice versa, matters of perspective (such as convergence and vanishing points) often are most interesting as they relate to the subject of shadows. This is why shadows in motion are often the chapter in 3D modeling references in which projection and perspective (whether matrix multiplication or spherical trigonometry) are covered. Today, mapping from 3D to 2D spheres and planes also has to take into account multiple perspectives, for example, a camera and a viewer. These are handled with sampling. A technique called Perspective aliasing relates to the (statistical sampling) parametrization of a shadow map, and can be used to convert the projection from camera to viewer projection perspectives. Both using point light sources on the Lénárt sphere and modeling in geometric software can demonstrate the effects of various samples.
Statistics teachers can use tools like the Lénárt sphere to demonstrate effects of perspective (such as shadow sampling), then use the deeper underlying trigonometry and linear algebra to calculate coordinates for aliasing. In that case, the perspective becomes the perspective of a function's Fourier transform – see the Nyquist–Shannon sampling theorem and the related article on Aliasing for details.
Game programmers can avoid both the trigonometry mapping of the Lénárt sphere and the underlying spherical trigonometry by simply calling an OpenGL depth clamp command, for example, to automatically create a projective matrix for, say, a 3D moving shadow. Underlying the OpenGL command, however, is a complex group of linear algebra functions. In fact, linear algebra, via vector and tensor arrays, matrix algebra and differential equations, were one of the first methods applied math was able to bypass projective and descriptive geometry altogether by solving 3D coordinates with direct matrix multiplication. These newer techniques, and their Fourier based algorithms, can give faster, more precise spherical triangle solutions than traditional historic approximations (see p. 241 and following of the Chauvenet reference).
Certain technical problems with the accuracy of this multiplication, however (e.g. z coordinates > 1) can be solved by hearkening back to spherical trigonometry, just as quaternions have been reprised from the 19th century for 4D computing in computer graphics. See Chapter 10 (p. 314 et al.) in the Lengyel reference for details.
3D spherical graphics and harmonics
Just as shadow motion projected on a sphere is an important educational application of the Lénárt sphere in computer graphics, other related applications include projection (and movement) of light sources on a sphere, indirect lighting, global illumination, precomputed radiance transfer, ambient occlusion, etc. which are topics generally falling under 3D computer graphics. Lénárt sphere overlays can be used both for positive and negative harmonics demonstrations and polar and spherical coordinate calculation.
Most of these applications originally fell under Laplace's equation and the Laplace transform at a time (1780's) when spherical trigonometry was the primary tool for demonstrating these effects and their computations. Today, Argand (1806) diagrams (falling in Wiki under complex plane and stereographic projection articles) also can be used with the Lénárt sphere to explore fast Fourier transforms and spherical-harmonic algorithms in the complex plane. Technically these find the angular part of solution sets to Laplace's transform equations in the special area of spherical harmonics.
In general, Laplace and Fourier transforms are related to each other and within signal processing and projection by resolving frequency, oscillation and time problems (and both have important spherical projection and harmonics aspects). On the sphere and via the addition theorem of spherical harmonics, one uses polynomials on the left side of a Laplace cosine trig identity equation and spherical harmonics on the right.
We then rotate spherical y vectors so they point along the z axis, set x = y, and via Unsold's theorem (primarily known for atomic spherical symmetry and stellar/solar calculations), project and generalize the sphere to a n = 2 dimension, so that higher dimensions are indexed to the volume of the sphere. These techniques expand the 3D graphics applications of spherical trig to newer applications in spectrum analysis, signal processing and fast Fourier transforms (specifically Rolchlin and Tygert, 2006—see Wiki fast Fourier transform article). Also see the Easton reference for numerous applications (12 page references) of the Argand diagram, which is a 2D graphic that can visually translate and demonstrate great circles and arcs on the sphere with real and imaginary phase angles on a unit circle (Easton is the inventor of GPS). Like quaternions revolving around i, Argand's diagram made complex numbers more palatable via visual representation.
In addition to computer graphics, projective geometry, while exhausted theoretically by the early 20th century, is finding new analytic applications in fields such as robotics and machine vision (See Mundy and Zisserman external link on machine vision).
Spherical tessellation and finite analysis
In computer graphics, tessellation is used to manage datasets of polygons and divide them into suitable structures for rendering. Especially for real-time rendering, data are tessellated into triangles, for example in DirectX 11 and OpenGL.
In computer-aided design the constructed design is represented by a boundary representation topological model, where analytical 3D surfaces and curves, limited to faces and edges, constitute a continuous boundary of a 3D body. Arbitrary 3D bodies are often too complicated to analyze directly. So they are approximated (tessellated) with a mesh of small, easy-to-analyze pieces of 3D volume—usually either irregular tetrahedra, or irregular hexahedra. The mesh is used for finite element analysis and creation (synthesis or modeling) of 3D graphics (spheres, planets, balls, stars, etc. in the case of Lenart).
The Lénárt sphere is extremely useful in modeling and demonstrating spherical tesselation techniques, especially as they apply to finite analysis problems. Using 3D graphics programs or Python code (see external reference link 8 for open source Python code examples vs. NURBS), greater and greater numbers of polygons can be projected to and from the sphere both for analysis of finite elements, and synthesis of objects and features on the sphere, such as the asteroid ridden planet in the example. In this case, the Lénárt sphere is useful for tesselation (tiling) as a simplification or approximation shortcut to the extremely complex differential math of finite analysis and construction (technically: modeling), especially of animated objects.
- Van Brummelen, Glen (2013). Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry. Princeton University Press. ISBN 978-0-691-14892-2.
- Lenart, Istvan (2003), Non-Euclidean Adventures on the Lenart Sphere: Activities Comparing Planar and Spherical Geometry, Key Curriculum Press, ISBN 978-1559531030
- Andersen, Kirsti (2007). The Geometry of an Art: The History of The Mathematical Theory of Perspective. Springer. p. 720. ISBN 978-0387-25961-1.
- Eisemann, Elmar (2012). Real-Time Shadows. CRC Press. p. 85. ISBN 978-1-56881-438-4.
- Chauvenet, William (1887). A treatise on plane and spherical geometry. Lippincott. ISBN 1-4069-6824-2.
- Lengyel, Eric (2009). Mathematics for 3D Game Programming and Computer Graphics. Cengage. ISBN 978-1-58450-277-7.
- Folland, Gerald (2000). Fourier Analysis and Its Applications. American Mathematical Society. p. 405. ISBN 978-0-8218-4790-9.
- Easton, Roger (2010). Fourier Methods in Imaging. Wiley. p. 54. ISBN 9780470689837.
- Mechtley, Adam (2011). Maya Python for Games and Film. Morgan Kaufmann. ISBN 978-0123785787.
-  Free download and information on Spherical Easel and other projective geometry educational programs and apps.
-  Pictures of the Sphere
-  Argand Diagram Images
-  Argand diagram history
-  Copyright free copy of reference 5
-  Copyright free version of Chauvenet on Astronomy applications
-  60 page preview of Van Brummelen, see p. xiii on Lenart (image redacted on Google)
-  Open source Python code for tesselation, nurbs, finite analysis, subdivision surfaces
- Projective Geometry for Machine Vision — tutorial by Joe Mundy and Andrew Zisserman.