![]() An isoclinic rotation will form a Villarceau circle on the torus, while a simple rotation will form a circle parallel or perpendicular to the central axis. For a rotation whose two rotation angles have a rational ratio, the paths will eventually reconnect while for an irrational ratio they will not. Half-lines from O in the axis-plane A are not displaced half-lines from O orthogonal to A are displaced through α all other half-lines are displaced through an angle less than α.ĭouble rotations Tesseract, in stereographic projection, in double rotation A 4D Clifford torus stereographically projected into 3D looks like a torus, and a double rotation can be seen as a helical path on that torus. All these 2D rotations have the same rotation angle α. For each such point P is the centre of the 2D rotation induced by R in B. Every plane B that is completely orthogonal to A intersects A in a certain point P. An "invariant plane" is a plane for which every vector in the plane, although it may be affected by the rotation, remains in the plane after the rotation.įour-dimensional rotations are of two types: simple rotations and double rotations.Ī simple rotation R about a rotation centre O leaves an entire plane A through O (axis-plane) fixed. For the sake of uniqueness, rotation angles are assumed to be in the segment except where mentioned or clearly implied by the context otherwise.Ī "fixed plane" is a plane for which every vector in the plane is unchanged after the rotation. In this article rotation means rotational displacement. The name comes from the fact that it is the special orthogonal group of order 4. In mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO(4). ![]()
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