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In two dimensions, a point reflection is the same as a rotation of 180 degrees. In three dimensions, a point reflection can be described as a 180-degree rotation composed with reflection across the plane of rotation, perpendicular to the axis of rotation. In dimension n, point reflections are orientation-preserving if n is even, and orientation ...
A point P has coordinates (x, y) with respect to the original system and coordinates (x′, y′) with respect to the new system. [1] 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.
A rotation in the plane can be formed by composing a pair of reflections. First reflect a point P to its image P′ on the other side of line L 1. Then reflect P′ to its image P′′ on the other side of line L 2. If lines L 1 and L 2 make an angle θ with one another, then points P and P′′ will make an angle 2θ around point O, the ...
If these are ω 1 and ω 2 then all points not in the planes rotate through an angle between ω 1 and ω 2. Rotations in four dimensions about a fixed point have six degrees of freedom. A four-dimensional direct motion in general position is a rotation about certain point (as in all even Euclidean dimensions), but screw operations exist also.
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 ...
Every rotation in three dimensions is defined by its axis (a vector along this axis is unchanged by the rotation), and its angle — the amount of rotation about that axis (Euler rotation theorem). There are several methods to compute the axis and angle from a rotation matrix (see also axis–angle representation ).
Rotational symmetry of order n, also called n-fold rotational symmetry, or discrete rotational symmetry of the n th order, with respect to a particular point (in 2D) or axis (in 3D) means that rotation by an angle of (180°, 120°, 90°, 72°, 60°, 51 3 ⁄ 7 °, etc.) does not change the object. A "1-fold" symmetry is no symmetry (all ...
A strobogrammatic number is a number whose numeral is rotationally symmetric, so that it appears the same when rotated 180 degrees. The numeral looks the same right-side up and upside down (e.g., 69, 96, 1001). [54] [55] [56] Some dates are natural numeral ambigrams. [57]