Affine Angles via Area Cross Ratio and Isoptic Hyperbolas

Affine geometry is usually regarded as a framework in which metric notions such as distance and angle are absent. However, just as projective geometry produces various metric geometries by introducing additional structures on the line at infinity, affine geometry can also serve as a natural basis for an angular geometry once certain directions at infinity are fixed. In this paper we introduce an affine angle determined by two fixed directions on the line at infinity and defined via an area cross ratio. This quantity is invariant under affine transformations preserving the chosen directions. We show that the locus of points from which a fixed segment is seen under a constant affine angle is a hyperbola whose asymptotes are parallel to the chosen directions. This provides an affine analogue of the classical fact that in Euclidean geometry the isoptic curve of a segment is a circle. Furthermore, we establish that this angle arises as a parabolic degeneration of the Cayley--Klein angle, and that the same quantity naturally appears in a power theorem associated with hyperbolas. These results provide a unified perspective linking affine angles, isoptic hyperbolas, and hyperbolic power through the area cross ratio.