A physical or a mathematical angle cannot simply exist by its own definition or simply by its own measurement. In other words angle is not subjected to a generalization. This is contrary to the notions of generalized coordinates, generalized momentum, and generalized force. Although as a physical coordinate, angle becomes generalized but of which the exact nature of the global mechanical or quantum mechanical system cannot be completely specified. However, locally the angle is transformed into a local infinitesimal gauge called the phase factor. This implies that the use of angle is always very specific, very precise, very accurate, and its measurement is always continuous. It is the one and only physical quantity that remains variably continuous everywhere and everywhen in the space-time continuum, both in quantum mechanics and in general relativity. In the history of scientific progress, the debate between general acceptances of competing theories had been decided by a tiny angle, this happened between Newtonian physics and Einstein’s physics. Wherever and whenever this angle’s measurement is expanded by infinite series, its perturbations (tiny changes) were able to prove the existence of antimatter, of quantized energy, of spin, or vacuum polarization, of charge conjugation, and of magnetic moment. The list goes on and on. Actually, there are at least five distinct but somewhat related definitions: (1) angle between lines in 3D space, (2) angle between lines in the plane, (3) angle between 2D planes, (4) angle between vectors or oriented surfaces, and (5) solid angles.

Definition 1 presupposed the existence of coordinated lines in 3D space with three defined direction ratios or direction cosines. Definition 2 presupposed a 2D Cartesian coordinate system as well as definitions for slopes and gradients. Definition 3 presupposed the existence of normal vectors. Definition 4 presupposed the existence of 3D vectors with definitions for magnitudes as absolute values of vectors as well as a definition for their scalar products. Definition 5 presupposed the existence of a unit sphere or spheres of arbitrary radii centered at the vertex of intersecting cones. In this case the angle measured in steradians that completely surrounding a singularity is 4p which is exactly the surface area of a sphere of unit radius. If a small normal area increment DA is at a distance R from the given singularity then the solid angle is simply DA/R². Clearly, this is a general covariance form of the inverse square law obeyed by Newton’s universal law of gravitation, Coulomb’s law of electrostatics, and Ampère’s law of magnetostatic force.