Structural classification of extremal angle configurations

Classify, up to the stated essential equivalences, all finite planar point sets that attain the sharp linear lower bound for the number of distinct angles, determining whether every such configuration arises essentially from either two points together with an arithmetic progression on their perpendicular bisector or the vertices of a regular polygon together with its centre.

Background

For every finite non-collinear planar point set P, the number of distinct angles determined by triples of points satisfies a sharp lower bound of order |P|. The paper describes two known constructions attaining this order: a configuration based on a perpendicular bisector and a regular polygon with its centre. A cited conjecture proposes that these examples essentially exhaust the extremal configurations, making this a structural classification problem rather than merely a quantitative lower-bound problem.