This paper covers a fundamental problem of local phase based signal processing: the isotropic generalization of the classical 1D analytic signal to two dimensions. The well known analytic signal enables the analysis of local phase and amplitude information of 1D signals. Local phase, amplitude and additional orientation information can be extracted by the 2D monogenic signal with the restriction to the subclass of intrinsically one dimensional signals. In case of 2D image signals the monogenic signal enables the rotationally invariant analysis of lines and edges. In this work we present the 2D analytic signal as a novel generalization of both the analytic signal and the 2D monogenic signal. In case of 2D image signals the 2D analytic signal enables the isotropic analysis of lines, edges, corners and junctions in one unified framework. Furthermore, we show that 2D signals exist per se in a 3D projective subspace of the homogeneous conformal space which delivers a descriptive geometric interpretation of signals providing new insights on the relation of geometry and 2D signals.