projective geometry, branch of geometry 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. 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.

non-Euclidean geometry, branch of geometry in which the fifth postulate of Euclidean geometry, which allows one and only one line parallel to a given line through a given external point, is replaced by one of two alternative postulates. Allowing two parallels through any external point, the first alternative to Euclid's fifth postulate, leads to the hyperbolic geometry developed by the Russian N. I. Lobachevsky in 1826 and independently by the Hungarian Janos Bolyai in 1832. The second alternative, which allows no parallels through any external point, leads to the elliptic geometry developed by the German Bernhard Riemann in 1854. The results of these two types of non-Euclidean geometry are identical with those of Euclidean geometry in every respect except those propositions involving parallel lines, either explicitly or implicitly (as in the theorem for the sum of the angles of a triangle).

Hyperbolic Geometry

In hyperbolic geometry the two rays extending out in either direction from a point P and not meeting a line L are considered distinct parallels to L; among the results of this geometry is the theorem that the sum of the angles of a triangle is less than 180°. One surprising result is that there is a finite upper limit on the area of a triangle, this maximum corresponding to a triangle all of whose sides are parallel and all of whose angles are zero. Lobachevsky's geometry is called hyperbolic because a line in the hyperbolic plane has two points at infinity, just as a hyperbola has two asymptotes. The analogy used in considering this geometry involves the lines and figures drawn on a saddleshaped surface.

Elliptic Geometry

In elliptic geometry there are no parallels to a given line L through an external point P, and the sum of the angles of a triangle is greater than 180°. Riemann's geometry is called elliptic because a line in the plane described by this geometry has no point at infinity, where parallels may intersect it, just as an ellipse has no asymptotes. An idea of the geometry on such a plane is obtained by considering the geometry on the surface of a sphere, which is a special case of an ellipsoid. The shortest distance between two points on a sphere is not a straight line but an arc of a great circle (a circle dividing the sphere exactly in half). Since any two great circles always meet (in not one but two points, on opposite sides of the sphere), no parallel lines are possible. The angles of a triangle formed by arcs of three great circles always add up to more than 180°, as can be seen by considering such a triangle on the earth's surface bounded by a portion of the equator and two meridians of longitude connecting its end points to one of the poles (the two angles at the equator are each 90°, so the amount by which the sum of the angles exceeds 180° is determined by the angle at which the meridians meet at the pole).

Non-Euclidean Geometry and Curved Space

What distinguishes the plane of Euclidean geometry from the surface of a sphere or a saddle surface is the curvature of each (see differential geometry); the plane has zero curvature, the surface of a sphere and other surfaces described by Riemann's geometry have positive curvature, and the saddle surface and other surfaces described by Lobachevsky's geometry have negative curvature. Similarly, in three dimensions the spaces corresponding to these three types of geometry also have zero, positive, or negative curvature, respectively.

As to which of these systems is a valid description of our own three-dimensional space (or four-dimensional space-time), the choice can be made only on the basis of measurements made over very large, cosmological distances of a billion light-years or more; the differences between a Euclidean universe of zero curvature and a non-Euclidean universe of very small positive or negative curvature are too small to be detected from ordinary measurements. One interesting feature of a universe described by Riemann's geometry is that it is finite but unbounded; straight lines ultimately form closed curves, so that a ray of light could eventually return to its source.

See cosmology; relativity.

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.

Types of Geometry

Euclidean geometry, elementary geometry of two and three dimensions (plane and solid geometry), is based largely on the Elements of the Greek mathematician Euclid (fl. c.300 B.C.). In 1637, René Descartes showed how numbers can be used to describe points in a plane or in space and to express geometric relations in algebraic form, thus founding analytic geometry, of which algebraic geometry is a further development (see Cartesian coordinates). The problem of representing three-dimensional objects on a two-dimensional surface was solved by Gaspard Monge, who invented descriptive geometry for this purpose in the late 18th cent. differential geometry, in which the concepts of the calculus are applied to curves, surfaces, and other geometrical objects, was founded by Monge and C. F. Gauss in the late 18th and early 19th cent. The modern period in geometry begins with the formulations of projective geometry by J. V. Poncelet (1822) and of non-Euclidean geometry by N. I. Lobachevsky (1826) and János Bolyai (1832). Another type of non-Euclidean geometry was discovered by Bernhard Riemann (1854), who also showed how the various geometries could be generalized to any number of dimensions.

Their Relationship to Each Other

The different geometries are classified and related to one another in various ways. The non-Euclidean geometries are exactly analogous to the geometry of Euclid, except that Euclid's postulate regarding parallel lines is replaced and all theorems depending on this postulate are changed accordingly. Both Euclidean and non-Euclidean geometry are types of metric geometry, in which the lengths of line segments and the sizes of angles may be measured and compared. Projective geometry, on the other hand, is more general and includes the metric geometries as a special case; pure projective geometry makes no reference to lengths or angle measurements.

The general metric geometry consisting of all of Euclidean geometry except that part dependent on the parallel postulate is called absolute geometry; its propositions are valid for both Euclidean and non-Euclidean geometry. Another type of geometry, called affine geometry, includes Euclid's parallel postulate but disregards two other postulates concerning circles and angle measurement; the propositions of affine geometry are also valid in the four-dimensional geometry of space-time used in the theory of relativity. Ordered geometry consists of all propositions common to both absolute geometry and affine geometry; this geometry includes the notion on intermediacy ("betweenness") but not that of measurement.

An important step in recognizing the connections between the different types of geometry was the Erlangen program, proposed by the German Felix Klein in his inaugural address at the Univ. of Erlangen (1872), according to which geometries are classified with respect to the geometrical properties that are left unchanged (invariant) under a given group of transformations. For example, Euclidean geometry is the study of properties unchanged by similarity transformations, affine geometry is concerned with properties invariant under the linear transformations (affine collineations) that preserve parallelism, and projective geometry studies invariants under the more general projective transformations (collineations and correlations). Topology, perhaps the most general type of geometry although often considered a separate branch of mathematics, is concerned with properties invariant under continuous transformations, which carry neighborhoods of points into neighborhoods of their images.

The Axiomatic Approach to Geometry

Euclid's Elements organized the geometry then known into a systematic presentation that is still used in many texts. Euclid first defined his basic terms, such as point and line, then stated without proof certain axioms and postulates about them that seemed to be self-evident or obvious truths, and finally derived a number of statements (theorems) from the postulates by means of deductive logic. This axiomatic method has since been adopted not only throughout mathematics but in many other fields as well. The close examination of the axioms and postulates of Euclidean geometry during the 19th cent. resulted in the realization that the logical basis of geometry was not as firm as had previously been supposed. New axiom and postulate systems were developed by various mathematicians, notably David Hilbert (1899).

fractal geometry, branch of mathematics concerned with irregular patterns made of parts that are in some way similar to the whole, e.g., twigs and tree branches, a property called self-similarity or self-symmetry. Unlike conventional geometry, which is concerned with regular shapes and whole-number dimensions, such as lines (one-dimensional) and cones (three-dimensional), fractal geometry deals with shapes found in nature that have non-integer, or fractal, dimensions—linelike rivers with a fractal dimension of about 1.2 and conelike mountains with a fractal dimension between 2 and 3.

Fractal geometry developed from Benoit Mandelbrot's study of complexity and chaos (see chaos theory). Beginning in 1961, he published a series of studies on fluctuations of the stock market, the turbulent motion of fluids, the distribution of galaxies in the universe, and on irregular shorelines on the English coast. By 1975 Mandelbrot had developed a theory of fractals that became a serious subject for mathematical study. Fractal geometry has been applied to such diverse fields as the stock market, chemical industry, meteorology, and computer graphics.

differential geometry, branch of geometry in which the concepts of the calculus are applied to curves, surfaces, and other geometric entities. The approach in classical differential geometry involves the use of coordinate geometry (see analytic geometry; Cartesian coordinates), although in the 20th cent. the methods of differential geometry have been applied in other areas of geometry, e.g., in projective geometry.

The Analysis of Curves

If a point r moves along a curve at arc length s from some fixed point, then t = dr/ds is a unit tangent vector to the curve at r. The normal vector n is perpendicular to the curve at the point and indicates the direction of the rate of change of t, i.e., the tendency of r to bend in the plane containing both r and t, and the binormal vector b is perpendicular to both t and n and indicates the tendency of the curve to twist out of the plane of t and n.

These three vectors are related by the three formulas of the French mathematician Jean Frédéric Frenet, which are fundamental to the study of space curves: dt/ds = κn; dn/ds = -κt + τb; db/ds = -τn, where the constants κ and τ are the curvature and the torsion of the curve, respectively. Of special interest are the curves called evolutes and involutes; the evolute of a curve is another curve whose tangents are the normals to the original curve, and an involute of a curve is a curve whose evolute is the given curve.

The Analysis of Surfaces

In the analysis of surfaces, points on a surface may be described not only with respect to the three-dimensional coordinates of the space in which the surface is considered but also with respect to an intrinsic coordinate system defined in terms of a system of curves on the surface itself. The curves on the surface that locally represent the shortest distances between points on the surface are called geodesics; geodesics on a plane are straight lines. Tangent and normal vectors are also defined for a surface, but the relationships between them are more complex than for a space curve (e.g., a surface has a whole circle of unit vectors tangent to it at a given point).

The results of the theory of surfaces are expressed most easily in the notation of tensors. It is found that the total, or Gaussian, curvature of a surface is a bending invariant, i.e., an intrinsic property of the surface itself, independent of the space in which the surface may be considered. Of particular importance are surfaces of constant curvature; planes, cylinders, cones, and other so-called developable surfaces have zero curvature, while the elliptic and hyperbolic planes of non-Euclidean geometry are surfaces of constant positive and negative curvature, respectively.

Development of Differential Geometry

Differential geometry was founded by Gaspard Monge and C. F. Gauss in the beginning of the 19th cent. Important contributions were made by many mathematicians during the 19th cent., including B. Riemann, E. B. Christoffel, and C. G. Ricci. This work was collected and systematized at the end of the century by J. G. Darboux and Luigi Bianchi. The importance of differential geometry may be seen from the fact that Einstein's general theory of relativity is formulated entirely in terms of the differential geometry, in tensor notation, of a four-dimensional manifold combining space and time.

descriptive geometry, branch of geometry concerned with the two-dimensional representation of three-dimensional objects; it was introduced in 1795 by Gaspard Monge. By means of such representations, geometrical problems in three dimensions may be solved in the plane. (Such problems arise in all branches of engineering.) Modern mechanical drawing and architectural drawing are based on the principles of descriptive geometry.

analytic geometry, branch of geometry in which points are represented with respect to a coordinate system, such as Cartesian coordinates, and in which the approach to geometric problems is primarily algebraic. Its most common application is in the representation of equations involving two or three variables as curves in two or three dimensions or surfaces in three dimensions. For example, the linear equation ax+by+c=0 represents a straight line in the xy-plane, and the linear equation ax+by+cz+d=0 represents a plane in space, where a, b, c, and d are constant numbers (coefficients). In this way a geometric problem can be translated into an algebraic problem and the methods of algebra brought to bear on its solution. Conversely, the solution of a problem in algebra, such as finding the roots of an equation or system of equations, can be estimated or sometimes given exactly by geometric means, e.g., plotting curves and surfaces and determining points of intersection.

In plane analytic geometry a line is frequently described in terms of its slope, which expresses its inclination to the coordinate axes; technically, the slope m of a straight line is the (trigonometric) tangent of the angle it makes with the x-axis. If the line is parallel to the x-axis, its slope is zero. Two or more lines with equal slopes are parallel to one another. In general, the slope of the line through the points (x₁, y₁) and (x₂, y₂) is given by m= (y₂-y₁) / (x₂-x₁). The conic sections are treated in analytic geometry as the curves corresponding to the general quadratic equation ax²+bxy+cy²+dx+ey+f=0, where a, b, … , f are constants and a, b, and c are not all zero.

In solid analytic geometry the orientation of a straight line is given not by one slope but by its direction cosines, λ, μ, and ν, the cosines of the angles the line makes with the x-, y-, and z-axes, respectively; these satisfy the relationship λ²+μ²+ν²= 1. In the same way that the conic sections are studied in two dimensions, the 17 quadric surfaces, e.g., the ellipsoid, paraboloid, and elliptic paraboloid, are studied in solid analytic geometry in terms of the general equation ax²+by²+cz²+dxy+exz+fyz+px+qy+rz+s=0.

The methods of analytic geometry have been generalized to four or more dimensions and have been combined with other branches of geometry. Analytic geometry was introduced by René Descartes in 1637 and was of fundamental importance in the development of the calculus by Sir Isaac Newton and G. W. Leibniz in the late 17th cent. More recently it has served as the basis for the modern development and exploitation of algebraic geometry.

algebraic geometry, branch of geometry, based on analytic geometry, that is concerned with geometric objects (loci) defined by algebraic relations among their coordinates (see Cartesian coordinates). In plane geometry an algebraic curve is the locus of all points satisfying the polynomial equation f(x,y)=0; in three dimensions the polynomial equation f(x,y,z)=0 defines an algebraic surface. In general, points in n-space are defined by ordered sequences of numbers (x₁,x₂,x₃, … x_n), where each n-tuple specifies a unique point and x₁, x₂, x₃, … x_n are members of a given field (e.g., the complex numbers). An algebraic hypersurface is the locus of all such points satisfying the polynomial equation f(x₁,x₂,x₃, … x_n)=0, whose coefficients are also chosen from the given field. The intersection of two or more algebraic hypersurfaces defines an algebraic set, or variety, a concept of particular importance in algebraic geometry.

affine geometry: see geometry.

Any theory of the nature of geometric space differing from the traditional view held since Euclid's time. These geometries arose in the 19th century when several mathematicians working independently explored the possibility of rejecting Euclid's parallel postulate. Different assumptions about how many lines through a point not on a given line could be parallel to that line resulted in hyperbolic geometry and elliptic geometry. Mathematicians were forced to abandon the idea of a single correct geometry; it became their task not to discover mathematical systems but to create them by selecting consistent axioms and studying the theorems that could be derived from them. The development of these alternative geometries had a profound impact on the notion of space and paved the way for the theory of relativity. Seealso Nikolay Lobachevsky, Bernhard Riemann.

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Study of points, lines, angles, surfaces, and solids based on Euclid's axioms. Its importance lies less in its results than in the systematic method Euclid used to develop and present them. This axiomatic method has been the model for many systems of rational thought, even outside mathematics, for over 2,000 years. From 10 axioms and postulates, Euclid deduced 465 theorems, or propositions, concerning aspects of plane and solid geometric figures. This work was long held to constitute an accurate description of the physical world and to provide a sufficient basis for understanding it. During the 19th century, rejection of some of Euclid's postulates resulted in two non-Euclidean geometries that proved just as valid and consistent.

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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.

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Non-Euclidean geometry, useful in modeling interstellar space, that rejects the parallel postulate, proposing instead that at least two lines through any point not on a given line are parallel to that line. Though many of its theorems are identical to those of Euclidean geometry, others differ. For example, two parallel lines converge in one direction and diverge in the other, and the angles of a triangle add up to less than 180°.

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In mathematics, the study of complex shapes with the property of self-similarity, known as fractals. Rather like holograms that store the entire image in each part of the image, any part of a fractal can be repeatedly magnified, with each magnification resembling all or part of the original fractal. This phenomenon can be seen in objects like snowflakes and tree bark. The term fractal was coined by Benoit B. Mandelbrot in 1975. This new system of geometry has had a significant impact on such diverse fields as physical chemistry, physiology, and fluid mechanics; fractals can describe irregularly shaped objects or spatially nonuniform phenomena that cannot be described by Euclidean geometry. Fractal simulations have been used to plot the distributions of galactic clusters and to generate lifelike images of complicated, irregular natural objects, including rugged terrains and foliage used in films. Seealso chaos theory.

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Non-Euclidean geometry that rejects Euclid's fifth postulate (the parallel postulate) and modifies his second postulate. It is also known as Riemannian geometry, after Bernhard Riemann. It asserts that no line passing through a point not on a given line is parallel to that line. It also states that while any straight line of finite length can be extended indefinitely, all straight lines are the same length. Though many of elliptic geometry's theorems are identical to those of Euclidean geometry, others differ (e.g., the angles in a triangle add up to more than 180°). It can most easily be pictured as geometry done on the surface of a sphere where all lines are great circles.

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Field of mathematics in which methods of calculus are applied to the local geometry of curves and surfaces (i.e., to a small portion of a surface or curve around a point). A simple example is finding the tangent line on a two-dimensional curve at a given point. Similar operations may be extended to calculate the curvature and length of a curve and to analogous properties of surfaces in any number of dimensions.

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Investigation of geometric objects using coordinate systems. Because René Descartes was the first to apply algebra to geometry, it is also known as Cartesian geometry. It springs from the idea that any point in two-dimensional space can be represented by two numbers and any point in three-dimensional space by three. Because lines, circles, spheres, and other figures can be thought of as collections of points in space that satisfy certain equations, they can be explored via equations and formulas rather than graphs. Most of analytic geometry deals with the conic sections. Because these are defined using the notion of fixed distance, each section can be represented by a general equation derived from the distance formula.

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Study of geometric objects expressed as equations and represented by graphs in a given coordinate system. In contrast to Euclidean geometry, algebraic geometry represents geometric objects using algebraic equations (e.g., a circle of radius math.r is defined by math.x² + math.y² = math.r²). Objects so defined can then be analyzed for symmetries, intercepts, and other properties without having to refer to a graph.

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