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The group Spin(3) is isomorphic to the special unitary group SU(2); it is also diffeomorphic to the unit 3-sphere S 3 and can be understood as the group of versors (quaternions with absolute value 1). The connection between quaternions and rotations, commonly exploited in computer graphics, is explained in quaternions and spatial rotations.
In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ∠ ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as part of the 31st proposition in the third book of Euclid 's Elements . [ 1 ]
Poloidal direction (red arrow) and toroidal direction (blue arrow) A torus of revolution in 3-space can be parametrized as: [2] (,) = (+ ) (,) = (+ ) (,) = using angular coordinates θ, φ ∈ [0, 2π), representing rotation around the tube and rotation around the torus's axis of revolution, respectively, where the major radius R is the distance from the center of the tube to ...
Another type of sphere arises from a 4-ball, whose three-dimensional surface is the 3-sphere: points equidistant to the origin of the euclidean space R 4. If a point has coordinates, P(x, y, z, w), then x 2 + y 2 + z 2 + w 2 = 1 characterizes those points on the unit 3-sphere centered at the origin.
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Here α, β, γ are the direction cosines and the Cartesian coordinates of the unit vector | |, and a, b, c are the direction angles of the vector v. The direction angles a , b , c are acute or obtuse angles , i.e., 0 ≤ a ≤ π , 0 ≤ b ≤ π and 0 ≤ c ≤ π , and they denote the angles formed between v and the unit basis vectors e x ...
Hannah Kobayashi, a Hawaiian woman who has been missing for weeks, walked freely into Mexico just after noon on Nov. 12, the Los Angeles Police Department said on Monday. The department said the ...
The "Alternatives to Detention" program is tracking more than 25,000 migrants using ankle and wrist-worn monitors, which costs taxpayers an average of nearly $80,000 each day, according to ICE data.
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