BilinearTransformation

Bilinear transformation, also known as a Möbius transformation, is a mapping of the extended complex plane given by f(z) = (az + b) / (cz + d), where a, b, c, d are complex numbers with ad − bc ≠ 0. It acts on the Riemann sphere, sending finite complex numbers to finite numbers and the point at infinity to a finite value, and it is defined wherever cz + d ≠ 0.

Bilinear transformations have several key geometric and algebraic properties. They map generalized circles (circles and lines) in the complex plane to generalized circles, and they preserve the cross-ratio of four points, a fundamental invariant in projective geometry. Composition corresponds to matrix multiplication: the map f is associated with the matrix [ [a, b], [c, d] ], defined up to a nonzero scalar multiple. The inverse transformation is f^{-1}(w) = (dw − b) / (−cw + a). The derivative f′(z) = (ad − bc) / (cz + d)^2 shows the map is conformal (angle-preserving) away from the pole z = −d/c, where it is not defined.

Special cases and interpretation are common in both complex analysis and geometry. If c = 0, the transformation reduces to an affine map z → (a/d)z + (b/d). In general, bilinear transformations form the automorphism group of the extended complex plane, meaning they are the most general rational maps of degree one that preserve the sphere’s structure.

Applications include conformal mapping techniques in complex analysis, such as mapping the unit disk to itself or to other domains, and in digital signal processing where the bilinear transform is used to convert analog filters to digital form by substituting s = (2/T)(z − 1)/(z + 1). This substitution preserves stability but introduces frequency warping.