mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a projective range is a set of points in

projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''p ...

considered in a unified fashion. A projective range may be a

projective line In projective geometry and mathematics more generally, a projective line is, roughly speaking, the extension of a usual line by a point called a '' point at infinity''. The statement and the proof of many theorems of geometry are simplified by the ...

or a conic. A projective range is the dual of a

pencil A pencil () is a writing or drawing implement with a solid pigment core in a protective casing that reduces the risk of core breakage and keeps it from marking the user's hand. Pencils create marks by physical abrasion, leaving a trail of ...

of lines on a given point. For instance, a

correlation In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics ...

interchanges the points of a projective range with the lines of a pencil. A

projectivity In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps line (geometry), lines to lines, and thus a collineati ...

is said to act from one range to another, though the two ranges may coincide as sets. A projective range expresses projective invariance of the relation of

projective harmonic conjugate In projective geometry, the harmonic conjugate point of a point on the real projective line with respect to two other points is defined by the following construction: :Given three collinear points , let be a point not lying on their join and le ...

s. 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. Such a quadruple of points is termed a harmonic range. In 1940 Julian Coolidge described this structure and identified its originator: :Two fundamental one-dimensional forms such as point ranges, pencils of lines, or of planes are defined as projective, when their members are in one-to-one correspondence, and a harmonic set of one ... corresponds to a harmonic set of the other. ... If two one-dimensional forms are connected by a train of projections and intersections, harmonic elements will correspond to harmonic elements, and they are projective in the sense of Von Staudt.

Conic ranges

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:Viktor Prasolov & Yuri Solovyev (1997) ''Elliptic Functions and Elliptic Integrals'', page one, Translations of Mathematical Monographs volume 170,

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

: Let ''A'' and ''B'' be in the range (conic) and ''AB'' the line connecting them. Let ''L'' be the line through ''E'' and parallel to ''AB''. The "sum of points ''A'' and ''B''", ''A'' + ''B'', is the intersection of ''L'' with the range. The

circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...

and

hyperbola In mathematics, a hyperbola is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected component ( ...

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

angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...

s on the circle and hyperbolic angles on the hyperbola.

References

{{Reflist * H. S. M. Coxeter (1955) ''The Real Projective Plane'',

University of Toronto Press The University of Toronto Press is a Canadian university press. Although it was founded in 1901, the press did not actually publish any books until 1911. The press originally printed only examination books and the university calendar. Its first s ...

, p 20 for line, p 101 for conic. Projective geometry