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In mathematics, a rotation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x′y′-Cartesian coordinate system in which the origin is kept fixed and the x′ and y′ axes are obtained by rotating the x and y axes counterclockwise through an angle .
Rotation formalisms are focused on proper (orientation-preserving) motions of the Euclidean space with one fixed point, that a rotation refers to.Although physical motions with a fixed point are an important case (such as ones described in the center-of-mass frame, or motions of a joint), this approach creates a knowledge about all motions.
Signed binary angle measurement. Black is traditional degrees representation, green is a BAM as a decimal number and red is hexadecimal 32-bit BAM. In this figure the 32-bit binary integers are interpreted as signed binary fixed-point values with scaling factor 2 −31, representing fractions between −1.0 (inclusive) and +1.0 (exclusive).
Figure 2: A rotation represented by an Euler axis and angle. In three dimensions, angular displacement is an entity with a direction and a magnitude. The direction specifies the axis of rotation, which always exists by virtue of the Euler's rotation theorem ; the magnitude specifies the rotation in radians about that axis (using the right-hand ...
The binary degree, also known as the binary radian (or brad), is 1 / 256 turn. [21] The binary degree is used in computing so that an angle can be represented to the maximum possible precision in a single byte. Other measures of angle used in computing may be based on dividing one whole turn into 2 n equal parts for other values of n. [22]
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.
is angular frequency, with unit radian per second or degree per second; For example, a stepper motor might turn exactly one complete revolution each second. Its angular frequency is 360 degrees per second (360°/s), or 2π radians per second (2π rad/s), while the rotational frequency is 60 rpm.
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 ...