₆ΛM 3d: Generating uniformly distributed points on a sphere

Roman Chijner, Jan 9, 2021

[b] ΛM -Lagrange Multipliers with One Constraint. Finding Estimators of location on a surface of the sphere as Critical points of the corresponding Lagrangian for a discrete set of points. There is some uniform distribution of points on the sphere. With respect to the points of some other concentric sphere of radius R, we define the distance function f(φ;θ). That is, the sum of all distances from the points of the selected distribution to the (R;φ;θ)- point on this sphere. The critical points: [color=#c51414]max[/color], [color=#1551b5]min[/color], [color=#0a971e]saddle[/color] points of this function form subsets of new "uniform" distributions. And, obviously, a subset of the points of relative minima coincides with the original distribution. Critical points can be found using Lagrange multipliers as finding the Extreme values of the function f(x,y,z) subject to a constraining equation g(x,y,z):=x²+y²+z²-R²=0. There is a system of equations: ∇f(x,y,z)= λ∇g(x,y,z). A local optimum occurs when ∇f(x,y,z) and ∇g(x,y,z) are parallel, and so ∇f is some multiple of ∇g. [/b] *From Book: Extended definitions of point location estimates [url]https://www.geogebra.org/m/hhmfbvde[/url] From: List of My Public Books on GeoGebra Topics: Constructing polyhedra -https://www.geogebra.org/m/eabstecp

Table of Contents

  1. A critical points scheme for Generating uniformly distributed points on a sphere. V=12 ●Icosahedron.
    1. Images to applet: Generating two different uniformly distributed points on a sphere from another uniform distribution.
    2. V=12 Icosahedron. Images: A critical points scheme for Generating uniformly distributed points on a sphere.
    3. Generating two different uniformly distributed points on a sphere using one uniform distribution: Icosahedron V=12.
    4. Icosahedron. Vertices 12.
    5. Dodecahedron. Vertices 20.
    6. Icosidodecahedron. Vertices 30.
  2. V=4: Tetrahedron
    1. n=4 Tetrahedron. Images: A critical points scheme for Generating uniformly distributed points on a sphere.
    2. Generating two different uniformly distributed points on a sphere using one uniform distribution: Regular Tetrahedron V=4.
  3. V=6: Octahedron.
    1. V=6 Octahedron. Images: A critical points scheme for Generating uniformly distributed points on a sphere
  4. V=8: Square antiprism.
    1. n=8 Square antiprism(anticube). Images: A critical points scheme for Generating uniformly distributed points on a sphere
  5. V=8: Cube.
    1. Generating two different uniformly distributed points on a sphere using one uniform distribution: Cube V=8.
    2. V=8 Cube. Images: A critical points scheme for Generating uniformly distributed points on a sphere.
  6. V=12: Cuboctahedron.
    1. V=12 Cuboctahedron. Images: A critical points scheme for Generating uniformly distributed points on a sphere
  7. V=14: Tetrakis hexahedron.
    1. V=14 Tetrakis hexahedron Images: A critical points scheme for Generating uniformly distributed points on a sphere
  8. V=20: Dodecahedron.
    1. V=20 Dodecahedron. Images: A critical points scheme for Generating uniformly distributed points on a sphere
  9. V=24: Rhombicuboctahedron.
    1. V=24 Rhombicuboctahedron. Images: A critical points scheme for Generating uniformly distributed points on a sphere
  10. V=30: Icosidodecahedron.
    1. n=30 Icosidodecahedron. Images: A critical points scheme for Generating uniformly distributed points on a sphere.
    2. Generating two different uniformly distributed points on a sphere using one uniform distribution: Icosidodecahedron V=30.
    3. ●Icosidodecahedron. Vertices 30.
    4. ●Pentakis Dodecahedron. Vertices 32.
    5. Generating Elements of mesh modeling the surfaces of a Pentakis Dodecahedron(V=32) and its dual Solid: Truncated Icosahedron(V=60)
    6. Pentakis Dodecahedron(V=32) and its dual image -Truncated Icosahedron(V=60): Coloring the edges and faces
    7. Rhombicosidodecahedron-c. Vertices 60.
    8. Generating Elements of mesh modeling the surfaces of a Rhombicosidodecahedron-c(V=60) and its dual Solid: Deltoidal Hexecontahedron(V=62)
    9. Rhombicosidodecahedron-c(V=60) and its dual image - Deltoidal Hexecontahedron(V=62): Coloring the edges and faces
  11. V=60: Biscribed Truncated Icosahedron
    1. Vertices 60. Biscribed Truncated Icosahedron(extreme distribution). Images: A critical points scheme for Generating uniformly distributed points on a sphere.
    2. Generating two different uniformly distributed points on a sphere using one uniform distribution: Biscribed Truncated Icosahedron V=60.
    3. Biscribed Truncated Icosahedron(extreme distribution). Vertices 60.
    4. Biscribed Truncated Icosahedron, n=60. Polyhedra with extreme distribution of equivalent vertices.
    5. Three-parameter model transformations of the Icosahedron. Extreme distributions.
    6. ●Pentakis Dodecahedron. Vertices 32.
    7. Generating Elements of mesh modeling the surfaces of a Pentakis Dodecahedron(V=32) and its dual Solid: Truncated Icosahedron(V=60)
    8. Pentakis Dodecahedron(V=32) and its dual image -Truncated Icosahedron(V=60): Coloring the edges and faces
    9. Generating Elements of mesh modeling the surfaces of a Rectified truncated icosahedron(V=90) and its dual Solid.
    10. Rectified truncated icosahedron(V=90) and its dual image: Coloring the edges and faces
    11. Rectified Truncated Icosahedron. Vertices 90.
    12. Five pointed Star and Star of David inscribed in a Rectified Truncated Icosahedron.
    13. as Rectified Truncated Icosahedron. Vertices 90.
  12. polyhedra
    1. Tetrahedron. Vertices 4.
    2. Octahedron. Vertices 6.
    3. Cube. Vertices 8.
    4. Square Antiprism. Vertices 8.
    5. Icosahedron. Vertices 12.
    6. Cuboctahedron. Vertices 12.
    7. Dodecahedron. Vertices 20.
    8. Tetrakis hexahedron. Vertices 14.
    9. Disdyakis dodecahedron. Vertices 26.
    10. Icosidodecahedron. Vertices 30.
    11. ●Icosidodecahedron. Vertices 30.
    12. ●Pentakis Dodecahedron. Vertices 32.
    13. Rhombicosidodecahedron-c. Vertices 60.
    14. Biscribed Truncated Icosahedron(extreme distribution). Vertices 60.
    15. Rectified Truncated Icosahedron. Vertices 90.
    16. Five pointed Star and Star of David inscribed in a Rectified Truncated Icosahedron.
    17. as Rectified Truncated Icosahedron. Vertices 90.
  13. Coloring edges and faces
    1. Images. n=4: Tetrahedron -Platonic Solid and n=5: Triangular bipyramid (Triangular Dipyramid) -Johnson solid J₁₂ and their Dual Polyhedrons
    2. Images. n=6: Octahedron -Platonic Solid and n=8: Square antiprism and their Dual Polyhedrons
    3. Images. n=12: Icosahedron -Platonic Solid and n=20. Their Dual Polyhedrons
    4. Copy of 3 in 1. Constructing, surface triangulation, visualizing polyhedron. New Version.
    5. Truncated Tetrahedron, n=12. Polyhedra with extreme distribution of equivalent vertices.
    6. Example of non-uniqueness of the extreme distribution of n=16 particles on the surface of a sphere.
    7. Truncated Cube, n=24. Polyhedra with extreme distribution of equivalent vertices.
    8. 3. Vertices n=24. Coloring the edges and faces of a polyhedron and its dual image.
    9. Images. n=24: Biscribed Snub Cube (laevo) and n=32. Their Dual Polyhedrons
    10. Biscribed Truncated Octahedron, n=24. Polyhedra with extreme distribution of equivalent vertices.
    11. Icosidodecahedron, n=30. Polyhedra with extreme distribution of equivalent vertices.
    12. Pentakis Dodecahedron(V=32) and its dual image -Truncated Icosahedron(V=60): Coloring the edges and faces
    13. Truncated Dodecahedron, n=60. Polyhedra with extreme distribution of equivalent vertices.
    14. Rhombicosidodecahedron-c(V=60) and its dual image - Deltoidal Hexecontahedron(V=62): Coloring the edges and faces
    15. Biscribed Truncated Icosahedron, n=60. Polyhedra with extreme distribution of equivalent vertices.
    16. Rhombicosidodecahedron, n=60. Polyhedra with extreme distribution of equivalent vertices.
    17. Biscribed Snub Dodecahedron, n=60. Polyhedra with extreme distribution of equivalent vertices.
    18. Coloring the edges and faces of a polyhedron(n=72) Pentakis Snub Dodecahedron (laevo) and its dual image.
    19. Biscribed Pentakis Snub Dodecahedron (V=72) and its dual image: Coloring the edges and faces
    20. Coloring of edges and faces of a polyhedron(n=72) extreme distribution and its dual image.
    21. Rectified truncated icosahedron(V=90) and its dual image: Coloring the edges and faces
    22. Biscribed Truncated Icosidodecahedron, n=120. Polyhedra with extreme distribution of equivalent vertices.
    23. Finding and coloring edges and faces of a polyhedron and its dual image. V=180
    24. Canonical Truncated Snub Cube (laevo)(V=120) and its dual image: Coloring the edges and faces
    25. Images: Canonical Truncated Snub Cube (laevo)(V=120) and its dual image: Coloring the edges and faces
    26. Biscribed Hexpropello Dodecahedron (dextro) (V=140) and its dual image: Coloring the edges and faces
    27. Biscribed Orthotruncated Propello Icosahedron (V=120) and its dual image: Coloring the edges and faces

1.

A critical points scheme for Generating uniformly distributed points on a sphere. V=12 ●Icosahedron.

  • 1. Images to applet: Generating two different uniformly distributed points on a sphere from another uniform distribution.
  • 2. V=12 Icosahedron. Images: A critical points scheme for Generating uniformly distributed points on a sphere.
  • 3. Generating two different uniformly distributed points on a sphere using one uniform distribution: Icosahedron V=12.
  • 4. Icosahedron. Vertices 12.
  • 5. Dodecahedron. Vertices 20.
  • 6. Icosidodecahedron. Vertices 30.

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V=4: Tetrahedron

  • 1. n=4 Tetrahedron. Images: A critical points scheme for Generating uniformly distributed points on a sphere.
  • 2. Generating two different uniformly distributed points on a sphere using one uniform distribution: Regular Tetrahedron V=4.

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V=6: Octahedron.

  • 1. V=6 Octahedron. Images: A critical points scheme for Generating uniformly distributed points on a sphere

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V=8: Square antiprism.

  • 1. n=8 Square antiprism(anticube). Images: A critical points scheme for Generating uniformly distributed points on a sphere

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V=8: Cube.

  • 1. Generating two different uniformly distributed points on a sphere using one uniform distribution: Cube V=8.
  • 2. V=8 Cube. Images: A critical points scheme for Generating uniformly distributed points on a sphere.

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V=12: Cuboctahedron.

  • 1. V=12 Cuboctahedron. Images: A critical points scheme for Generating uniformly distributed points on a sphere

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V=14: Tetrakis hexahedron.

  • 1. V=14 Tetrakis hexahedron Images: A critical points scheme for Generating uniformly distributed points on a sphere

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V=20: Dodecahedron.

  • 1. V=20 Dodecahedron. Images: A critical points scheme for Generating uniformly distributed points on a sphere

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V=24: Rhombicuboctahedron.

  • 1. V=24 Rhombicuboctahedron. Images: A critical points scheme for Generating uniformly distributed points on a sphere

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V=30: Icosidodecahedron.

  • 1. n=30 Icosidodecahedron. Images: A critical points scheme for Generating uniformly distributed points on a sphere.
  • 2. Generating two different uniformly distributed points on a sphere using one uniform distribution: Icosidodecahedron V=30.
  • 3. ●Icosidodecahedron. Vertices 30.
  • 4. ●Pentakis Dodecahedron. Vertices 32.
  • 5. Generating Elements of mesh modeling the surfaces of a Pentakis Dodecahedron(V=32) and its dual Solid: Truncated Icosahedron(V=60)
  • 6. Pentakis Dodecahedron(V=32) and its dual image -Truncated Icosahedron(V=60): Coloring the edges and faces
  • 7. Rhombicosidodecahedron-c. Vertices 60.
  • 8. Generating Elements of mesh modeling the surfaces of a Rhombicosidodecahedron-c(V=60) and its dual Solid: Deltoidal Hexecontahedron(V=62)
  • 9. Rhombicosidodecahedron-c(V=60) and its dual image - Deltoidal Hexecontahedron(V=62): Coloring the edges and faces

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V=60: Biscribed Truncated Icosahedron

  • 1. Vertices 60. Biscribed Truncated Icosahedron(extreme distribution). Images: A critical points scheme for Generating uniformly distributed points on a sphere.
  • 2. Generating two different uniformly distributed points on a sphere using one uniform distribution: Biscribed Truncated Icosahedron V=60.
  • 3. Biscribed Truncated Icosahedron(extreme distribution). Vertices 60.
  • 4. Biscribed Truncated Icosahedron, n=60. Polyhedra with extreme distribution of equivalent vertices.
  • 5. Three-parameter model transformations of the Icosahedron. Extreme distributions.
  • 6. ●Pentakis Dodecahedron. Vertices 32.
  • 7. Generating Elements of mesh modeling the surfaces of a Pentakis Dodecahedron(V=32) and its dual Solid: Truncated Icosahedron(V=60)
  • 8. Pentakis Dodecahedron(V=32) and its dual image -Truncated Icosahedron(V=60): Coloring the edges and faces
  • 9. Generating Elements of mesh modeling the surfaces of a Rectified truncated icosahedron(V=90) and its dual Solid.
  • 10. Rectified truncated icosahedron(V=90) and its dual image: Coloring the edges and faces
  • 11. Rectified Truncated Icosahedron. Vertices 90.
  • 12. Five pointed Star and Star of David inscribed in a Rectified Truncated Icosahedron.
  • 13. as Rectified Truncated Icosahedron. Vertices 90.

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polyhedra

  • 1. Tetrahedron. Vertices 4.
  • 2. Octahedron. Vertices 6.
  • 3. Cube. Vertices 8.
  • 4. Square Antiprism. Vertices 8.
  • 5. Icosahedron. Vertices 12.
  • 6. Cuboctahedron. Vertices 12.
  • 7. Dodecahedron. Vertices 20.
  • 8. Tetrakis hexahedron. Vertices 14.
  • 9. Disdyakis dodecahedron. Vertices 26.
  • 10. Icosidodecahedron. Vertices 30.
  • 11. ●Icosidodecahedron. Vertices 30.
  • 12. ●Pentakis Dodecahedron. Vertices 32.
  • 13. Rhombicosidodecahedron-c. Vertices 60.
  • 14. Biscribed Truncated Icosahedron(extreme distribution). Vertices 60.
  • 15. Rectified Truncated Icosahedron. Vertices 90.
  • 16. Five pointed Star and Star of David inscribed in a Rectified Truncated Icosahedron.
  • 17. as Rectified Truncated Icosahedron. Vertices 90.

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Coloring edges and faces

  • 1. Images. n=4: Tetrahedron -Platonic Solid and n=5: Triangular bipyramid (Triangular Dipyramid) -Johnson solid J₁₂ and their Dual Polyhedrons
  • 2. Images. n=6: Octahedron -Platonic Solid and n=8: Square antiprism and their Dual Polyhedrons
  • 3. Images. n=12: Icosahedron -Platonic Solid and n=20. Their Dual Polyhedrons
  • 4. Copy of 3 in 1. Constructing, surface triangulation, visualizing polyhedron. New Version.
  • 5. Truncated Tetrahedron, n=12. Polyhedra with extreme distribution of equivalent vertices.
  • 6. Example of non-uniqueness of the extreme distribution of n=16 particles on the surface of a sphere.
  • 7. Truncated Cube, n=24. Polyhedra with extreme distribution of equivalent vertices.
  • 8. 3. Vertices n=24. Coloring the edges and faces of a polyhedron and its dual image.
  • 9. Images. n=24: Biscribed Snub Cube (laevo) and n=32. Their Dual Polyhedrons
  • 10. Biscribed Truncated Octahedron, n=24. Polyhedra with extreme distribution of equivalent vertices.
  • 11. Icosidodecahedron, n=30. Polyhedra with extreme distribution of equivalent vertices.
  • 12. Pentakis Dodecahedron(V=32) and its dual image -Truncated Icosahedron(V=60): Coloring the edges and faces
  • 13. Truncated Dodecahedron, n=60. Polyhedra with extreme distribution of equivalent vertices.
  • 14. Rhombicosidodecahedron-c(V=60) and its dual image - Deltoidal Hexecontahedron(V=62): Coloring the edges and faces
  • 15. Biscribed Truncated Icosahedron, n=60. Polyhedra with extreme distribution of equivalent vertices.
  • 16. Rhombicosidodecahedron, n=60. Polyhedra with extreme distribution of equivalent vertices.
  • 17. Biscribed Snub Dodecahedron, n=60. Polyhedra with extreme distribution of equivalent vertices.
  • 18. Coloring the edges and faces of a polyhedron(n=72) Pentakis Snub Dodecahedron (laevo) and its dual image.
  • 19. Biscribed Pentakis Snub Dodecahedron (V=72) and its dual image: Coloring the edges and faces
  • 20. Coloring of edges and faces of a polyhedron(n=72) extreme distribution and its dual image.
  • 21. Rectified truncated icosahedron(V=90) and its dual image: Coloring the edges and faces
  • 22. Biscribed Truncated Icosidodecahedron, n=120. Polyhedra with extreme distribution of equivalent vertices.
  • 23. Finding and coloring edges and faces of a polyhedron and its dual image. V=180
  • 24. Canonical Truncated Snub Cube (laevo)(V=120) and its dual image: Coloring the edges and faces
  • 25. Images: Canonical Truncated Snub Cube (laevo)(V=120) and its dual image: Coloring the edges and faces
  • 26. Biscribed Hexpropello Dodecahedron (dextro) (V=140) and its dual image: Coloring the edges and faces
  • 27. Biscribed Orthotruncated Propello Icosahedron (V=120) and its dual image: Coloring the edges and faces

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