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Transformation of three phase electrical quantities to two phase quantities is a usual practice to simplify analysis of three phase electrical circuits. Polyphase a.c machines can be represented by an equivalent two phase model provided the rotating polyphases winding in rotor and the stationary polyphase windings in stator can be expressed in a fictitious two axes coils.
The case of θ = 0, φ ≠ 0 is called a simple rotation, with two unit eigenvalues forming an axis plane, and a two-dimensional rotation orthogonal to the axis plane. Otherwise, there is no axis plane. The case of θ = φ is called an isoclinic rotation, having eigenvalues e ±iθ repeated twice, so every vector is rotated through an angle θ.
With these conventions; the rotation matrix rotates any 3d vector about the axis e through angle θ anticlockwise (an active transformation), which has the equivalent effect of rotating the coordinate frame clockwise about the same axis through the same angle (a passive transformation). The rotation matrix is an orthogonal matrix, its transpose ...
Rotation of an object in two dimensions around a point O. Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point.
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 .
Transformations with reflection are represented by matrices with a determinant of −1. This allows the concept of rotation and reflection to be generalized to higher dimensions. In finite-dimensional spaces, the matrix representation (with respect to an orthonormal basis) of an orthogonal transformation is an orthogonal matrix.
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.
One author expressed the importance of group theory to transformation geometry as follows: I have gone to some trouble to develop from first principles all the group theory that I need, with the intention that my book can serve as a first introduction to transformation groups, and the notions of abstract group theory if you have never seen ...