Squeeze mapping

In linear algebra, a squeeze mapping is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is not a rotation or shear mapping. In linear algebra, a squeeze mapping is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is not a rotation or shear mapping. For a fixed positive real number a, the mapping is the squeeze mapping with parameter a. Since is a hyperbola, if u = ax and v = y/a, then uv = xy and the points of the image of the squeeze mapping are on the same hyperbola as (x,y) is. For this reason it is natural to think of the squeeze mapping as a hyperbolic rotation, as did Émile Borel in 1914, by analogy with circular rotations, which preserve circles. The squeeze mapping sets the stage for development of the concept of logarithms. The problem of finding the area bounded by a hyperbola (such as xy = 1) is one of quadrature. The solution, found by Grégoire de Saint-Vincent and Alphonse Antonio de Sarasa in 1647, required the natural logarithm function, a new concept. Some insight into logarithms comes through hyperbolic sectors that are permuted by squeeze mappings while preserving their area. The area of a hyperbolic sector is taken as a measure of a hyperbolic angle associated with the sector. The hyperbolic angle concept is quite independent of the ordinary circular angle, but shares a property of invariance with it: whereas circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping. Both circular and hyperbolic angle generate invariant measures but with respect to different transformation groups. The hyperbolic functions, which take hyperbolic angle as argument, perform the role that circular functions play with the circular angle argument. If r and s are positive real numbers, the composition of their squeeze mappings is the squeeze mapping of their product. Therefore, the collection of squeeze mappings forms a one-parameter group isomorphic to the multiplicative group of positive real numbers. An additive view of this group arises from consideration of hyperbolic sectors and their hyperbolic angles. From the point of view of the classical groups, the group of squeeze mappings is SO+(1,1), the identity component of the indefinite orthogonal group of 2 × 2 real matrices preserving the quadratic form u2 − v2. This is equivalent to preserving the form xy via the change of basis and corresponds geometrically to preserving hyperbolae. The perspective of the group of squeeze mappings as hyperbolic rotation is analogous to interpreting the group SO(2) (the connected component of the definite orthogonal group) preserving quadratic form x2 + y2 as being circular rotations. Note that the 'SO+' notation corresponds to the fact that the reflections