In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform is a homography used in real analysis, complex analysis, and quaternionic analysis. In the theory of Hilbert spaces, the Cayley transform is a mapping between linear operators . Real homography The Cayley transform is an automorphism of the real projective line that permutes the elements of {1, 0, −1, ∞} in sequence. For example, it maps the positive real numbers to the interval [−1, 1]. Thus the Cayley transform is used to adapt Legendre polynomials for use with functions on the positive real numbers with Legendre rational functions. As a real homography, points are described with projective coordinates, and the mapping is Complex homography On the Riemann sphere, the Cayley transform is: Since {∞, 1, –1 } is mapped to {1, –i, i }, and Möbius transformations permute the generalised circles in the complex plane, f maps the real line to the unit circle. Furthermore, since f is continuous and i is taken to 0 by f, the upper half-plane is mapped to the unit disk. In terms of the models of hyperbolic geometry, this Cayley transform relates the Poincaré half-plane model to the Poincaré disk model. In electrical engineering the Cayley transform has been used to map a reactance half-plane to the Smith chart used for impedance matching of transmission lines. Quaternion homography In the four-dimensional space of quaternions q = a + b i + c j + d k, the versors form the unit 3-sphere. Since quaternions are non-commutative, elements of its projective line have homogeneous coordinates written U(a,b) to indicate that the homogeneous factor multiplies on the left. The quaternion transform is The real and complex homographies described above are instances of the quaternion homography where θ is zero or π/2, respectively. Evidently the transform takes u → 0 → –1 and takes –u → ∞ → 1. Evaluating this homography at q = 1 maps the versor u into its axis: But Thus In this form the Cayley transform has been described as a rational parametrization of rotation: Let t = tan φ/2 in the complex number identity where the right hand side is the transform of t i and the left hand side represents the rotation of