Related Research Articles
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, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points to Euclidean points, and vice-versa.
In mathematics, affine geometry is what remains of Euclidean geometry when not using the metric notions of distance and angle.
A finite geometry is any geometric system that has only a finite number of points. The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the pixels are considered to be the points, would be a finite geometry. While there are many systems that could be called finite geometries, attention is mostly paid to the finite projective and affine spaces because of their regularity and simplicity. Other significant types of finite geometry are finite Möbius or inversive planes and Laguerre planes, which are examples of a general type called Benz planes, and their higher-dimensional analogs such as higher finite inversive geometries.
Synthetic geometry is the study of geometry without the use of coordinates or formulae. It relies on the axiomatic method and the tools directly related to them, that is, compass and straightedge, to draw conclusions and solve problems.
In finite geometry, the Fano plane is the finite projective plane of order 2. It is the finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point. The standard notation for this plane, as a member of a family of projective spaces, is PG(2, 2) where PG stands for "projective geometry", the first parameter is the geometric dimension and the second parameter is the order.
In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.
In projective geometry, Pascal's theorem states that if six arbitrary points are chosen on a conic and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon meet at three points which lie on a straight line, called the Pascal line of the hexagon. It is named after Blaise Pascal.
In painting, photography, graphical perspective and descriptive geometry, a picture plane is an image plane located between the "eye point" and the object being viewed and is usually coextensive to the material surface of the work. It is ordinarily a vertical plane perpendicular to the sightline to the object of interest.
In geometry, the notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects, which are often described in terms of two points or referred to using a single letter.
In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An incidence structure is what is obtained when all other concepts are removed and all that remains is the data about which points lie on which lines. Even with this severe limitation, theorems can be proved and interesting facts emerge concerning this structure. Such fundamental results remain valid when additional concepts are added to form a richer geometry. It sometimes happens that authors blur the distinction between a study and the objects of that study, so it is not surprising to find that some authors refer to incidence structures as incidence geometries.
Karl Georg Christian von Staudt was a German mathematician who used synthetic geometry to provide a foundation for arithmetic.
In mathematics, specifically in incidence geometry and especially in projective geometry, a complete quadrangle is a system of geometric objects consisting of any four points in a plane, no three of which are on a common line, and of the six lines connecting the six pairs of points. Dually, a complete quadrilateral is a system of four lines, no three of which pass through the same point, and the six points of intersection of these lines. The complete quadrangle was called a tetrastigm by Lachlan (1893), and the complete quadrilateral was called a tetragram; those terms are occasionally still used.
In geometry, a pencil is a family of geometric objects with a common property, for example the set of lines that pass through a given point in a plane, or the set of circles that pass through two given points in a plane.
George Bruce Halsted, usually cited as G. B. Halsted, was an American mathematician who explored foundations of geometry and introduced non-Euclidean geometry into the United States through his own work and his many important translations. Especially noteworthy were his translations and commentaries relating to non-Euclidean geometry, including works by Bolyai, Lobachevski, Saccheri, and Poincaré. He wrote an elementary geometry text, Rational Geometry, based on Hilbert's axioms, which was translated into French, German, and Japanese.
In geometry, a centre of an object is a point in some sense in the middle of the object. According to the specific definition of center taken into consideration, an object might have no center. If geometry is regarded as the study of isometry groups then a center is a fixed point of all the isometries which move the object onto itself.
In geometry, a homothetic center is a point from which at least two geometrically similar figures can be seen as a dilation or contraction of one another. If the center is external, the two figures are directly similar to one another; their angles have the same rotational sense. If the center is internal, the two figures are scaled mirror images of one another; their angles have the opposite sense.
In projective geometry, the harmonic conjugate point of an ordered triple of points on the real projective line is defined by the following construction:
In geometry, the pole and polar are respectively a point and a line that have a unique reciprocal relationship with respect to a given conic section.
Foundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries. These are fundamental to the study and of historical importance, but there are a great many modern geometries that are not Euclidean which can be studied from this viewpoint. The term axiomatic geometry can be applied to any geometry that is developed from an axiom system, but is often used to mean Euclidean geometry studied from this point of view. The completeness and independence of general axiomatic systems are important mathematical considerations, but there are also issues to do with the teaching of geometry which come into play.
In geometry, two diameters of a conic section are said to be conjugate if each chord parallel to one diameter is bisected by the other diameter. For example, two diameters of a circle are conjugate if and only if they are perpendicular.
References
- G. B. Halsted (1906) Synthetic Projective Geometry, Introduction, page 7 via Internet Archive
- Edward V. Huntington and Kurt E. Rosinger (1932) "Postulates for Separation of Point-Pairs (Reversible order on a closed line)", Proceedings of the American Academy of Arts and Sciences 67(4): 61-145 via JSTOR
- Bertrand Russell (1903) The Principles of Mathematics, Separation of couples via Internet Archive