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57-413: In any geometry, the set of points on a line are said to be collinear . In Euclidean geometry this relation is intuitively visualized by points lying in a row on a "straight line". However, in most geometries (including Euclidean) a line is typically a primitive (undefined) object type , so such visualizations will not necessarily be appropriate. A model for the geometry offers an interpretation of how

114-467: A collinear (or co-linear) antenna array is an array of dipole antennas mounted in such a manner that the corresponding elements of each antenna are parallel and aligned, that is they are located along a common line or axis. The collinearity equations are a set of two equations, used in photogrammetry and computer stereo vision , to relate coordinates in an image ( sensor ) plane (in two dimensions) to object coordinates (in three dimensions). In

171-415: A rectangle with a width of 3 and a length of 4 has an area that represents the product, 12. Because this geometrical interpretation of multiplication was limited to three dimensions, there was no direct way of interpreting the product of four or more numbers, and Euclid avoided such products, although they are implied, for example in the proof of book IX, proposition 20. Euclid refers to a pair of lines, or

228-402: A 2x6 rectangle and a 3x4 rectangle are equal but not congruent, and the letter R is congruent to its mirror image. Figures that would be congruent except for their differing sizes are referred to as similar . Corresponding angles in a pair of similar shapes are equal and corresponding sides are in proportion to each other. Because of Euclidean geometry's fundamental status in mathematics, it

285-577: A colorful figure about whom many historical anecdotes are recorded, is remembered along with Euclid as one of the greatest of ancient mathematicians. Although the foundations of his work were put in place by Euclid, his work, unlike Euclid's, is believed to have been entirely original. He proved equations for the volumes and areas of various figures in two and three dimensions, and enunciated the Archimedean property of finite numbers. Apollonius of Perga ( c.  240 BCE  – c.  190 BCE )

342-480: A data set. More commonly, the issue of multicollinearity arises when there is a "strong linear relationship" among two or more independent variables, meaning that where the variance of ε i {\displaystyle \varepsilon _{i}} is relatively small. The concept of lateral collinearity expands on this traditional view, and refers to collinearity between explanatory and criteria (i.e., explained) variables. In telecommunications ,

399-407: A midpoint). Concurrent lines In geometry , lines in a plane or higher-dimensional space are concurrent if they intersect at a single point . The set of all lines through a point is called a pencil , and their common intersection is called the vertex of the pencil. In any affine space (including a Euclidean space ) the set of lines parallel to a given line (sharing

456-427: A pair of planar or solid figures, as "equal" (ἴσος) if their lengths, areas, or volumes are equal respectively, and similarly for angles. The stronger term " congruent " refers to the idea that an entire figure is the same size and shape as another figure. Alternatively, two figures are congruent if one can be moved on top of the other so that it matches up with it exactly. (Flipping it over is allowed.) Thus, for example,

513-404: A ray as an object that extends to infinity in one direction, would normally use locutions such as "if the line is extended to a sufficient length", although he occasionally referred to "infinite lines". A "line" for Euclid could be either straight or curved, and he used the more specific term "straight line" when necessary. The pons asinorum ( bridge of asses ) states that in isosceles triangles

570-469: A row . A mapping of a geometry to itself which sends lines to lines is called a collineation ; it preserves the collinearity property. The linear maps (or linear functions) of vector spaces , viewed as geometric maps, map lines to lines; that is, they map collinear point sets to collinear point sets and so, are collineations. In projective geometry these linear mappings are called homographies and are just one type of collineation. In any triangle

627-445: A statement such as "Find the greatest common measure of ..." Euclid often used proof by contradiction . Points are customarily named using capital letters of the alphabet. Other figures, such as lines, triangles, or circles, are named by listing a sufficient number of points to pick them out unambiguously from the relevant figure, e.g., triangle ABC would typically be a triangle with vertices at points A, B, and C. Angles whose sum

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684-447: A steep bridge that only a sure-footed donkey could cross. Triangles are congruent if they have all three sides equal (SSS), two sides and the angle between them equal (SAS), or two angles and a side equal (ASA) (Book I, propositions 4, 8, and 26). Triangles with three equal angles (AAA) are similar, but not necessarily congruent. Also, triangles with two equal sides and an adjacent angle are not necessarily equal or congruent. The sum of

741-483: A triangle are concurrent as well. For example: According to the Rouché–Capelli theorem , a system of equations is consistent if and only if the rank of the coefficient matrix is equal to the rank of the augmented matrix (the coefficient matrix augmented with a column of intercept terms), and the system has a unique solution if and only if that common rank equals the number of variables. Thus with two variables

798-453: Is constructive . Postulates 1, 2, 3, and 5 assert the existence and uniqueness of certain geometric figures, and these assertions are of a constructive nature: that is, we are not only told that certain things exist, but are also given methods for creating them with no more than a compass and an unmarked straightedge . In this sense, Euclidean geometry is more concrete than many modern axiomatic systems such as set theory , which often assert

855-403: Is a right angle are called complementary . Complementary angles are formed when a ray shares the same vertex and is pointed in a direction that is in between the two original rays that form the right angle. The number of rays in between the two original rays is infinite. Angles whose sum is a straight angle are supplementary . Supplementary angles are formed when a ray shares the same vertex and

912-418: Is an exact linear relationship between the two, so the correlation between them is equal to 1 or −1. That is, X 1 and X 2 are perfectly collinear if there exist parameters λ 0 {\displaystyle \lambda _{0}} and λ 1 {\displaystyle \lambda _{1}} such that, for all observations i , we have This means that if

969-438: Is impractical to give more than a representative sampling of applications here. As suggested by the etymology of the word, one of the earliest reasons for interest in and also one of the most common current uses of geometry is surveying . In addition it has been used in classical mechanics and the cognitive and computational approaches to visual perception of objects . Certain practical results from Euclidean geometry (such as

1026-466: Is in contrast to analytic geometry , introduced almost 2,000 years later by René Descartes , which uses coordinates to express geometric properties by means of algebraic formulas . The Elements is mainly a systematization of earlier knowledge of geometry. Its improvement over earlier treatments was rapidly recognized, with the result that there was little interest in preserving the earlier ones, and they are now nearly all lost. There are 13 books in

1083-512: Is mainly known for his investigation of conic sections. René Descartes (1596–1650) developed analytic geometry , an alternative method for formalizing geometry which focused on turning geometry into algebra. In this approach, a point on a plane is represented by its Cartesian ( x , y ) coordinates, a line is represented by its equation, and so on. In Euclid's original approach, the Pythagorean theorem follows from Euclid's axioms. In

1140-416: Is of rank 2 or less, the points are collinear. In particular, for three points in the plane ( n = 2 ), the above matrix is square and the points are collinear if and only if its determinant is zero; since that 3 × 3 determinant is plus or minus twice the area of a triangle with those three points as vertices, this is equivalent to the statement that the three points are collinear if and only if

1197-444: Is pointed in a direction that is in between the two original rays that form the straight angle (180 degree angle). The number of rays in between the two original rays is infinite. In modern terminology, angles would normally be measured in degrees or radians . Modern school textbooks often define separate figures called lines (infinite), rays (semi-infinite), and line segments (of finite length). Euclid, rather than discussing

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1254-485: Is proved that there are infinitely many prime numbers. Books XI–XIII concern solid geometry . A typical result is the 1:3 ratio between the volume of a cone and a cylinder with the same height and base. The platonic solids are constructed. Euclidean geometry is an axiomatic system , in which all theorems ("true statements") are derived from a small number of simple axioms. Until the advent of non-Euclidean geometry , these axioms were considered to be obviously true in

1311-403: Is that physical space itself is not Euclidean, and Euclidean space is a good approximation for it only over short distances (relative to the strength of the gravitational field ). Euclidean geometry is an example of synthetic geometry , in that it proceeds logically from axioms describing basic properties of geometric objects such as points and lines, to propositions about those objects. This

1368-499: Is the determination of packing arrangements , such as the problem of finding the most efficient packing of spheres in n dimensions. This problem has applications in error detection and correction . Geometry is used extensively in architecture . Geometry can be used to design origami . Some classical construction problems of geometry are impossible using compass and straightedge , but can be solved using origami . Archimedes ( c.  287 BCE  – c.  212 BCE ),

1425-432: Is the plane dual notion to collinearity. Given a partial geometry P , where two points determine at most one line, a collinearity graph of P is a graph whose vertices are the points of P , where two vertices are adjacent if and only if they determine a line in P . In statistics , collinearity refers to a linear relationship between two explanatory variables . Two variables are perfectly collinear if there

1482-715: The Elements : Books I–IV and VI discuss plane geometry. Many results about plane figures are proved, for example, "In any triangle, two angles taken together in any manner are less than two right angles." (Book I proposition 17) and the Pythagorean theorem "In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle." (Book I, proposition 47) Books V and VII–X deal with number theory , with numbers treated geometrically as lengths of line segments or areas of surface regions. Notions such as prime numbers and rational and irrational numbers are introduced. It

1539-448: The k lines in the plane, associated with a set of k equations, are concurrent if and only if the rank of the k × 2 coefficient matrix and the rank of the k × 3 augmented matrix are both 2. In that case only two of the k equations are independent , and the point of concurrency can be found by solving any two mutually independent equations simultaneously for the two variables. In projective geometry , in two dimensions concurrency

1596-550: The right angle as his basic unit, so that, for example, a 45- degree angle would be referred to as half of a right angle. The distance scale is relative; one arbitrarily picks a line segment with a certain nonzero length as the unit, and other distances are expressed in relation to it. Addition of distances is represented by a construction in which one line segment is copied onto the end of another line segment to extend its length, and similarly for subtraction. Measurements of area and volume are derived from distances. For example,

1653-409: The triangle inequality d ( AC ) ≤ d ( AB ) + d ( BC ) holds with equality. Two numbers m and n are not coprime —that is, they share a common factor other than 1—if and only if for a rectangle plotted on a square lattice with vertices at (0, 0), ( m , 0), ( m , n ), (0, n ) , at least one interior point is collinear with (0, 0) and ( m, n ) . In various plane geometries

1710-680: The Cartesian approach, the axioms are the axioms of algebra, and the equation expressing the Pythagorean theorem is then a definition of one of the terms in Euclid's axioms, which are now considered theorems. The equation defining the distance between two points P = ( p x , p y ) and Q = ( q x , q y ) is then known as the Euclidean metric , and other metrics define non-Euclidean geometries . In terms of analytic geometry,

1767-480: The adjective "Euclidean" was unnecessary because Euclid's axioms seemed so intuitively obvious (with the possible exception of the parallel postulate ) that theorems proved from them were deemed absolutely true, and thus no other sorts of geometry were possible. Today, however, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Albert Einstein 's theory of general relativity

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1824-406: The angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another . Its name may be attributed to its frequent role as the first real test in the Elements of the intelligence of the reader and as a bridge to the harder propositions that followed. It might also be so named because of the geometrical figure's resemblance to

1881-426: The angles of a triangle is equal to a straight angle (180 degrees). This causes an equilateral triangle to have three interior angles of 60 degrees. Also, it causes every triangle to have at least two acute angles and up to one obtuse or right angle . The celebrated Pythagorean theorem (book I, proposition 47) states that in any right triangle, the area of the square whose side is the hypotenuse (the side opposite

1938-407: The area of a circle and the volume of a parallelepipedal solid. Euclid determined some, but not all, of the relevant constants of proportionality. For instance, it was his successor Archimedes who proved that a sphere has 2/3 the volume of the circumscribing cylinder. Euclidean geometry has two fundamental types of measurements: angle and distance . The angle scale is absolute, and Euclid uses

1995-404: The cube and squaring the circle . In the case of doubling the cube, the impossibility of the construction originates from the fact that the compass and straightedge method involve equations whose order is an integral power of two, while doubling a cube requires the solution of a third-order equation. Euler discussed a generalization of Euclidean geometry called affine geometry , which retains

2052-730: The existence of objects without saying how to construct them, or even assert the existence of objects that cannot be constructed within the theory. Strictly speaking, the lines on paper are models of the objects defined within the formal system, rather than instances of those objects. For example, a Euclidean straight line has no width, but any real drawn line will have. Though nearly all modern mathematicians consider nonconstructive proofs just as sound as constructive ones, they are often considered less elegant , intuitive, or practically useful. Euclid's constructive proofs often supplanted fallacious nonconstructive ones, e.g. some Pythagorean proofs that assumed all numbers are rational, usually requiring

2109-446: The existence of the constructed objects, in his reasoning he also implicitly assumes them to be unique. The Elements also include the following five "common notions": Modern scholars agree that Euclid's postulates do not provide the complete logical foundation that Euclid required for his presentation. Modern treatments use more extensive and complete sets of axioms. To the ancients, the parallel postulate seemed less obvious than

2166-423: The fifth postulate unmodified while weakening postulates three and four in a way that eliminates the notions of angle (whence right triangles become meaningless) and of equality of length of line segments in general (whence circles become meaningless) while retaining the notions of parallelism as an equivalence relation between lines, and equality of length of parallel line segments (so line segments continue to have

2223-403: The following sets of points are collinear: In coordinate geometry , in n -dimensional space, a set of three or more distinct points are collinear if and only if, the matrix of the coordinates of these vectors is of rank 1 or less. For example, given three points if the matrix is of rank 1 or less, the points are collinear. Equivalently, for every subset of X, Y, Z , if the matrix

2280-439: The manner of Euclid Book III, Prop. 31. In modern terminology, the area of a plane figure is proportional to the square of any of its linear dimensions, A ∝ L 2 {\displaystyle A\propto L^{2}} , and the volume of a solid to the cube, V ∝ L 3 {\displaystyle V\propto L^{3}} . Euclid proved these results in various special cases such as

2337-407: The notion of interchanging the roles of "points" and "lines" while preserving the relationship between them is called plane duality . Given a set of collinear points, by plane duality we obtain a set of lines all of which meet at a common point. The property that this set of lines has (meeting at a common point) is called concurrency , and the lines are said to be concurrent lines . Thus, concurrency

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2394-487: The number of special cases is reduced. Geometers of the 18th century struggled to define the boundaries of the Euclidean system. Many tried in vain to prove the fifth postulate from the first four. By 1763, at least 28 different proofs had been published, but all were found incorrect. Leading up to this period, geometers also tried to determine what constructions could be accomplished in Euclidean geometry. For example,

2451-443: The optical centre. Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid , which he described in his textbook on geometry , Elements . Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions ( theorems ) from these. Although many of Euclid's results had been stated earlier, Euclid

2508-466: The others, as evidenced by the organization of the Elements : his first 28 propositions are those that can be proved without it. Many alternative axioms can be formulated which are logically equivalent to the parallel postulate (in the context of the other axioms). For example, Playfair's axiom states: The "at most" clause is all that is needed since it can be proved from the remaining axioms that at least one parallel line exists. Euclidean Geometry

2565-462: The others. They aspired to create a system of absolutely certain propositions, and to them, it seemed as if the parallel line postulate required proof from simpler statements. It is now known that such a proof is impossible since one can construct consistent systems of geometry (obeying the other axioms) in which the parallel postulate is true, and others in which it is false. Euclid himself seems to have considered it as being qualitatively different from

2622-412: The photography setting, the equations are derived by considering the central projection of a point of the object through the optical centre of the camera to the image in the image (sensor) plane. The three points, object point, image point and optical centre, are always collinear. Another way to say this is that the line segments joining the object points with their image points are all concurrent at

2679-410: The physical world, so that all the theorems would be equally true. However, Euclid's reasoning from assumptions to conclusions remains valid independently from the physical reality. Near the beginning of the first book of the Elements , Euclid gives five postulates (axioms) for plane geometry, stated in terms of constructions (as translated by Thomas Heath): Although Euclid explicitly only asserts

2736-428: The points, lines and other object types relate to one another and a notion such as collinearity must be interpreted within the context of that model. For instance, in spherical geometry , where lines are represented in the standard model by great circles of a sphere, sets of collinear points lie on the same great circle. Such points do not lie on a "straight line" in the Euclidean sense, and are not thought of as being in

2793-437: The problem of trisecting an angle with a compass and straightedge is one that naturally occurs within the theory, since the axioms refer to constructive operations that can be carried out with those tools. However, centuries of efforts failed to find a solution to this problem, until Pierre Wantzel published a proof in 1837 that such a construction was impossible. Other constructions that were proved impossible include doubling

2850-533: The restriction of classical geometry to compass and straightedge constructions means a restriction to first- and second-order equations, e.g., y = 2 x + 1 (a line), or x + y = 7 (a circle). Also in the 17th century, Girard Desargues , motivated by the theory of perspective , introduced the concept of idealized points, lines, and planes at infinity. The result can be considered as a type of generalized geometry, projective geometry , but it can also be used to produce proofs in ordinary Euclidean geometry in which

2907-413: The right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle). Thales' theorem , named after Thales of Miletus states that if A, B, and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle. Cantor supposed that Thales proved his theorem by means of Euclid Book I, Prop. 32 after

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2964-428: The right-angle property of the 3-4-5 triangle) were used long before they were proved formally. The fundamental types of measurements in Euclidean geometry are distances and angles, both of which can be measured directly by a surveyor. Historically, distances were often measured by chains, such as Gunter's chain , and angles using graduated circles and, later, the theodolite . An application of Euclidean solid geometry

3021-416: The same direction ) is also called a pencil , and the vertex of each pencil of parallel lines is a distinct point at infinity ; including these points results in a projective space in which every pair of lines has an intersection. In a triangle , four basic types of sets of concurrent lines are altitudes , angle bisectors , medians , and perpendicular bisectors : Other sets of lines associated with

3078-458: The square of the area of a triangle with side lengths d ( AB ), d ( BC ), d ( AC ) ; so checking if this determinant equals zero is equivalent to checking whether the triangle with vertices A, B, C has zero area (so the vertices are collinear). Equivalently, a set of at least three distinct points are collinear if and only if, for every three of those points A, B, C with d ( AC ) greater than or equal to each of d ( AB ) and d ( BC ) ,

3135-422: The triangle with those points as vertices has zero area. A set of at least three distinct points is called straight , meaning all the points are collinear, if and only if, for every three of those points A, B, C , the following determinant of a Cayley–Menger determinant is zero (with d ( AB ) meaning the distance between A and B , etc.): This determinant is, by Heron's formula , equal to −16 times

3192-437: The various observations ( X 1 i , X 2 i ) are plotted in the ( X 1 , X 2 ) plane, these points are collinear in the sense defined earlier in this article. Perfect multicollinearity refers to a situation in which k ( k ≥ 2) explanatory variables in a multiple regression model are perfectly linearly related, according to for all observations i . In practice, we rarely face perfect multicollinearity in

3249-547: Was the first to organize these propositions into a logical system in which each result is proved from axioms and previously proved theorems. The Elements begins with plane geometry , still taught in secondary school (high school) as the first axiomatic system and the first examples of mathematical proofs . It goes on to the solid geometry of three dimensions . Much of the Elements states results of what are now called algebra and number theory , explained in geometrical language. For more than two thousand years,

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