projective geometry, branch of geometry geometry [Gr.,=earth measuring], branch of mathematics concerned with the properties of and relationships between points, lines, planes, and figures and with generalizations of these concepts.
..... Click the link for more information. concerned with those properties of geometric figures that remain invariant under projection. The basic elements are points, lines, and planes, and the following statements are usually taken as assumptions: (1) two points lie in a unique line; (2) three points not on the same line determine a plane; (3) two lines in a plane intersect in a point; (4) two planes intersect in a line; (5) three planes not containing the same line intersect in a point. The basic elements retain their character under projection; e.g., the projection of a line is another line, and the point of intersection of two lines is projected into another point that is the intersection of the projections of the two original lines. However, lengths and ratios of lengths are not invariant under projection, nor are angles or the shapes of figures. The concept of parallelism does not appear at all in projective geometry; any pair of distinct lines intersects in a point, and if these lines are parallel in the sense of Euclidean geometry, then their point of intersection is at infinity infinity, in mathematics, that which is not finite. A sequence of numbers, a₁, a₂, a₃, … , is said to "approach infinity" if the numbers eventually become arbitrarily large, i.e.
..... Click the link for more information. . The plane that includes the ideal line, or line at infinity, consisting of all such ideal points, is called the projective plane. Two properties that are invariant under projection are the order of three or more points on a line and the harmonic relationship, or cross ratio, among four points, A, B, C, D, i.e., AC/BC : AD/BD. One important concept in projective geometry is that of duality. In the plane, the terms point and line are dual and can be interchanged in any valid statement to yield another valid statement, e.g., statements (1) and (3) above; in space, the terms plane, line, and point are interchanged with point, line, and plane, respectively, to yield dual statements (sometimes with slight changes in wording) as in statements (2) and (5) and statements (1) and (4) above. The origins of projective geometry are found in the work of Pappus, Gérard Desargues, and others. It first emerged as a discipline in its own right with the work of J. V. Poncelet (1822) and was placed on an axiomatic basis by K. G. C. von Staudt (1847), both these mathematicians adopting the pure, or synthetic, approach, in which algebraic and analytic methods are avoided and the treatment is purely geometric, in contrast to the approach of A. F. Möbius, Julius Plücker, and others. Projective geometry is more general than the familiar Euclidean geometry and includes the metric geometries (both Euclidean and non-Euclidean) as special cases.

projective geometry

Branch of mathematics that deals with the relationships between geometric figures and the images (mappings) of them that result from projection. Examples of projections include motion pictures, maps of the Earth's surface, and shadows cast by objects. One stimulus for the subject's development was the need to understand perspective in drawing and painting. Every point of the projected object and the corresponding point of its image must lie on the projection ray, a line that passes through the centre of projection. Modern projective geometry emphasizes the mathematical properties (such as straightness of lines and points of intersection) preserved in projections despite the distortion of lengths, angles, and shapes.

projective geometry [prə′jek·tiv jē′äm·ə·trē]

(mathematics)

The study of those properties of geometric objects which are invariant under projection.