A projective range is the dual of a pencil of lines on a given point.
For instance, a correlation interchanges the points of a projective range with the lines of a pencil.
Indeed, three points on a projective line determine a fourth by this relation.
Application of a projectivity to this quadruple results in four points likewise in the harmonic relation.
In 1940 Julian Coolidge described this structure and identified its originator:[1] When a conic is chosen for a projective range, and a particular point E on the conic is selected as origin, then addition of points may be defined as follows:[2] The circle and hyperbola are instances of a conic and the summation of angles on either can be generated by the method of "sum of points", provided points are associated with angles on the circle and hyperbolic angles on the hyperbola.