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In the Euclidean plane, a point reflection is the same as a half-turn rotation (180° or π radians), while in three-dimensional Euclidean space a point reflection is an improper rotation which preserves distances but reverses orientation. A point reflection is an involution: applying it twice is the identity transformation.
A 180° rotation (middle) followed by a positive 90° rotation (left) is equivalent to a single negative 90° (positive 270°) rotation (right). Each of these figures depicts the result of a rotation relative to an upright starting position (bottom left) and includes the matrix representation of the permutation applied by the rotation (center ...
The set of all reflections in lines through the origin and rotations about the origin, together with the operation of composition of reflections and rotations, forms a group. The group has an identity: Rot(0). Every rotation Rot(φ) has an inverse Rot(−φ). Every reflection Ref(θ) is its own inverse. Composition has closure and is ...
The sign of the azimuth is determined by designating the rotation that is the positive sense of turning about the zenith. This choice is arbitrary, and is part of the coordinate system definition. (If the inclination is either zero or 180 degrees (= π radians), the azimuth is arbitrary. If the radius is zero, both azimuth and inclination are ...
In the new coordinate system, the point P will appear to have been rotated in the opposite direction, that is, clockwise through the angle . A rotation of axes in more than two dimensions is defined similarly. [2] [3] A rotation of axes is a linear map [4] [5] and a rigid transformation.
The circle defined by this slice will be very small, corresponding to the small angle of the rotation. As the rotation angles become larger, the slice moves in the negative z direction, and the circles become larger until the equator of the sphere is reached, which will correspond to a rotation angle of 180 degrees. Continuing southward, the ...
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The corresponding rotation axis must be defined to point in a direction that limits the rotation angle to not exceed 180 degrees. (This can always be done because any rotation of more than 180 degrees about an axis m {\displaystyle m} can always be written as a rotation having 0 ≤ α ≤ 180 ∘ {\displaystyle 0\leq \alpha \leq 180^{\circ ...