Projective plane

In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines intersect in a single point, but there are some pairs of lines that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect, thus any two distinct lines in a projective plane intersect in only one point. Renaissance artists, in developing the techniques of drawing in perspective, laid the groundwork for this mathematical topic; the archetypical example is the real projective plane known as the extended Euclidean plane. This example, in different guises, is important in algebraic geometry and projective geometry where it may be denoted variously by PG, RP2, or P2, among other notations. There are many other projective planes, both infinite, such as the complex projective plane, finite, such as the Fano plane. A projective plane is a 2-dimensional projective space, but not all projective planes can be embedded in 3-dimensional projective spaces.

Such embeddability is a consequence of a property known as Desargues' theorem, not shared by all projective planes. A projective plane consists of a set of lines, a set of points, a relation between points and lines called incidence, having the following properties: The second condition means that there are no parallel lines; the last condition excludes the so-called degenerate cases. The term "incidence" is used to emphasize the symmetric nature of the relationship between points and lines, thus the expression "point P is incident with line ℓ " is used instead of either "P is on ℓ " or "ℓ passes through P ". To turn the ordinary Euclidean plane into a projective plane proceed as follows: To each set of mutually parallel lines add a single new point; that point is considered incident with each line of this set. The point added; these new points are called points at infinity. Add a new line, considered incident with all the points at infinity; this line is called the line at infinity. The extended structure is a projective plane and is called the extended Euclidean plane or the real projective plane.

The process outlined above, used to obtain it, is called "projective completion" or projectivization. This plane can be constructed by starting from R3 viewed as a vector space, see § Vector space construction below; the points of the Moulton plane are the points of the Euclidean plane, with coordinates in the usual way. To create the Moulton plane from the Euclidean plane some of the lines are redefined; that is, some of their point sets will be changed. Redefine all the lines with negative slopes so that they look like "bent" lines, meaning that these lines keep their points with negative x-coordinates, but the rest of their points are replaced with the points of the line with the same y-intercept but twice the slope wherever their x-coordinate is positive; the Moulton plane is an affine plane. It can be projectivized, as in the previous example. Desargues' theorem is not a valid theorem in either the Moulton plane or the projective Moulton plane; this example has just thirteen points and thirteen lines.

We label the points P1... P13 and the lines m1...m13. The incidence relation can be given by the following incidence matrix; the rows are labelled by the points and the columns are labelled by the lines. A 1 in row i and column j means that the point Pi is on the line mj, while a 0 means that they are not incident; the matrix is in Paige-Wexler normal form. To verify the conditions that make this a projective plane, observe that every two rows have one common column in which 1's appear and that every two columns have one common row in which 1's appear. Among many possibilities, the points P1,P4,P5,and P8, for example, will satisfy the third condition; this example is known as the projective plane of order three. Though the line at infinity of the extended real plane may appear to have a different nature than the other lines of that projective plane, this is not the case. Another construction of the same projective plane shows that no line can be distinguished from any other. In this construction, each "point" of the real projective plane is the one-dimensional subspace through the origin in a 3-dimensional vector space, a "line" in the projective plane arises from a plane through the origin in the 3-space.

This idea can be made more precise as follows. Let K be any division ring. Let K3 denote the set of all triples x = of elements of K. For any nonzero x in K3, the minimal subspace of K3 containing x is the subset of K3. Let x and y be linearly independent elements of K3, meaning that kx + my = 0 implies that k = m = 0; the minimal subspace of K3 containing x and y is the subset of K3. This 2-dimensional subspace contains various 1-di

Pencil (optics)

In optics, a pencil or pencil of rays is a geometric construct used to describe a beam or portion of a beam of electromagnetic radiation or charged particles in the form of a narrow cone or cylinder. Antennas which bundle in azimuth and elevation are described as "pencil-beam" antennas. For example, a phased array antenna can send out a beam, thin; such antennas are used for tracking radar. See Beamforming for further details. In optics, the focusing action of a lens is described in terms of pencils of rays. In addition to conical and cylindrical pencils, optics deals with astigmatic pencils as well. In electron optics, scanning electron microscopes use narrow pencil beams to achieve a deep depth of field. Ionizing radiation used in radiation therapy, whether photons or charged particles, such as proton therapy and electron therapy machines, is sometimes delivered through the use of pencil beam scanning. In Backscatter X-ray imaging a pencil beam of x-ray radiation is used the scan over an object to create an intensity image of the Compton-scattered radiation.

Collimated beam Pencil, a family of geometric objects having a common property such as passage through a given point. Fan beam Pencil beam scanning Microwave transmission

Parallel (geometry)

In geometry, parallel lines are lines in a plane which do not meet. By extension, a line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel. However, two lines in three-dimensional space which do not meet must be in a common plane to be considered parallel. Parallel planes are planes in the same three-dimensional space. Parallel lines are the subject of Euclid's parallel postulate. Parallelism is a property of affine geometries and Euclidean geometry is a special instance of this type of geometry. In some other geometries, such as hyperbolic geometry, lines can have analogous properties that are referred to as parallelism; the parallel symbol is ∥. For example, A B ∥ C D indicates that line AB is parallel to line CD. In the Unicode character set, the "parallel" and "not parallel" signs have codepoints U+2225 and U+2226, respectively. In addition, U+22D5 represents the relation "equal and parallel to". Given parallel straight lines l and m in Euclidean space, the following properties are equivalent: Every point on line m is located at the same distance from line l.

Line m is in the same plane as line l but does not intersect l. When lines m and l are both intersected by a third straight line in the same plane, the corresponding angles of intersection with the transversal are congruent. Since these are equivalent properties, any one of them could be taken as the definition of parallel lines in Euclidean space, but the first and third properties involve measurement, so, are "more complicated" than the second. Thus, the second property is the one chosen as the defining property of parallel lines in Euclidean geometry; the other properties are consequences of Euclid's Parallel Postulate. Another property that involves measurement is that lines parallel to each other have the same gradient; the definition of parallel lines as a pair of straight lines in a plane which do not meet appears as Definition 23 in Book I of Euclid's Elements. Alternative definitions were discussed by other Greeks as part of an attempt to prove the parallel postulate. Proclus attributes a definition of parallel lines as equidistant lines to Posidonius and quotes Geminus in a similar vein.

Simplicius mentions Posidonius' definition as well as its modification by the philosopher Aganis. At the end of the nineteenth century, in England, Euclid's Elements was still the standard textbook in secondary schools; the traditional treatment of geometry was being pressured to change by the new developments in projective geometry and non-Euclidean geometry, so several new textbooks for the teaching of geometry were written at this time. A major difference between these reform texts, both between themselves and between them and Euclid, is the treatment of parallel lines; these reform texts were not without their critics and one of them, Charles Dodgson, wrote a play and His Modern Rivals, in which these texts are lambasted. One of the early reform textbooks was James Maurice Wilson's Elementary Geometry of 1868. Wilson based his definition of parallel lines on the primitive notion of direction. According to Wilhelm Killing the idea may be traced back to Leibniz. Wilson, without defining direction since it is a primitive, uses the term in other definitions such as his sixth definition, "Two straight lines that meet one another have different directions, the difference of their directions is the angle between them."

Wilson In definition 15 he introduces parallel lines in this way. Wilson Augustus De Morgan reviewed this text and declared it a failure on the basis of this definition and the way Wilson used it to prove things about parallel lines. Dodgson devotes a large section of his play to denouncing Wilson's treatment of parallels. Wilson edited this concept out of the third and higher editions of his text. Other properties, proposed by other reformers, used as replacements for the definition of parallel lines, did not fare much better; the main difficulty, as pointed out by Dodgson, was that to use them in this way required additional axioms to be added to the system. The equidistant line definition of Posidonius, expounded by Francis Cuthbertson in his 1874 text Euclidean Geometry suffers from the problem that the points that are found at a fixed given distance on one side of a straight line must be shown to form a straight line; this must be assumed to be true. The corresponding angles formed by a transversal property, used by W. D. Cooley in his 1860 text, The Elements of Geometry and explained requires a proof of the fact that if one transversal meets a pair of lines in congruent corresponding angles all transversals must do so.

Again, a new axiom is needed to justify this statement. The three properties above lead to three different methods of construction of parallel lines; because parallel lines in a Euclidean plane are equidistant there is a unique distance between the two parallel lines. Given the equations of two non-vertical, non-horizontal parallel lines, y = m x + b 1 y = m x + b 2

Projective geometry

Projective geometry is a topic in mathematics. It is the study of geometric properties that are invariant with respect to projective transformations; this means that, compared to elementary geometry, projective geometry has a different setting, projective space, a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, that geometric transformations are permitted that transform the extra points to Euclidean points, vice versa. Properties meaningful for projective geometry are respected by this new idea of transformation, more radical in its effects than can be expressed by a transformation matrix and translations; the first issue for geometers is. It is not possible to refer to angles in projective geometry as it is in Euclidean geometry, because angle is an example of a concept not invariant with respect to projective transformations, as is seen in perspective drawing. One source for projective geometry was indeed the theory of perspective.

Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, once the concept is translated into projective geometry's terms. Again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See projective plane for the basics of projective geometry in two dimensions. While the ideas were available earlier, projective geometry was a development of the 19th century; this included the theory of complex projective space, the coordinates used being complex numbers. Several major types of more abstract mathematics were based on projective geometry, it was a subject with a large number of practitioners for its own sake, as synthetic geometry. Another topic that developed from axiomatic studies of projective geometry is finite geometry; the topic of projective geometry is itself now divided into many research subtopics, two examples of which are projective algebraic geometry and projective differential geometry.

Projective geometry is an elementary non-metrical form of geometry, meaning that it is not based on a concept of distance. In two dimensions it begins with the study of configurations of lines; that there is indeed some geometric interest in this sparse setting was first established by Desargues and others in their exploration of the principles of perspective art. In higher dimensional spaces there are considered hyperplanes, other linear subspaces, which exhibit the principle of duality; the simplest illustration of duality is in the projective plane, where the statements "two distinct points determine a unique line" and "two distinct lines determine a unique point" show the same structure as propositions. Projective geometry can be seen as a geometry of constructions with a straight-edge alone. Since projective geometry excludes compass constructions, there are no circles, no angles, no measurements, no parallels, no concept of intermediacy, it was realised. For example, the different conic sections are all equivalent in projective geometry, some theorems about circles can be considered as special cases of these general theorems.

During the early 19th century the work of Jean-Victor Poncelet, Lazare Carnot and others established projective geometry as an independent field of mathematics. Its rigorous foundations were addressed by Karl von Staudt and perfected by Italians Giuseppe Peano, Mario Pieri, Alessandro Padoa and Gino Fano during the late 19th century. Projective geometry, like affine and Euclidean geometry, can be developed from the Erlangen program of Felix Klein. After much work on the large number of theorems in the subject, the basics of projective geometry became understood; the incidence structure and the cross-ratio are fundamental invariants under projective transformations. Projective geometry can be modeled by the affine plane plus a line "at infinity" and treating that line as "ordinary". An algebraic model for doing projective geometry in the style of analytic geometry is given by homogeneous coordinates. On the other hand, axiomatic studies revealed the existence of non-Desarguesian planes, examples to show that the axioms of incidence can be modelled by structures not accessible to reasoning through homogeneous coordinate systems.

In a foundational sense, projective geometry and ordered geometry are elementary since they involve a minimum of axioms and either can be used as the foundation for affine and Euclidean geometry. Projective geometry is not ``; the first geometrical properties of a projective nature were discovered during the 3rd century by Pappus of Alexandria. Filippo Brunelleschi started investigating the geometry of perspective during 1425. Johannes Kepler and Gérard Desargues independently developed the concept of the "point at infinity". Desarg

Emil Artin

Emil Artin was an Austrian mathematician of Armenian descent. Artin was one of the leading mathematicians of the twentieth century, he is best known for his work on algebraic number theory, contributing to class field theory and a new construction of L-functions. He contributed to the pure theories of rings and fields. Emil Artin was born in Vienna to parents Emma Maria, née Laura, a soubrette on the operetta stages of Austria and Germany, Emil Hadochadus Maria Artin, Austrian-born of mixed Austrian and Armenian descent. Several documents, including Emil's birth certificate, list the father's occupation as “opera singer” though others list it as “art dealer.” It seems at least plausible. They were married in St. Stephen's Parish on July 24, 1895. Artin entered school in September 1904 in Vienna. By his father was suffering symptoms of advanced syphilis, among them increasing mental instability, was institutionalized at the established insane asylum at Mauer Öhling, 125 kilometers west of Vienna.

It is notable that neither wife nor child contracted this infectious disease. Artin's father died there July 20, 1906. Young Artin was eight. On July 15, 1907, Artin's mother remarried to a man named Rudolf Hübner: a prosperous manufacturing entrepreneur in the German-speaking city called Reichenberg, Bohemia. Documentary evidence suggests that Emma had been a resident in Reichenberg the previous year, in deference to her new husband, she had abandoned her vocal career. Hübner deemed a life in the theater unseemly unfit for the wife of a man of his position. In September, 1907, Artin entered the Volksschule in Strobnitz, a small town in southern Czechoslovakia near the Austrian border. For that year, he lived away from home; the following year, he returned to the home of his mother and stepfather, entered the Realschule in Reichenberg, where he pursued his secondary education until June, 1916. In Reichenberg, Artin formed a lifelong friendship with a young neighbor, Arthur Baer, who became an astronomer, teaching for many years at Cambridge University.

Astronomy was an interest the two boys shared at this time. They each had telescopes, they rigged a telegraph between their houses, over which once Baer excitedly reported to his friend an astronomical discovery he thought he had made—perhaps a supernova, he thought—and told Artin where in the sky to look. Artin tapped back the terse reply “A-N-D-R-O-M-E-D-A N-E-B-E-L.” Artin's academic performance in the first years at the Realschule was spotty. Up to the end of the 1911–1912 school year, for instance, his grade in mathematics was “genügend,”. Of his mathematical inclinations at this early period he wrote, “Meine eigene Vorliebe zur Mathematik zeigte sich erst im sechzehnten Lebensjahr, während vorher von irgendeiner Anlage dazu überhaupt nicht die Rede sein konnte.” His grade in French for 1912 was “nicht genügend”. He did rather better work in chemistry, but from 1910 to 1912, his grade for “Comportment” was “nicht genügend.” Artin spent the school year 1912–1913 away from home, in France, a period he spoke of as one of the happiest of his life.

He lived that year with the family of Edmond Fritz, in the vicinity of Paris, attended a school there. When he returned from France to Reichenberg, his academic work markedly improved, he began receiving grades of “gut” or “sehr gut” in all subjects—including French and “Comportment.” By the time he completed studies at the Realschule in June, 1916, he was awarded the Reifezeugnis that affirmed him “reif mit Auszeichnung” for graduation to a technical university. Now that it was time to move on to university studies, Artin was no doubt content but to leave Reichenberg, for relations with his stepfather were clouded. According to him, Hübner reproached him “day and night” with being a financial burden, when Artin became a university lecturer and a professor, Hübner deprecated his academic career as self-indulgent and belittled its paltry emolument. In October, 1916, Artin matriculated at the University of Vienna, having focused by now on mathematics, he studied there with Philipp Furtwängler, took courses in astrophysics and Latin.

Studies at Vienna were interrupted when Artin was drafted in 1918 into the Austrian army. Assigned to the K.u. K. 44th Infantry Regiment, he was stationed northwest of Venice at Primolano, on the Italian front in the foothills of the Dolomites. To his great relief, Artin managed to avoid combat by volunteering for service as a translator—his ignorance of Italian notwithstanding, he did know French, of course, some Latin, was a quick study, was motivated by a rational fear in a theater of that war that had proven a meat-grinder. In his scramble to learn at least some Italian, Artin had recourse to an encyclopedia, which he once consulted for help in dealing with the cockroaches that infested the Austrian barracks. At some length, the article described a variety of technical methods, concluding with—Artin laughingly recalled in years—“la caccia diretta". Indeed, “la caccia diretta” was the straightforward method he and his fe