In mathematics, 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, ...

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

projective line In mathematics, 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 resultant elimination of special cases; ...

or a

conic In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...

. 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 lines on a given point. For instance, a correlation 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 lines to lines, and thus a collineation. In general, ...

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 an ordered triple of points on the real projective line is defined by the following construction: :Given three collinear points , let be a point not lying on their join and let any line t ...

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 Julian Lowell Coolidge (September 28, 1873 – March 5, 1954) was an American mathematician, historian and a professor and chairman of the Harvard University Mathematics Department. Biography Born in Brookline, Massachusetts, he graduated from Har ...

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 points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...

and

hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth 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, ca ...

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 is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...

s on the circle and

hyperbolic angle In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of ''xy'' = 1 in Quadrant I of the Cartesian plane. The hyperbolic angle parametrises the unit hyperbola, which has hyperbolic function ...

s 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 founded in 1901. 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 cale ...

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