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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 . A point P has coordinates ( x, y) with respect to the ...
Rotate the vector v = (X, Y, Z) around the rotation vector Q = (X, Y, Z). The angle of rotation will be θ = ‖ Q ‖. Calculate the cosine of the angle times the vector to rotate, plus sine of the angle times the axis of rotation, plus one minus cosine of the angle times the dot product of the vector and rotation axis times the axis of rotation.
Euler angles (in 3-2-1 sequence) to quaternion conversion. By combining the quaternion representations of the Euler rotations we get for the Body 3-2-1 sequence, where the airplane first does yaw (Body-Z) turn during taxiing onto the runway, then pitches (Body-Y) during take-off, and finally rolls (Body-X) in the air.
Rotation matrix. In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix. rotates points in the xy plane counterclockwise through an angle θ about the origin of a two-dimensional Cartesian coordinate system.
Helmert transformation. The transformation from a reference frame 1 to a reference frame 2 can be described with three translations Δx, Δy, Δz, three rotations Rx, Ry, Rz and a scale parameter μ. The Helmert transformation (named after Friedrich Robert Helmert, 1843–1917) is a geometric transformation method within a three-dimensional space.
v. t. e. Rotation around a fixed axis or axial rotation is a special case of rotational motion around an axis of rotation fixed, stationary, or static in three-dimensional space. This type of motion excludes the possibility of the instantaneous axis of rotation changing its orientation and cannot describe such phenomena as wobbling or precession.
In mathematics and mechanics, the Euler–Rodrigues formula describes the rotation of a vector in three dimensions. It is based on Rodrigues' rotation formula, but uses a different parametrization. The rotation is described by four Euler parameters due to Leonhard Euler. The Rodrigues formula (named after Olinde Rodrigues ), a method of ...
The product G(i, j, θ)x represents a counterclockwise rotation of the vector x in the (i, j) plane of θ radians, hence the name Givens rotation. The main use of Givens rotations in numerical linear algebra is to transform vectors or matrices into a special form with zeros in certain coefficients. This effect can, for example, be employed for ...