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2. Charts on SO (3) - Wikipedia

   en.wikipedia.org/wiki/Charts_on_SO(3)

   The 3-ball with this identification of the boundary is P 3 (R). A similar situation holds for applying a Cayley transform to the skew-symmetric matrix. Axis angle gives parameters in S 2 × S 1; if we replace the unit vector by the actual axis of rotation, so that n and −n give the same axis line, the set of axis becomes P 2 (R), the real ...

3. Rotation formalisms in three dimensions - Wikipedia

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

   The most external matrix rotates the other two, leaving the second rotation matrix over the line of nodes, and the third one in a frame comoving with the body. There are 3 × 3 × 3 = 27 possible combinations of three basic rotations but only 3 × 2 × 2 = 12 of them can be used for representing arbitrary 3D rotations as Euler angles. These 12 ...

4. Three-dimensional space - Wikipedia

   en.wikipedia.org/wiki/Three-dimensional_space

   A sphere in 3-space (also called a 2-sphere because it is a 2-dimensional object) consists of the set of all points in 3-space at a fixed distance r from a central point P. The solid enclosed by the sphere is called a ball (or, more precisely a 3-ball). The volume of the ball is given by

5. Axis–angle representation - Wikipedia

   en.wikipedia.org/wiki/Axis–angle_representation

   The angle θ and axis unit vector e define a rotation, concisely represented by the rotation vector θe.. In mathematics, the axis–angle 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 ...

6. Translation of axes - Wikipedia

   en.wikipedia.org/wiki/Translation_of_axes

   For an xyz-Cartesian coordinate system in three dimensions, suppose that a second Cartesian coordinate system is introduced, with axes x', y' and z' so located that the x' axis is parallel to the x axis and h units from it, the y' axis is parallel to the y axis and k units from it, and the z' axis is parallel to the z axis and l units from it.

7. Rotation matrix - Wikipedia

   en.wikipedia.org/wiki/Rotation_matrix

   In 3-space n = 3, the axis of a non-null proper rotation is always a unique line, and a rotation around this axis by angle θ has eigenvalues λ = 1, e iθ, e −iθ. In 4-space n = 4 , the four eigenvalues are of the form e ± iθ , e ± iφ .

8. Goat grazing problem - Wikipedia

   en.wikipedia.org/wiki/Goat_grazing_problem

   The goat problems do not yield any new mathematical insights; rather they are primarily exercises in how to artfully deconstruct problems in order to facilitate solution. Three-dimensional analogues and planar boundary/area problems on other shapes, including the obvious rectangular barn and/or field, have been proposed and solved. [ 1 ]

9. Cartesian coordinate system - Wikipedia

   en.wikipedia.org/wiki/Cartesian_coordinate_system

   A Cartesian coordinate system in two dimensions (also called a rectangular coordinate system or an orthogonal coordinate system [8]) is defined by an ordered pair of perpendicular lines (axes), a single unit of length for both axes, and an orientation for each axis. The point where the axes meet is taken as the origin for both, thus turning ...