Let be the Euclidean plane equipped with the Cartesian coordinate system. Recall that given a circle centered at the origin , one can define an “ordinary” rotation to be a linear transformation that takes any point on to another point on . In other words, .
Since a hyperbolic rotation is defined as a linear transformation, let us see what it looks like in matrix form. We start with the simple case when a rectangular hyperbola has the form , where is a non-negative real number.
Suppose denotes a hyperbolic rotation such that . Set
where is the matrix representation of , and . Solving for and we get and . In other words, with respect to rectangular hyperbolas of the form , the matrix representation of a hyperbolic rotation looks like
Since the matrix is non-singular, we see that in fact .
Now that we know the matrix form of a hyperbolic rotation when the rectangular hyperbolas have the form , it is not hard to solve the general case. Since the two asymptotes of any rectangular hyperbola are perpendicular, by an appropriate change of bases (ordinary rotation), can be transformed into a rectangular hyperbola whose asymptotes are the and axes, so that has the algebraic form . As a result, the matrix representation of a hyperbolic rotation with respect to has the form
Below are some simple properties:
Unlike an ordinary rotation , where fixes any circle centered at , a hyperbolic rotation fixing one rectangular hyperbola centered at may not fix another hyperbola of the same kind (as implied by the discussion above).
Let be the pencil of all rectangular hyperbolas centered at . For each , let be the subset of containing all hyperbolas whose asymptotes are same as the asymptotes for . If a hyperbolic rotation fixing , then for any .
defined above partitions into disjoint subsets. Call each of these subset a sub-pencil. Let be a sub-pencil of . Call fixes if fixes any element of . Let be sub-pencils of . Then fixes iff does not fix .
Let be sub-pencils of . Let be hyperbolic rotations such that fixes and fixes . Then is a hyperbolic rotation iff .
In other words, the set of all hyperbolic rotations fixing a sub-pencil is closed under composition. In fact, it is a group.
Suppose fixes the unit hyperbola . Let . Then fixes the (measure of) hyperbolic angle between and . In other words, if is the measure of the hyperbolic angle between and and, by abuse of notation, let be the measure of the hyperbolic angle between and . Then .
|Date of creation||2013-03-22 17:24:34|
|Last modified on||2013-03-22 17:24:34|
|Last modified by||CWoo (3771)|