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2. 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

3. Right-hand rule - Wikipedia

   en.wikipedia.org/wiki/Right-hand_rule

   In mathematics, a rotating body is commonly represented by a pseudovector along the axis of rotation. The length of the vector gives the speed of rotation and the direction of the axis gives the direction of rotation according to the right-hand rule: right fingers curled in the direction of rotation and the right thumb pointing in the positive ...

4. Rotation formalisms in three dimensions - Wikipedia

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

   The Rodrigues vector (sometimes called the Gibbs vector, with coordinates called Rodrigues parameters) [3] [4] can be expressed in terms of the axis and angle of the rotation as follows: = ^ ⁡ This representation is a higher-dimensional analog of the gnomonic projection , mapping unit quaternions from a 3-sphere onto the 3-dimensional pure ...

5. 3D rotation group - Wikipedia

   en.wikipedia.org/wiki/3D_rotation_group

   The group SO(3) can therefore be identified with the group of these matrices under matrix multiplication. These matrices are known as "special orthogonal matrices", explaining the notation SO(3). The group SO(3) is used to describe the possible rotational symmetries of an object, as well as the possible orientations of an object in space.

6. 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 ...

7. 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 ...