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  2. Rotation formalisms in three dimensions - Wikipedia

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

    The angle θ which appears in the eigenvalue expression corresponds to the angle of the Euler axis and angle representation. The eigenvector corresponding to the eigenvalue of 1 is the accompanying Euler axis, since the axis is the only (nonzero) vector which remains unchanged by left-multiplying (rotating) it with the rotation matrix.

  3. Euler's rotation theorem - Wikipedia

    en.wikipedia.org/wiki/Euler's_rotation_theorem

    A spatial rotation is a linear map in one-to-one correspondence with a 3 × 3 rotation matrix R that transforms a coordinate vector x into X, that is Rx = X. Therefore, another version of Euler's theorem is that for every rotation R , there is a nonzero vector n for which Rn = n ; this is exactly the claim that n is an eigenvector of R ...

  4. Axis–angle representation - Wikipedia

    en.wikipedia.org/wiki/Axisangle_representation

    The angle θ and axis unit vector e define a rotation, concisely represented by the rotation vector θe.. In mathematics, the axisangle representation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector e indicating the direction of an axis of rotation, and an angle of rotation θ describing the magnitude and sense (e.g., clockwise) of the ...

  5. Horizontal coordinate system - Wikipedia

    en.wikipedia.org/wiki/Horizontal_coordinate_system

    The horizontal coordinate system is a celestial coordinate system that uses the observer's local horizon as the fundamental plane to define two angles of a spherical coordinate system: altitude and azimuth. Therefore, the horizontal coordinate system is sometimes called the az/el system, [1] the alt/az system, or the alt-azimuth system, among

  6. Conversion between quaternions and Euler angles - Wikipedia

    en.wikipedia.org/wiki/Conversion_between...

    The resulting orientation of Body 3-2-1 sequence (around the capitalized axis in the illustration of Tait–Bryan angles) is equivalent to that of lab 1-2-3 sequence (around the lower-cased axis), where the airplane is rolled first (lab-x axis), and then nosed up around the horizontal lab-y axis, and finally rotated around the vertical lab-z ...

  7. Three-dimensional space - Wikipedia

    en.wikipedia.org/wiki/Three-dimensional_space

    They are usually labeled x, y, and z. Relative to these axes, the position of any point in three-dimensional space is given by an ordered triple of real numbers, each number giving the distance of that point from the origin measured along the given axis, which is equal to the distance of that point from the plane determined by the other two ...

  8. Spherical coordinate system - Wikipedia

    en.wikipedia.org/wiki/Spherical_coordinate_system

    And these systems of the mathematics convention may measure the azimuthal angle counterclockwise (i.e., from the south direction x-axis, or 180°, towards the east direction y-axis, or +90°)—rather than measure clockwise (i.e., from the north direction x-axis, or 0°, towards the east direction y-axis, or +90°), as done in the horizontal ...

  9. Davenport chained rotations - Wikipedia

    en.wikipedia.org/wiki/Davenport_chained_rotations

    The X axis is now at angle α with respect to the x axis. The XYZ system rotates again about the x axis by β. The Z axis is now at angle β with respect to the z axis. The XYZ system rotates a third time about the z axis by γ. In sum, the three elemental rotations occur about z, x and z. Indeed, this sequence is often denoted z-x-z (or 3-1-3 ...