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Uniform Distribution of Points on Hypersphere and Smooth Spherical Motion of Solid Body

This research focuses on the uniform distribution of points on a hypersphere by fitting centrosymmetric regular polyhedra. It also explores the smooth rotation of a solid body using unit quaternions.

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Uniform Distribution of Points on Hypersphere and Smooth Spherical Motion of Solid Body

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  1. The finite subgroups of SO(3), regular polyhedronsin R4 and the smooth spherical motion of asolid body Evgenii A. Mityushov, Natalia E. Misyura and Svetlana A. Berestova

  2. Consider the group of unit quaternions - Sp(1). The elements оf this group definethe point on the surface of a unit radius hypersphere . Consider the rotation group SO(3). The group Sp(1)is a two-sheeted cover for the group SO(3). We pose the problem of uniform distribution of points on the hypersphere . We solve this problem by fitting centrosymmetricregular polyhedrons into the hypersphere

  3. Forexample, considertheregular24-cellpolyhedron Its apex , , , , Wediscard the mirror-symmetric vertices, and for the remaining vertices we present the corresponding unit quaternions 1 - , 2 - , 3 - , 4 - , 5 - 6 - 7 - 8 - 9 - 10 - 11 - 12 - .

  4. We define the angular distance between the quaternions Where there exists a scalar product of the quaternions

  5. These are the angular distances between discrete orientations

  6. There are different routes, which now connect 12 vertices. Wechosetheshortest The angular distances are discovered to be the same and equal to . The result: We were able to get the 12 uniformly distributed points on the hypersphere obtained, which corresponds to a finite uniform subgroup of the group SO (3).

  7. We construct the law of smooth rotation of a solid body We numbered points on the hypersphere as future interpolation nodes We have constructed a smooth non-linear interpolation of unit quaternions. We used the soft start and deceleration function

  8. We used the representation of the matrix of rotation through the coordinates of the unit quaternion: We have the representation of unit quaternion by coordinates of points on the hypersphere, which are the same - (the elements of a finite uniform subgroup of the group SO (3)), with the use of interpolation (3)).

  9. We obtain the law of smooth rotation of a solid body along given shortest trajectories in the orientation space. The frames are the animation of the rotation of a solid body according to the law https://www.youtube.com/watch?v=_k00jJIBqWY

  10. Thank you for your attention!

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