Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a geometry, not Euclidean. Two practical applications of the principles of spherical geometry are astronomy. In plane geometry, the basic concepts are lines. On a sphere, points are defined in the usual sense; the equivalents of lines are not defined in the usual sense of "straight line" in Euclidean geometry, but in the sense of "the shortest paths between points", which are called geodesics. On a sphere, the geodesics are the great circles. Thus, in spherical geometry, angles are defined between great circles, resulting in a spherical trigonometry that differs from ordinary trigonometry in many respects. Spherical geometry is rather a subset of elliptic geometry. For example, it shares with that geometry the property that a line has no parallels through a given point. Contrast this with Euclidean geometry, in which a line has one parallel through a given point, hyperbolic geometry, in which a line has two parallels and an infinite number of ultraparallels through a given point.
An important geometry related to that of the sphere is that of the real projective plane. Locally, the projective plane has all the properties of spherical geometry, but it has different global properties. In particular, it one-sided. Concepts of spherical geometry may be applied to the oblong sphere, though minor modifications must be implemented on certain formulas. Higher-dimensional spherical geometries exist; the earliest mathematical work of antiquity to come down to our time is On the rotating sphere by Autolycus of Pitane, who lived at the end of the fourth century BC. Spherical trigonometry was studied by early Greek mathematicians such as Theodosius of Bithynia, a Greek astronomer and mathematician who wrote the Sphaerics, a book on the geometry of the sphere, Menelaus of Alexandria, who wrote a book on spherical trigonometry called Sphaerica and developed Menelaus' theorem; the book of unknown arcs of a sphere written by the Islamic mathematician Al-Jayyani is considered to be the first treatise on spherical trigonometry.
The book contains formulae for right-handed triangles, the general law of sines, the solution of a spherical triangle by means of the polar triangle. The book On Triangles by Regiomontanus, written around 1463, is the first pure trigonometrical work in Europe. However, Gerolamo Cardano noted a century that much of its material on spherical trigonometry was taken from the twelfth-century work of the Andalusi scholar Jabir ibn Aflah. Euler published a series of important memoirs on spherical geometry: L. Euler, Principes de la trigonométrie sphérique tirés de la méthode des plus grands et des plus petits, Mémoires de l'Académie des Sciences de Berlin 9, 1755, p. 233–257. XXVII, p. 277–308. L. Euler, Eléments de la trigonométrie sphéroïdique tirés de la méthode des plus grands et des plus petits, Mémoires de l'Académie des Sciences de Berlin 9, 1755, p. 258–293. XXVII, p. 309–339. L. Euler, De curva rectificabili in superficie sphaerica, Novi Commentarii academiae scientiarum Petropolitanae 15, 1771, pp. 195–216.
L. Euler, De mensura angulorum solidorum, Acta academiae scientarum imperialis Petropolitinae 2, 1781, p. 31–54. XXVI, p. 204–223. L. Euler, Problematis cuiusdam Pappi Alexandrini constructio, Acta academiae scientarum imperialis Petropolitinae 4, 1783, p. 91–96. XXVI, p. 237–242. L. Euler, Geometrica et sphaerica quaedam, Mémoires de l'Académie des Sciences de Saint-Pétersbourg 5, 1815, p. 96–114. XXVI, p. 344–358. L. Euler, Trigonometria sphaerica universa, ex primis principiis breviter et dilucide derivata, Acta academiae scientarum imperialis Petropolitinae 3, 1782, p. 72–86. XXVI, p. 224–236. L. Euler, Variae speculationes super area triangulorum sphaericorum, Nova Acta academiae scientarum imperialis Petropolitinae 10, 1797, p. 47–62. XXIX, p. 253–266. With points defined as the points on a sphere and lines as the great circles of that sphere, a spherical geometry has the following properties: Any two lines intersect in two diametrically opposite points, called antipodal points. Any two points that are not antipodal points determine a unique line.
There is a natural unit of length and a natural unit of area. Each line is associated with a pair of antipodal points, called the poles of the line, which are the common intersections of the set of lines perpendicular to the given line; each point is associated with a unique line, called the polar line of the point, the line on the plane through the centre of the sphere and perpendicular to the diameter of the sphere through the given point. As there are two arcs determined by a pair of points, which are not antipodal, on the line they determine, three non-collinear points do not determine a unique triangle. However, if we only consider triangles whose sides are minor arcs of great circles, we have the following properties: The angle sum of a triangle is greater than 180° and less than 270°; the area of a triangle is proportional to the excess of its angle sum over 180
In geometry, a polyhedron is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from as poly - + - hedron. A convex polyhedron is the convex hull of finitely many points on the same plane. Cubes and pyramids are examples of convex polyhedra. A polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions. Convex polyhedra are well-defined, with several equivalent standard definitions. However, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of "polyhedron" have been given within particular contexts, some more rigorous than others, there is not universal agreement over which of these to choose; some of these definitions exclude shapes that have been counted as polyhedra or include shapes that are not considered as valid polyhedra. As Branko Grünbaum observed, "The Original Sin in the theory of polyhedra goes back to Euclid, through Kepler, Poinsot and many others... at each stage... the writers failed to define what are the polyhedra".
There is general agreement that a polyhedron is a solid or surface that can be described by its vertices, edges and sometimes by its three-dimensional interior volume. One can distinguish among these different definitions according to whether they describe the polyhedron as a solid, whether they describe it as a surface, or whether they describe it more abstractly based on its incidence geometry. A common and somewhat naive definition of a polyhedron is that it is a solid whose boundary can be covered by finitely many planes or that it is a solid formed as the union of finitely many convex polyhedra. Natural refinements of this definition require the solid to be bounded, to have a connected interior, also to have a connected boundary; the faces of such a polyhedron can be defined as the connected components of the parts of the boundary within each of the planes that cover it, the edges and vertices as the line segments and points where the faces meet. However, the polyhedra defined in this way do not include the self-crossing star polyhedra, their faces may not form simple polygons, some edges may belong to more than two faces.
Definitions based on the idea of a bounding surface rather than a solid are common. For instance, O'Rourke defines a polyhedron as a union of convex polygons, arranged in space so that the intersection of any two polygons is a shared vertex or edge or the empty set and so that their union is a manifold. If a planar part of such a surface is not itself a convex polygon, O'Rourke requires it to be subdivided into smaller convex polygons, with flat dihedral angles between them. Somewhat more Grünbaum defines an acoptic polyhedron to be a collection of simple polygons that form an embedded manifold, with each vertex incident to at least three edges and each two faces intersecting only in shared vertices and edges of each. Cromwell gives a similar definition but without the restriction of three edges per vertex. Again, this type of definition does not encompass the self-crossing polyhedra. Similar notions form the basis of topological definitions of polyhedra, as subdivisions of a topological manifold into topological disks whose pairwise intersections are required to be points, topological arcs, or the empty set.
However, there exist topological polyhedra. One modern approach is based on the theory of abstract polyhedra; these can be defined as ordered sets whose elements are the vertices and faces of a polyhedron. A vertex or edge element is less than an edge or face element when the vertex or edge is part of the edge or face. Additionally, one may include a special bottom element of this partial order and a top element representing the whole polyhedron. If the sections of the partial order between elements three levels apart have the same structure as the abstract representation of a polygon these ordered sets carry the same information as a topological polyhedron. However, these requirements are relaxed, to instead require only that sections between elements two levels apart have the same structure as the abstract representation of a line segment. Geometric polyhedra, defined in other ways, can be described abstractly in this way, but it is possible to use abstract polyhedra as the basis of a definition of geometric polyhedra.
A realization of an abstract polyhedron is taken to be a mapping from the vertices of the abstract polyhedron to geometric points, such that the points of each face are coplanar. A geometric polyhedron can be defined as a realization of an abstract polyhedron. Realizations that forgo the requirement of planarity, that impose additional requirements of symmetry, or that map the vertices to higher dimensional spaces have been considered. Unlike the solid-based and surface-based definitions, this works well for star polyhedra. However, without additional restrictions, this definition allows degenerate or unfaithful polyhedra (for instance, by mapp
The tilde is a grapheme with several uses. The name of the character came into English from Spanish and from Portuguese, which in turn came from the Latin titulus, meaning "title" or "superscription"; the reason for the name was that it was written over a letter as a scribal abbreviation, as a "mark of suspension", shown as a straight line when used with capitals. Thus the used words Anno Domini were abbreviated to Ao Dñi, an elevated terminal with a suspension mark placed over the "n"; such a mark could denote the omission of several letters. This saved on the cost of vellum and ink. Medieval European charters written in Latin are made up of such abbreviated words with suspension marks and other abbreviations; the tilde has since been applied to a number of other uses as a diacritic mark or a character in its own right. These are encoded in Unicode at U+0303 ◌̃ COMBINING TILDE and U+007E ~ TILDE, there are additional similar characters for different roles. In lexicography, the latter kind of tilde and the swung dash are used in dictionaries to indicate the omission of the entry word.
This symbol informally means "approximately", "about", or "around", such as "~30 minutes before", meaning "approximately 30 minutes before". It can mean "similar to", including "of the same order of magnitude as", such as: "x ~ y" meaning that x and y are of the same order of magnitude. Another approximation symbol is the double-tilde ≈, meaning "approximately equal to", the critical difference being the subjective level of accuracy: ≈ indicates a value which can be considered functionally equivalent for a calculation within an acceptable degree of error, whereas ~ is used to indicate a larger significant, degree of error; the tilde is used to indicate "equal to" or "approximately equal to" by placing it over the "=" symbol, like so: ≅. In the computing field in Unix based systems, the tilde indicates the user's home directory; the text of the Domesday Book of 1086, relating for example, to the manor of Molland in Devon, is abbreviated as indicated by numerous tildes. The text with abbreviations expanded is as follows: Mollande tempore regis Edwardi geldabat pro quattuor hidis et uno ferling.
Terra est quadraginta carucae. In dominio sunt tres carucae et decem servi et triginta villani et viginta bordarii cum sedecim carucis. Ibi duodecim acrae prati et quindecim acrae silvae. Pastura tres leugae in longitudine et latitudine. Libras ad pensam. Huic manerio est adjuncta Blachepole. Elwardus tenebat tempore regis Edwardi pro manerio et geldabat pro dimidia hida. Terra est duae carucae. Ibi sunt quinque villani cum uno servo. Valet viginti solidos arsuram. Eidem manerio est injuste adjuncta Nimete et valet quindecim solidos. Ipsi manerio pertinet tercius denarius de Hundredis Nortmoltone et Badentone et Brantone et tercium animal pasturae morarum; the incorporation of the tilde into ASCII is a direct result of its appearance as a distinct character on mechanical typewriters in the late nineteenth century. When all character sets were pieces of metal permanently installed, number of characters much more limited than in typography, the question of which languages and markets required which characters was an important one.
Any good typewriter store had a catalog of alternative keyboards that could be specified for machines ordered from the factory. At that time, the tilde was used only in Portuguese typewriters. In Modern Spanish, the tilde is used only with ñ and Ñ. Both were conveniently assigned to a single mechanical typebar, which sacrificed a key, felt to be less important the ½—¼ key. Portuguese, uses not ñ but nh, it uses the tilde on the vowels a and o. So as not to sacrifice two of the limited keys to ã Ã õ Õ, the decision was made to make the ~ a separate "dead" character in which the carriage holding the paper did not move. Dead keys, which had a notch cut out to avoid hitting a mechanical linkage that triggered carriage movement, were used for characters that were intended to be combined. On mechanical typewriters, Spanish keyboards had a dead key, which contained the acute accent, used over any vowel, the dieresis, used only over u, it was a simple matter to create a dead key for a Portuguese keyboard to be overstruck with a and o and so the ~ was born as a typographical character, which did not exist as a type or hot-lead printing character.
That was a product of the first and leading manufacturer of typewriters, Remington. As indicated by the etymological origin of the word "tilde" in English, this symbol has been associated with the Spanish language; the connection stems from the use of the tilde above the letter "n" to form "ñ" in Spanish, a feature shared by only a few other languages, all connected to Spanish. This peculiarity can help non-native speakers identify a text as being written in Spanish with little chance of error. In addition, most native speakers, although not all, use the word "español" to refer to their language. During the 1990s, Spanish-speaking intellectuals and news outlets demonstrated support for the language and the culture by defending this letter against globalisation and computerisation trends that threatened to remove it from keyboards and other standardised products and codes; the Instituto Cervantes, founded by Spain's government to promote the Spanish language internationally, chose as its logo a stylised Ñ with a large tilde.
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A mirror image is a reflected duplication of an object that appears identical, but is reversed in the direction perpendicular to the mirror surface. As an optical effect it results from reflection off of substances such as water, it is a concept in geometry and can be used as a conceptualization process for 3-D structures. In geometry, the mirror image of an object or two-dimensional figure is the virtual image formed by reflection in a plane mirror. Two-dimensional mirror images can be seen in the reflections of mirrors or other reflecting surfaces, or on a printed surface seen inside-out. If we look at an object, two-dimensional and turn it towards a mirror, the object turns through an angle of 180º and we see a left-right reversal in the mirror. In this example, it is the change in orientation rather than the mirror itself that causes the observed reversal. Another example is when we stand with our backs to the mirror and face an object that's in front of the mirror. We compare the object with its reflection by turning ourselves 180º, towards the mirror.
Again we perceive a left-right reversal due to a change in orientation. So, in these examples the mirror does not cause the observed reversals; the concept of reflection can be extended to three-dimensional objects, including the inside parts if they are not transparent. The term relates to structural as well as visual aspects. A three-dimensional object is reversed in the direction perpendicular to the mirror surface. In physics, mirror images are investigated in the subject called geometrical optics. In chemistry, two versions of a molecule, one a "mirror image" of the other, are called enantiomers if they are not "superposable" on each other; that is an example of chirality. In general, an object and its mirror image are called enantiomorphs. If a point of an object has coordinates the image of this point has coordinates, thus reflection is a reversal of the coordinate axis perpendicular to the mirror's surface. Although a plane mirror reverses an object only in the direction normal to the mirror surface, there is a perception of a left-right reversal.
Hence, the reversal is called "lateral inversion". The perception of a left-right reversal is because the left and right of an object are defined by its perceived top and front, but there is still some debate about the explanation amongst psychologists; the psychology of the perceived left-right reversal is discussed in "Much ado about mirrors" by Professor Michael Corballis. Reflection in a mirror does result in a change in chirality, more from a right-handed to a left-handed coordinate system; as a consequence, if one looks in a mirror and lets two axes coincide with those in the mirror this gives a reversal of the third axis. If a person stands side-on to a mirror and right will be reversed directly by the mirror, because the person's left-right axis is normal to the mirror plane. However, it's important to understand that there are always only two enantiomorphs, the object and its image. Therefore, no matter how the object is oriented towards the mirror, all the resulting images are fundamentally identical.
In the picture of the mountain reflected in the lake, the reversal normal to the reflecting surface is obvious. Notice that there is no obvious front-back or left-right of the mountain. In the example of the urn and mirror, the urn is symmetrical front-back. Thus, no obvious reversal of any sort can be seen in the mirror image of the urn. A mirror image appears more three-dimensional if the observer moves, or if the image is viewed using binocular vision; this is because the relative position of objects changes as the observer's perspective changes, or is differently viewed with each eye. Looking through a mirror from different positions is like looking at the 3D mirror image of space. A mirror does not just produce an image of. A mirror hanging on the wall makes the room brighter because additional light sources appear in the mirror image. However, the appearance of additional light does not violate the conservation of energy principle, because some light no longer reaches behind the mirror, as the mirror re-directs the light energy.
In terms of the light distribution, the virtual mirror image has the same appearance and the same effect as a real, symmetrically arranged half-space behind a window. Shadows may extend from the mirror into the halfspace before it, vice versa. In mirror writing a text is deliberately displayed in mirror image, in order to be read through a mirror. For example, emergency vehicles such as ambulances or fire engines use mirror images in order to be read from a driver's rear-view mirror; some movie theaters take advantage of mirror writing in a Rear Window Captioning System used to assist individuals with heari
Euclidean plane isometry
In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length. There are four types: translations, rotations and glide reflections; the set of Euclidean plane isometries forms a group under composition: the Euclidean group in two dimensions. It is generated by reflections in lines, every element of the Euclidean group is the composite of at most three distinct reflections. Informally, a Euclidean plane isometry is any way of transforming the plane without "deforming" it. For example, suppose that the Euclidean plane is represented by a sheet of transparent plastic sitting on a desk. Examples of isometries include: Shifting the sheet one inch to the right. Rotating the sheet by ten degrees around some marked point. Turning the sheet over to look at it from behind. Notice that if a picture is drawn on one side of the sheet after turning the sheet over, we see the mirror image of the picture.
These are examples of translations and reflections respectively. There is one further type of isometry, called a glide reflection. However, cutting, or melting the sheet are not considered isometries. Neither are less drastic alterations like stretching, or twisting. An isometry of the Euclidean plane is a distance-preserving transformation of the plane; that is, it is a map M: R 2 → R 2 such that for any points p and q in the plane, d = d, where d is the usual Euclidean distance between p and q. It can be shown. Reflections, or mirror isometries, denoted by Fc,v, where c is a point in the plane and v is a unit vector in R2. have the effect of reflecting the point p in the line L, perpendicular to v and that passes through c. The line L is called the associated mirror. To find a formula for Fc,v, we first use the dot product to find the component t of p − c in the v direction, t = ⋅ v = v x + v y, we obtain the reflection of p by subtraction, F c, v = p − 2 t v; the combination of rotations about the origin and reflections about a line through the origin is obtained with all orthogonal matrices forming orthogonal group O.
In the case of a determinant of −1 we have: R 0, θ =. Which is a reflection in the x-axis followed by a rotation by an angle θ, or equivalently, a reflection in a line making an angle of θ/2 with the x-axis. Reflection in a parallel line corresponds to adding a vector perpendicular to it. Translations, denoted by Tv, where v is a vector in R2 have the effect of shifting the plane in the direction of v; that is, for any point p in the plane, T v = p + v, or in terms of coordinates, T v =. A translation can be seen as a composite of two parallel reflections. Rotations, denoted by Rc,θ, where c is a point in the plane, θ is the angle of rotation. In terms of coordinates, rotations are most expressed by breaking them up into two operations. First, a rotation around the origin is given by R 0, θ = ( cos θ − sin θ sin θ cos θ
In mathematics, the Pythagorean theorem known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides; the theorem can be written as an equation relating the lengths of the sides a, b and c called the "Pythagorean equation": a 2 + b 2 = c 2, where c represents the length of the hypotenuse and a and b the lengths of the triangle's other two sides. Although it is argued that knowledge of the theorem predates him, the theorem is named after the ancient Greek mathematician Pythagoras as it is he who, by tradition, is credited with its first proof, although no evidence of it exists. There is some evidence that Babylonian mathematicians understood the formula, although little of it indicates an application within a mathematical framework. Mesopotamian and Chinese mathematicians all discovered the theorem independently and, in some cases, provided proofs for special cases.
The theorem has been given numerous proofs – the most for any mathematical theorem. They are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years; the theorem can be generalized in various ways, including higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, indeed, to objects that are not triangles at all, but n-dimensional solids. The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power; the Pythagorean theorem was known long before Pythagoras, but he may well have been the first to prove it. In any event, the proof attributed to him is simple, is called a proof by rearrangement; the two large squares shown in the figure each contain four identical triangles, the only difference between the two large squares is that the triangles are arranged differently. Therefore, the white space within each of the two large squares must have equal area.
Equating the area of the white space yields the Pythagorean theorem, Q. E. D; that Pythagoras originated this simple proof is sometimes inferred from the writings of the Greek philosopher and mathematician Proclus. Several other proofs of this theorem are described below. If c denotes the length of the hypotenuse and a and b denote the lengths of the other two sides, the Pythagorean theorem can be expressed as the Pythagorean equation: a 2 + b 2 = c 2. If the length of both a and b are known c can be calculated as c = a 2 + b 2. If the length of the hypotenuse c and of one side are known the length of the other side can be calculated as a = c 2 − b 2 or b = c 2 − a 2; the Pythagorean equation relates the sides of a right triangle in a simple way, so that if the lengths of any two sides are known the length of the third side can be found. Another corollary of the theorem is that in any right triangle, the hypotenuse is greater than any one of the other sides, but less than their sum. A generalization of this theorem is the law of cosines, which allows the computation of the length of any side of any triangle, given the lengths of the other two sides and the angle between them.
If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation. This theorem may have more known proofs than any other; this proof is based on the proportionality of the sides of two similar triangles, that is, upon the fact that the ratio of any two corresponding sides of similar triangles is the same regardless of the size of the triangles. Let ABC represent a right triangle, with the right angle located at C. Draw the altitude from point C, call H its intersection with the side AB. Point H divides the length of the hypotenuse c into parts d and e; the new triangle ACH is similar to triangle ABC, because they both have a right angle, they share the angle at A, meaning that the third angle will be the same in both triangles as well, marked as θ in the figure. By a similar reasoning, the triangle CBH is similar to ABC; the proof of similarity of the triangles requires the triangle postulate: the sum of the angles in a triangle is two right angles, is equivalent to the parallel postulate.
Similarity of the triangles leads to the equality of ratios of corresponding sides: B C A B = B H B C and A C A B = A H A C. The first result equates
Corresponding sides and corresponding angles
In geometry, the tests for congruence and similarity involve comparing corresponding sides and corresponding angles of polygons. In these tests, each side and each angle in one polygon is paired with a side or angle in the second polygon, taking care to preserve the order of adjacency. For example, if one polygon has sequential sides a, b, c, d, e and the other has sequential sides v, w, x, y, z, if b and w are corresponding sides side a must correspond to either v or x. If a and v correspond to each other c corresponds to x, d corresponds to y, e corresponds to z. On the other hand, if in addition to b corresponding to w we have c corresponding to v the ith element of abcde corresponds to the ith element of the reverse sequence xwvzy. Congruence tests look for all pairs of corresponding sides to be equal in length, though except in the case of the triangle this is not sufficient to establish congruence. Similarity tests look at whether the ratios of the lengths of each pair of corresponding sides are equal, though again this is not sufficient.
In either case equality of corresponding angles is necessary. The corresponding angles as well as the corresponding sides are defined as appearing in the same sequence, so for example if in a polygon with the side sequence abcde and another with the corresponding side sequence vwxyz we have vertex angle A appearing between sides a and b its corresponding vertex angle V must appear between sides v and w