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An antenna mast with four collinear directional arrays. In telecommunications, a collinear (or co-linear) antenna array is an array of dipole antennas mounted in such a manner that the corresponding elements of each antenna are parallel and aligned, that is they are located along a common line or axis.
Let X be an affine space over a field k, and V be its associated vector space. An affine transformation is a bijection f from X onto itself that is an affine map; this means that a linear map g from V to V is well defined by the equation () = (); here, as usual, the subtraction of two points denotes the free vector from the second point to the first one, and "well-defined" means that ...
However, parallel (non-crossing) pairs of lines are less restricted in hyperbolic line arrangements than in the Euclidean plane: in particular, the relation of being parallel is an equivalence relation for Euclidean lines but not for hyperbolic lines. [51] The intersection graph of the lines in a hyperbolic arrangement can be an arbitrary ...
If u and v are collinear (parallel or antiparallel along the same line of relative motion), the boost matrices commute: B(v)B(u) = B(u)B(v). This composite transformation happens to be another boost, B(w), where w is collinear with u and v. If u and v are not
The lines in any parallel class form a partition the points of the affine plane. Each of the n + 1 lines that pass through a single point lies in a different parallel class. The parallel class structure of an affine plane of order n may be used to construct a set of n − 1 mutually orthogonal latin squares. Only the incidence relations are ...
This form may be more useful when two vectors defining a plane are involved. An example in physics is the Thomas precession which includes the rotation given by Rodrigues' formula, in terms of two non-collinear boost velocities, and the axis of rotation is perpendicular to their plane.
Intersecting, parallel and ultra parallel lines through a with respect to l in the hyperbolic plane. The parallel lines appear to intersect l just off the image. This is just an artifact of the visualisation. On a real hyperbolic plane the lines will get closer to each other and 'meet' in infinity.
(If one of the four points is the line's point at infinity, then the two distances involving that point are dropped from the formula.) The point D is the harmonic conjugate of C with respect to A and B precisely if the cross-ratio of the quadruple is −1 , called the harmonic ratio .