In topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids. Other surfaces arise as graphs of functions of two variables. However, surfaces can be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space. Topological surfaces are sometimes equipped with additional information, such as a Riemannian metric or a complex structure, that connects them to other disciplines within mathematics, such as differential geometry and complex analysis; the various mathematical notions of surface can be used to model surfaces in the physical world. In mathematics, a surface is a geometrical shape; the most familiar examples arise as boundaries of solid objects in ordinary three-dimensional Euclidean space R3, such as spheres. The exact definition of a surface may depend on the context. In algebraic geometry, a surface may cross itself, while, in topology and differential geometry, it may not.
A surface is a two-dimensional space. In other words, around every point, there is a coordinate patch on which a two-dimensional coordinate system is defined. For example, the surface of the Earth resembles a two-dimensional sphere, latitude and longitude provide two-dimensional coordinates on it; the concept of surface is used in physics, computer graphics, many other disciplines in representing the surfaces of physical objects. For example, in analyzing the aerodynamic properties of an airplane, the central consideration is the flow of air along its surface. A surface is a topological space in which every point has an open neighbourhood homeomorphic to some open subset of the Euclidean plane E2; such a neighborhood, together with the corresponding homeomorphism, is known as a chart. It is through this chart that the neighborhood inherits the standard coordinates on the Euclidean plane; these coordinates are known as local coordinates and these homeomorphisms lead us to describe surfaces as being locally Euclidean.
In most writings on the subject, it is assumed, explicitly or implicitly, that as a topological space a surface is nonempty, second countable, Hausdorff. It is often assumed that the surfaces under consideration are connected; the rest of this article will assume, unless specified otherwise, that a surface is nonempty, second countable, connected. More a surface with boundary is a Hausdorff topological space in which every point has an open neighbourhood homeomorphic to some open subset of the closure of the upper half-plane H2 in C; these homeomorphisms are known as charts. The boundary of the upper half-plane is the x-axis. A point on the surface mapped via a chart to the x-axis is termed a boundary point; the collection of such points is known as the boundary of the surface, a one-manifold, that is, the union of closed curves. On the other hand, a point mapped to above the x-axis is an interior point; the collection of interior points is the interior of the surface, always non-empty. The closed disk is a simple example of a surface with boundary.
The boundary of the disc is a circle. The term surface used without qualification refers to surfaces without boundary. In particular, a surface with empty boundary is a surface in the usual sense. A surface with empty boundary, compact is known as a'closed' surface; the two-dimensional sphere, the two-dimensional torus, the real projective plane are examples of closed surfaces. The Möbius strip is a surface on which the distinction between clockwise and counterclockwise can be defined locally, but not globally. In general, a surface is said to be orientable if it does not contain a homeomorphic copy of the Möbius strip. For example, the sphere and torus are orientable. In differential and algebraic geometry, extra structure is added upon the topology of the surface; this added structures can be a smoothness structure, a Riemannian metric, a complex structure, or an algebraic structure. Surfaces were defined as subspaces of Euclidean spaces; these surfaces were the locus of zeros of certain functions polynomial functions.
Such a definition considered the surface as part of a larger space, as such was termed extrinsic. In the previous section, a surface is defined as a topological space with certain properties, namely Hausdorff and locally Euclidean; this topological space is not considered a subspace of another space. In this sense, the definition given above, the definition that mathematicians use at present, is intrinsic. A surface defined as intrinsic is not required to satisfy the added constraint of being a subspace of Euclidean space, it may seem possible for some surfaces defined intrinsically to not be surfaces in the
In set theory, a Cartesian product is a mathematical operation that returns a set from multiple sets. That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs where a ∈ A and b ∈ B. Products can be specified using e.g.. A × B =. A table can be created by taking the Cartesian product of a set of columns. If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form. More a Cartesian product of n sets known as an n-fold Cartesian product, can be represented by an array of n dimensions, where each element is an n-tuple. An ordered pair is a couple; the Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, further generalized in terms of direct product. An illustrative example is the standard 52-card deck; the standard playing card ranks form a 13-element set. The card suits form a four-element set; the Cartesian product of these sets returns a 52-element set consisting of 52 ordered pairs, which correspond to all 52 possible playing cards.
Ranks × Suits returns a set of the form. Suits × Ranks returns a set of the form. Both sets are distinct disjoint; the main historical example is the Cartesian plane in analytic geometry. In order to represent geometrical shapes in a numerical way and extract numerical information from shapes' numerical representations, René Descartes assigned to each point in the plane a pair of real numbers, called its coordinates; such a pair's first and second components are called its x and y coordinates, respectively. The set of all such pairs is thus assigned to the set of all points in the plane. A formal definition of the Cartesian product from set-theoretical principles follows from a definition of ordered pair; the most common definition of ordered pairs, the Kuratowski definition, is =. Under this definition, is an element of P, X × Y is a subset of that set, where P represents the power set operator. Therefore, the existence of the Cartesian product of any two sets in ZFC follows from the axioms of pairing, power set, specification.
Since functions are defined as a special case of relations, relations are defined as subsets of the Cartesian product, the definition of the two-set Cartesian product is prior to most other definitions. Let A, B, C, D be sets; the Cartesian product A × B is not commutative, A × B ≠ B × A, because the ordered pairs are reversed unless at least one of the following conditions is satisfied: A is equal to B, or A or B is the empty set. For example: A =. × C ≠ A × If for example A = × A = ≠ = A ×. The Cartesian product behaves nicely with respect to intersections. × = ∩. × ≠ ∪ In fact, we have that: ∪ = ∪ ∪ [ ( B
A dihedral angle is the angle between two intersecting planes. In chemistry it is the angle between planes through two sets of three atoms, having two atoms in common. In solid geometry it is defined as the union of a line and two half-planes that have this line as a common edge. In higher dimension, a dihedral angle represents the angle between two hyperplanes; when the two intersecting planes are described in terms of Cartesian coordinates by the two equations a 1 x + b 1 y + c 1 z + d 1 = 0 a 2 x + b 2 y + c 2 z + d 2 = 0 the dihedral angle, φ between them is given by: cos φ = | a 1 a 2 + b 1 b 2 + c 1 c 2 | a 1 2 + b 1 2 + c 1 2 a 2 2 + b 2 2 + c 2 2. An alternative method is to calculate the angle between the vectors, nA and nB, which are normal to the planes. Cos φ = | n A ⋅ n B | | n A | | n B | where nA · nB is the dot product of the vectors and |nA| |nB| is the product of their lengths. Any plane can be described by two non-collinear vectors lying in that plane. Thus, a dihedral angle can be defined by three vectors, b1, b2 and b3, forming two pairs of non-collinear vectors.
Φ = atan2 . In chemistry, a torsion angle is defined as a particular example of a dihedral angle, describing the geometric relation of two parts of a molecule joined by a chemical bond; every set of three not-colinear atoms of a molecule defines a plane. When two such planes intersect, the angle between them is a dihedral angle. Dihedral angles are used to specify the molecular conformation. Stereochemical arrangements corresponding to angles between 0° and ±90° are called syn, those corresponding to angles between ±90° and 180° anti. Arrangements corresponding to angles between 30° and 150° or between −30° and −150° are called clinal and those between 0° and ±30° or ±150° and 180° are called periplanar; the two types of terms can be combined so as to define four ranges of angle. The synperiplanar conformation is known as the syn- or cis-conformation. For example, with n-butane two planes can be specified in terms of the two central carbon atoms and either of the methyl carbon atoms; the syn-conformation shown above, with a dihedral angle of 60° is less stable than the anti-conformation with a dihedral angle of 180°.
For macromolecular usage the symbols T, C, G+, G−, A+ and A− are recommended. A Ramachandran plot developed in 1963 by G. N. Ramachandran, C. Ramakrishnan, V. Sasisekharan, is a way to visualize energetically allowed regions for backbone dihedral angles ψ against φ of amino acid residues in protein st
In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, braid groups, it can be regarded as a part of geometric topology. It may be used to refer to the study of topological spaces of dimension 1, though this is more considered part of continuum theory. A number of advances starting in the 1960s had the effect of emphasising low dimensions in topology; the solution by Smale, in 1961, of the Poincaré conjecture in higher dimensions made dimensions three and four seem the hardest. Thurston's geometrization conjecture, formulated in the late 1970s, offered a framework that suggested geometry and topology were intertwined in low dimensions, Thurston's proof of geometrization for Haken manifolds utilized a variety of tools from only weakly linked areas of mathematics. Vaughan Jones' discovery of the Jones polynomial in the early 1980s not only led knot theory in new directions but gave rise to still mysterious connections between low-dimensional topology and mathematical physics.
In 2002 Grigori Perelman announced a proof of the three-dimensional Poincaré conjecture, using Richard S. Hamilton's Ricci flow, an idea belonging to the field of geometric analysis. Overall, this progress has led to better integration of the field into the rest of mathematics. A surface is a topological manifold; the most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 – for example, the surface of a ball. On the other hand, there are surfaces, such as the Klein bottle, that cannot be embedded in three-dimensional Euclidean space without introducing singularities or self-intersections; the classification theorem of closed surfaces states that any connected closed surface is homeomorphic to some member of one of these three families: the sphere. The surfaces in the first two families are orientable, it is convenient to combine the two families by regarding the sphere as the connected sum of 0 tori. The number g of tori involved is called the genus of the surface.
The sphere and the torus have Euler characteristics 2 and 0 and in general the Euler characteristic of the connected sum of g tori is 2 − 2g. The surfaces in the third family are nonorientable; the Euler characteristic of the real projective plane is 1, in general the Euler characteristic of the connected sum of k of them is 2 − k. In mathematics, the Teichmüller space TX of a topological surface X, is a space that parameterizes complex structures on X up to the action of homeomorphisms that are isotopic to the identity homeomorphism; each point in TX may be regarded as an isomorphism class of'marked' Riemann surfaces where a'marking' is an isotopy class of homeomorphisms from X to X. The Teichmüller space is the universal covering orbifold of the moduli space. Teichmüller space has a wealth of natural metrics; the underlying topological space of Teichmüller space was studied by Fricke, the Teichmüller metric on it was introduced by Oswald Teichmüller. In mathematics, the uniformization theorem says that every connected Riemann surface is conformally equivalent to one of the three domains: the open unit disk, the complex plane, or the Riemann sphere.
In particular it admits a Riemannian metric of constant curvature. This classifies Riemannian surfaces as elliptic and hyperbolic according to their universal cover; the uniformization theorem is a generalization of the Riemann mapping theorem from proper connected open subsets of the plane to arbitrary connected Riemann surfaces. A topological space X is a 3-manifold if every point in X has a neighbourhood, homeomorphic to Euclidean 3-space; the topological, piecewise-linear, smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds. Phenomena in three dimensions can be strikingly different from phenomena in other dimensions, so there is a prevalence of specialized techniques that do not generalize to dimensions greater than three; this special role has led to the discovery of close connections to a diversity of other fields, such as knot theory, geometric group theory, hyperbolic geometry, number theory, Teichmüller theory, topological quantum field theory, gauge theory, Floer homology, partial differential equations.
3-manifold theory is considered a part of low-dimensional topology or geometric topology. Knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R3 upon itself.
In geometry, a surface S is ruled if through every point of S there is a straight line that lies on S. Examples include the plane, the curved surface of a cylinder or cone, a conical surface with elliptical directrix, the right conoid, the helicoid, the tangent developable of a smooth curve in space. A ruled surface can be described as the set of points swept by a moving straight line. For example, a cone is formed by keeping one point of a line fixed whilst moving another point along a circle. A surface is doubly ruled if through every one of its points there are two distinct lines that lie on the surface; the hyperbolic paraboloid and the hyperboloid of one sheet are doubly. The plane is the only surface which contains at least three distinct lines through each of its points; the properties of being ruled or doubly ruled are preserved by projective maps, therefore are concepts of projective geometry. In algebraic geometry ruled surfaces are sometimes considered to be surfaces in affine or projective space over a field, but they are sometimes considered as abstract algebraic surfaces without an embedding into affine or projective space, in which case "straight line" is understood to mean an affine or projective line.
DefinitionA two dimensional differentiable manifold is called ruled surface, if it is the union of a one parametric family of lines. The lines of this family are the generators of the ruled surface. Parametric representationA ruled surface can be described by a parametric representation of the form x = c + v r, v ∈ R. Any curve v ↦ x with fixed parameter u = u 0 is a generator and the curve u ↦ c is the directrix of the representation; the vectors r ≠ 0 describe the directions of the generators. The directrix may collapse to a point. Alternatively the ruled surface can be described by x = c + v d with the second directrix d = c + r. Alternatively, one can start with two non intersecting curves c, d as directrices, get by a ruled surface with line directions r = d − c. For the generation of a ruled surface by two directrices not only the geometric shape of these curves are essential but the special parametric representations of them influence the shape of the ruled surface, d)). For theoretical investigations representation is more advantageous, because the parameter v appears only once.
A) Right circular cylinder x 2 + y 2 = a 2: x = T = T + v T = T + v T. with c = ( a cos
In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a line and three-dimensional space. Planes can arise as subspaces of some higher-dimensional space, as with a room's walls extended infinitely far, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry; when working in two-dimensional Euclidean space, the definite article is used, so, the plane refers to the whole space. Many fundamental tasks in mathematics, trigonometry, graph theory, graphing are performed in a two-dimensional space, or, in other words, in the plane. Euclid set forth the first great landmark of mathematical thought, an axiomatic treatment of geometry, he selected a small core of undefined terms and postulates which he used to prove various geometrical statements. Although the plane in its modern sense is not directly given a definition anywhere in the Elements, it may be thought of as part of the common notions.
Euclid never used numbers to measure angle, or area. In this way the Euclidean plane is not quite the same as the Cartesian plane. A plane is a ruled surface; this section is concerned with planes embedded in three dimensions: in R3. In a Euclidean space of any number of dimensions, a plane is uniquely determined by any of the following: Three non-collinear points. A line and a point not on that line. Two distinct but intersecting lines. Two parallel lines; the following statements hold in three-dimensional Euclidean space but not in higher dimensions, though they have higher-dimensional analogues: Two distinct planes are either parallel or they intersect in a line. A line intersects it at a single point, or is contained in the plane. Two distinct lines perpendicular to the same plane must be parallel to each other. Two distinct planes perpendicular to the same line must be parallel to each other. In a manner analogous to the way lines in a two-dimensional space are described using a point-slope form for their equations, planes in a three dimensional space have a natural description using a point in the plane and a vector orthogonal to it to indicate its "inclination".
Let r0 be the position vector of some point P0 =, let n = be a nonzero vector. The plane determined by the point P0 and the vector n consists of those points P, with position vector r, such that the vector drawn from P0 to P is perpendicular to n. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the desired plane can be described as the set of all points r such that n ⋅ = 0. Expanded this becomes a + b + c = 0, the point-normal form of the equation of a plane; this is just a linear equation a x + b y + c z + d = 0, where d = −. Conversely, it is shown that if a, b, c and d are constants and a, b, c are not all zero the graph of the equation a x + b y + c z + d = 0, is a plane having the vector n = as a normal; this familiar equation for a plane is called the general form of the equation of the plane. Thus for example a regression equation of the form y = d + ax + cz establishes a best-fit plane in three-dimensional space when there are two explanatory variables.
Alternatively, a plane may be described parametrically as the set of all points of the form r = r 0 + s v + t w, where s and t range over all real numbers, v and w are given linearly independent vectors defining the plane, r0 is the vector representing the position of an arbitrary point on the plane. The vectors v and w can be visualized as vectors starting at r0 and pointing in different directions along the plane. Note that v and w can be perpendicular, but cannot be parallel. Let p1=, p2=, p3= be non-collinear points; the plane passing through p1, p2, p3 can be described as the set of all points that satisfy the following determinant equations: | x − x 1 y − y 1 z − z 1 x 2 − x 1 y 2 − y
Euclid, sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician referred to as the "founder of geometry" or the "father of geometry". He was active in Alexandria during the reign of Ptolemy I, his Elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics from the time of its publication until the late 19th or early 20th century. In the Elements, Euclid deduced the theorems of what is now called Euclidean geometry from a small set of axioms. Euclid wrote works on perspective, conic sections, spherical geometry, number theory, mathematical rigour; the English name Euclid is the anglicized version of the Greek name Εὐκλείδης, which means "renowned, glorious". Few original references to Euclid survive, so little is known about his life, he was born c. 325 BC, although the place and circumstances of both his birth and death are unknown and may only be estimated relative to other people mentioned with him.
He is mentioned by name by other Greek mathematicians from Archimedes onward, is referred to as "ὁ στοιχειώτης". The few historical references to Euclid were written by Proclus c. 450 AD, centuries after Euclid lived. A detailed biography of Euclid is given by Arabian authors, for example, a birth town of Tyre; this biography is believed to be fictitious. If he came from Alexandria, he would have known the Serapeum of Alexandria, the Library of Alexandria, may have worked there during his time. Euclid's arrival in Alexandria came about ten years after its founding by Alexander the Great, which means he arrived c. 322 BC. Proclus introduces Euclid only in his Commentary on the Elements. According to Proclus, Euclid belonged to Plato's "persuasion" and brought together the Elements, drawing on prior work of Eudoxus of Cnidus and of several pupils of Plato Proclus believes that Euclid is not much younger than these, that he must have lived during the time of Ptolemy I because he was mentioned by Archimedes.
Although the apparent citation of Euclid by Archimedes has been judged to be an interpolation by editors of his works, it is still believed that Euclid wrote his works before Archimedes wrote his. Proclus retells a story that, when Ptolemy I asked if there was a shorter path to learning geometry than Euclid's Elements, "Euclid replied there is no royal road to geometry." This anecdote is questionable since it is similar to a story told about Menaechmus and Alexander the Great. Euclid died c. 270 BC in Alexandria. In the only other key reference to Euclid, Pappus of Alexandria mentioned that Apollonius "spent a long time with the pupils of Euclid at Alexandria, it was thus that he acquired such a scientific habit of thought" c. 247–222 BC. Because the lack of biographical information is unusual for the period, some researchers have proposed that Euclid was not a historical personage, that his works were written by a team of mathematicians who took the name Euclid from Euclid of Megara. However, this hypothesis is not well accepted by scholars and there is little evidence in its favor.
Although many of the results in Elements originated with earlier mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework, making it easy to use and easy to reference, including a system of rigorous mathematical proofs that remains the basis of mathematics 23 centuries later. There is no mention of Euclid in the earliest remaining copies of the Elements, most of the copies say they are "from the edition of Theon" or the "lectures of Theon", while the text considered to be primary, held by the Vatican, mentions no author; the only reference that historians rely on of Euclid having written the Elements was from Proclus, who in his Commentary on the Elements ascribes Euclid as its author. Although best known for its geometric results, the Elements includes number theory, it considers the connection between perfect numbers and Mersenne primes, the infinitude of prime numbers, Euclid's lemma on factorization, the Euclidean algorithm for finding the greatest common divisor of two numbers.
The geometrical system described in the Elements was long known as geometry, was considered to be the only geometry possible. Today, that system is referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century; the Papyrus Oxyrhynchus 29 is a fragment of the second book of the Elements of Euclid, unearthed by Grenfell and Hunt 1897 in Oxyrhynchus. More recent scholarship suggests a date of 75–125 AD; the classic translation of T. L. Heath, reads: If a straight line be cut into equal and unequal segments, the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section is equal to the square on the half. In addition to the Elements, at least five works of Euclid have survived to the present day, they follow the same logical structure with definitions and proved propositions. Data deals with the nature and implications of "given" information in geometrical problems.