Lénárt sphere

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István Lénárt demonstrating a number of Lénárt spheres.

Lua error in package.lua at line 80: module 'strict' not found. A Lénárt sphere is a teaching and educational research model for spherical geometry. The Lénárt sphere is a modern replacement of a "spherical blackboard".[1] It can be used for use in visualizing spherical polygons (especially triangles) showing the relationships between the sides and the angles.

Basic set

The Lenart Sphere Basic Set includes:[2]

  • A transparent plastic, eight-inch sphere
  • A ring-shaped support, called a torus, to place under the sphere
  • Hemispherical transparencies that fit over the sphere so that students can draw on these with colored markers and cut out shapes with scissors
  • A spherical ruler with two scaled edges for drawing and measuring arcs, angles, and great circles on the sphere
  • A spherical compass and center locator for drawing circles on the sphere
  • A set of transparency markers for writing and drawing on the sphere and transparencies
  • A hanger for displaying your spherical constructions and designs
  • A 16-page booklet, Getting Started on the Lenart Sphere, with activities you can begin as soon as you open the box
  • A four-color polyconic projection of the earth that you can cut out and transform into your own globe

Related products

Equilateral triangle constructed on a Lenart sphere.

Other products for studying spherical geometry include visualization software programs such as The Geometer's Sketchpad, GeoGebra and Spherical Easel [3](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). Where these products only work on a flat plane the Lénárt sphere is giving hands-on experience of spherical geometry

History

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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.[4]

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.


Spherical tessellation

The Lénárt sphere is 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.[5]

See also

Column-generating template families

The templates listed here are not interchangeable. For example, using {{col-float}} with {{col-end}} instead of {{col-float-end}} would leave a HTML "div" (division) open, potentially harming any subsequent formatting.

Column templates
Type Family
Handles wiki
 table code?dagger
Responsive/
Mobile suited
Start template Column divider End template
Float "Col-float" Yes Yes {{Col-float}} {{Col-float-break}} {{Col-float-end}}
"Columns-start" Yes Yes {{Columns-start}} {{Column}} {{Columns-end}}
Columns "Div col" Yes Yes {{Div col}} {{Div col end}}
"Columns-list" No Yes {{Columns-list}} (wraps div col)
Flexbox "Flex columns" No Yes {{Flex columns}}
Table "Col" Yes No {{Col-begin}},
{{Col-begin-fixed}} or
{{Col-begin-small}}
{{Col-break}} or
{{Col-2}} .. {{Col-5}}
{{Col-end}}

dagger Can template handle the basic wiki markup {| | || |- |} used to create tables? If not, special templates that produce these elements (such as {{(!}}, {{!}}, {{!!}}, {{!-}}, {{!)}})—or HTML tags (<table>...</table>, <tr>...</tr>, etc.)—need to be used instead.

References

  1. Lua error in package.lua at line 80: module 'strict' not found.
  2. Lua error in package.lua at line 80: module 'strict' not found.
  3. http://merganser.math.gvsu.edu/easel/
  4. Lua error in package.lua at line 80: module 'strict' not found.
  5. Lua error in package.lua at line 80: module 'strict' not found.

External links

  • [1] Free download and information on Spherical Easel and other projective geometry educational programs and apps.
  • [2] Pictures of the Sphere
  • [3] Argand Diagram Images
  • [4] Argand diagram history
  • [5] Copyright free copy of reference 5
  • [6] Copyright free version of Chauvenet on Astronomy applications
  • [7] 60 page preview of Van Brummelen, see p. xiii on Lenart (image redacted on Google)
  • [8] Open source Python code for tesselation, nurbs, finite analysis, subdivision surfaces
  • Projective Geometry for Machine Vision — tutorial by Joe Mundy and Andrew Zisserman.