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  2. Rotation formalisms in three dimensions - Wikipedia

    en.wikipedia.org/wiki/Rotation_formalisms_in...

    Rotation formalisms are focused on proper (orientation-preserving) motions of the Euclidean space with one fixed point, that a rotation refers to.Although physical motions with a fixed point are an important case (such as ones described in the center-of-mass frame, or motions of a joint), this approach creates a knowledge about all motions.

  3. Möbius strip - Wikipedia

    en.wikipedia.org/wiki/Möbius_strip

    A 2D object traversing once around the Möbius strip returns in mirrored form. The Möbius strip has several curious properties. It is a non-orientable surface: if an asymmetric two-dimensional object slides one time around the strip, it returns to its starting position as its mirror image. In particular, a curved arrow pointing clockwise (↻ ...

  4. 3D rotation group - Wikipedia

    en.wikipedia.org/wiki/3D_rotation_group

    Given the above, for every point in this ball there is a rotation, with axis through the point and the origin, and rotation angle equal to the distance of the point from the origin. The identity rotation corresponds to the point at the center of the ball. Rotations through an angle 𝜃 between 0 and π (not including either) are on the same ...

  5. Rotation matrix - Wikipedia

    en.wikipedia.org/wiki/Rotation_matrix

    The case of θ = φ is called an isoclinic rotation, having eigenvalues e ±iθ repeated twice, so every vector is rotated through an angle θ. The trace of a rotation matrix is equal to the sum of its eigenvalues. For n = 2, a rotation by angle θ has trace 2 cos θ. For n = 3, a rotation around any axis by angle θ has trace 1 + 2 cos θ.

  6. Spherical coordinate system - Wikipedia

    en.wikipedia.org/wiki/Spherical_coordinate_system

    the radial distance r along the line connecting the point to a fixed point called the origin; the polar angle θ between this radial line and a given polar axis; [a] and; the azimuthal angle φ, which is the angle of rotation of the radial line around the polar axis. [b] (See graphic regarding the "physics convention".)

  7. Rotation - Wikipedia

    en.wikipedia.org/wiki/Rotation

    A sphere rotating (spinning) about an axis. Rotation or rotational motion is the circular movement of an object around a central line, known as an axis of rotation.A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersecting anywhere inside or outside the figure at a center of rotation.

  8. Torus - Wikipedia

    en.wikipedia.org/wiki/Torus

    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 ...

  9. Sphere - Wikipedia

    en.wikipedia.org/wiki/Sphere

    Any rotation about a line through the origin can be expressed as a combination of rotations around the three-coordinate axis (see Euler angles). Therefore, a three-parameter family of rotations exists such that each rotation transforms the sphere onto itself; this family is the rotation group SO(3).