Curve
In mathematics, a curve is speaking, an object similar to a line but that need not be straight. Thus, a curve is a generalization of a line. Various disciplines within mathematics have given the term different meanings depending on the area of study, so the precise meaning depends on context. However, many of these meanings are special instances of the definition. A curve is a topological space, locally homeomorphic to a line. In everyday language, this means that a curve is a set of points which, near each of its points, looks like a line, up to a deformation. A simple example of a curve is the parabola, shown to the right. A large number of other curves have been studied in multiple mathematical fields. A closed curve is a curve that forms a path whose starting point is its ending point—that is, a path from any of its points to the same point. Related meanings include the graph of a function and a two-dimensional graph. Interest in curves began; this can be seen in numerous examples of their decorative use in art and on everyday objects dating back to prehistoric times.
Curves, or at least their graphical representations, are simple to create, for example by a stick in the sand on a beach. The term line was used in place of the more modern term curve. Hence the phrases straight line and right line were used to distinguish what are today called lines from curved lines. For example, in Book I of Euclid's Elements, a line is defined as a "breadthless length", while a straight line is defined as "a line that lies evenly with the points on itself". Euclid's idea of a line is clarified by the statement "The extremities of a line are points,". Commentators further classified lines according to various schemes. For example: Composite lines Incomposite lines Determinate Indeterminate The Greek geometers had studied many other kinds of curves. One reason was their interest in solving geometrical problems that could not be solved using standard compass and straightedge construction; these curves include: The conic sections studied by Apollonius of Perga The cissoid of Diocles, studied by Diocles and used as a method to double the cube.
The conchoid of Nicomedes, studied by Nicomedes as a method to both double the cube and to trisect an angle. The Archimedean spiral, studied by Archimedes as a method to trisect an angle and square the circle; the spiric sections, sections of tori studied by Perseus as sections of cones had been studied by Apollonius. A fundamental advance in the theory of curves was the advent of analytic geometry in the seventeenth century; this enabled a curve to be described using an equation rather than an elaborate geometrical construction. This not only allowed new curves to be defined and studied, but it enabled a formal distinction to be made between curves that can be defined using algebraic equations, algebraic curves, those that cannot, transcendental curves. Curves had been described as "geometrical" or "mechanical" according to how they were, or could be, generated. Conic sections were applied in astronomy by Kepler. Newton worked on an early example in the calculus of variations. Solutions to variational problems, such as the brachistochrone and tautochrone questions, introduced properties of curves in new ways.
The catenary gets its name as the solution to the problem of a hanging chain, the sort of question that became accessible by means of differential calculus. In the eighteenth century came the beginnings of the theory of plane algebraic curves, in general. Newton had studied the cubic curves, in the general description of the real points into'ovals'; the statement of Bézout's theorem showed a number of aspects which were not directly accessible to the geometry of the time, to do with singular points and complex solutions. Since the nineteenth century there has not been a separate theory of curves, but rather the appearance of curves as the one-dimensional aspect of projective geometry, differential geometry; the era of the space-filling curves provoked the modern definitions of curve. In general, a curve is defined through a continuous function γ: I → X from an interval I of the real numbers into a topological space X. Depending on the context, it is either γ or its image γ, called a curve. In general topology, when non-differentiable functions are considered, it is the map γ, called a curve, because its image may look differently from what is called a curve.
For example, the image of the Peano curve fills the square. On the other hand, when one considers curves defined by a differentiable function, this is the image of the function, called a curve; the curve is said to be simple, or a Jordan arc, if γ is injective, i.e. if for all x, y in I, we have γ = γ
Synonym
A synonym is a word or phrase that means or nearly the same as another lexeme in the same language. Words that are synonyms are said to be synonymous, the state of being a synonym is called synonymy. For example, the words begin, start and initiate are all synonyms of one another. Words are synonymous in one particular sense: for example and extended in the context long time or extended time are synonymous, but long cannot be used in the phrase extended family. Synonyms with the exact same meaning share a seme or denotational sememe, whereas those with inexactly similar meanings share a broader denotational or connotational sememe and thus overlap within a semantic field; the former are sometimes called cognitive synonyms and the latter, near-synonyms, plesionyms or poecilonyms. Some lexicographers claim that no synonyms have the same meaning because etymology, phonic qualities, ambiguous meanings, so on make them unique. Different words that are similar in meaning differ for a reason: feline is more formal than cat.
Synonyms are a source of euphemisms. Metonymy can sometimes be a form of synonymy: the White House is used as a synonym of the administration in referring to the U. S. executive branch under a specific president. Thus a metonym is a type of synonym, the word metonym is a hyponym of the word synonym; the analysis of synonymy, polysemy and hypernymy is inherent to taxonomy and ontology in the information-science senses of those terms. It has applications in pedagogy and machine learning, because they rely on word-sense disambiguation; the word comes from ónoma. Synonyms can be any part of speech. Examples: noun drink and beverage verb buy and purchase adjective big and large adverb and speedily preposition on and upon"glass" and"cup"Synonyms are defined with respect to certain senses of words: pupil as the aperture in the iris of the eye is not synonymous with student; such like, he expired means the same as he died, yet my passport has expired cannot be replaced by my passport has died. In English, many synonyms emerged after the Norman conquest of England.
While England's new ruling class spoke Norman French, the lower classes continued to speak Old English. Thus, today we have synonyms like the Norman-derived people and archer, the Saxon-derived folk and bowman. For more examples, see the list of Germanic and Lat Latinate equivalents in English. A thesaurus lists related words; the word poecilonym is a rare synonym of the word synonym. It is not entered in most major dictionaries and is a curiosity or piece of trivia for being an autological word because of its meta quality as a synonym of synonym. Antonyms are words with nearly opposite meanings. For example: hot ↔ cold, large ↔ small, thick ↔ thin, synonym ↔ antonym Hypernyms and hyponyms are words that refer to a general category and a specific instance of that category. For example, vehicle is a hypernym of car, car is a hyponym of vehicle. Homophones are words that have different meanings. For example and which are homophones in most accents. Homographs are words that have different pronunciations.
For example, one can keep a record of documents. Homonyms are words that have different meanings. For example and rose are homonyms. -onym Cognitive synonymy Elegant variation, the gratuitous use of a synonym in prose Synonym ring Synonomy in Japanese Tools which graph words relations: Graph Words – Online tool for visualization word relations Synonyms.net – Online reference resource that provides instant synonyms and antonyms definitions including visualizations, voice pronunciations and translations English/French Semantic Atlas – Graph words relations in English and gives cross representations for translations – offers 500 searches per user per day. Plain words synonyms finder: Synonym Finder – Synonym finder including hypernyms in search result Thesaurus – Online synonyms in English, Italian and German Woxikon Synonyms – Over 1 million synonyms – English, Spanish, Italian, Portuguese and Dutch FindMeWords Synonyms – Online Synonym Dictionary with definitions Classic Thesaurus - Crowdsourced Synonym Dictionary Power Thesaurus - Synonym dictionary with definitions and examples
Speed skating rink
A speed skating rink is an ice rink in which a speed skating competition is held. A standard long track speed skating track is, according to the regulations of the International Skating Union, a double-laned track with two curved ends each of 180°, in which the radius of the inner curve is not less than 25 metres and not more than 26 metres; the width of the competition lanes is 4 metres. At the opposite straight of the finishing line, there is a crossing area, where the skaters must change lane. - Rule 203 At international competitions, the track must be 400 metres long, with a warm-up lane at least 4 metres wide inside the competition lanes. For Olympic competitions, the track must be enclosed within a building; the design and dimensions of a speed skating track have remained more or less unchanged since the foundation of ISU in 1892. The speed skating track is used for the sports of Icetrack cycling and Ice speedway The measurement of the track is made half a meter into the lane; the total length of the track is the distance a competitor skates each lap, i.e. the length of two straights, one inner curve and one outer curve, in addition to the extra distance skated when changing lanes in the cross-over area, which on a standard track equals 7 centimeters.
A 400 m track with inner radius 25.0 m has 113.57 m long straights A 400 m track with inner radius 25.5 m has 112.00 m long straights A 400 m track with inner radius 26.0 m has 110.43 m long straightsThe demarcation of the competition lanes are made by painted lines in the ice and movable blocks of rubber. On outdoor tracks, snow may be used for demarcation of the competition lanes. Although ISU regulations state that minimum measures for a standard speed skating track, alternative track lengths may be used for competition; the minimum requirements are track length on 200 meters, radius of inner curve of 15 meters and width of the competition lanes 2 meters. Short track speed skating tracks have a length of 111.111 metres. The rink is 60 metres long by 30 metres wide, the same size as an international-sized ice hockey rink. Many speed skating venues have no ice area at all inside the oval. A few are suitable for bandy, like Hamar Olympic Hall, Ice Palace Krylatskoye, Medeu; the National Speed Skating Oval in Beijing, in the process of being built for the 2022 Winter Olympics, is designed appropriately for that sport.
There is a growing cooperation between International Skating Union and Federation of International Bandy, since both have an interest in more indoor venues with large ice surfaces being built. In Norway there is an agreement in place, stating that an indoor arena intended for either bandy or long track speed skating, shall have ice surface for the other sport as well. Below is a complete list of the indoor 400 m speed skating tracks around the world; the data presented are retrieved from the online database Speed Skating News. Note: The Richmond Olympic Oval was dismantled upon completion of the 2010 Winter Olympics and is no longer used for speed skating. However, if the need arises the speed skating rink can be reinstalled. In the table below, some of the world's major outdoor speed skating tracks still in use are listed; this is not a complete list of speed skating venues, but lists most of the outdoor tracks used for world cup competitions and championships the past years. The data in the table are retrieved from the Speed Skating News database.
Long track speed skating Figure skating rink Ice hockey rink Ice rink
Plane (geometry)
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
Egg as food
Some eggs are laid by female animals of many different species, including birds, amphibians and fish, have been eaten by humans for thousands of years. Bird and reptile eggs consist of a protective eggshell and vitellus, contained within various thin membranes; the most consumed eggs are chicken eggs. Other poultry eggs including those of duck and quail are eaten. Fish eggs are called caviar. Egg yolks and whole eggs store significant amounts of protein and choline, are used in cookery. Due to their protein content, the United States Department of Agriculture categorized eggs as Meats within the Food Guide Pyramid. Despite the nutritional value of eggs, there are some potential health issues arising from cholesterol content, salmonella contamination, allergy to egg proteins. Chickens and other egg-laying creatures are kept throughout the world and mass production of chicken eggs is a global industry. In 2009, an estimated 62.1 million metric tons of eggs were produced worldwide from a total laying flock of 6.4 billion hens.
There are issues of regional variation in demand and expectation, as well as current debates concerning methods of mass production. In 2012, the European Union banned battery husbandry of chickens. Bird eggs have been valuable foodstuffs since prehistory, in both hunting societies and more recent cultures where birds were domesticated; the chicken was domesticated for its eggs before 7500 BCE. Chickens were brought to Sumer and Egypt by 1500 BCE, arrived in Greece around 800 BCE, where the quail had been the primary source of eggs. In Thebes, the tomb of Haremhab, dating to 1420 BCE, shows a depiction of a man carrying bowls of ostrich eggs and other large eggs those of the pelican, as offerings. In ancient Rome, eggs were preserved using a number of methods and meals started with an egg course; the Romans crushed the shells in their plates to prevent evil spirits from hiding there. In the Middle Ages, eggs were forbidden during Lent because of their richness; the word mayonnaise was derived from moyeu, the medieval French word for the yolk, meaning center or hub.
Egg scrambled. The dried egg industry developed in the nineteenth century, before the rise of the frozen egg industry. In 1878, a company in St. Louis, Missouri started to transform egg yolk and egg white into a light-brown, meal-like substance by using a drying process; the production of dried eggs expanded during World War II, for use by the United States Armed Forces and its allies. In 1911, the egg carton was invented by Joseph Coyle in Smithers, British Columbia, to solve a dispute about broken eggs between a farmer in Bulkley Valley and the owner of the Aldermere Hotel. Early egg cartons were made of paper. Bird eggs are a common one of the most versatile ingredients used in cooking, they are important in many branches of the modern food industry. The most used bird eggs are those from the chicken and goose eggs. Smaller eggs, such as quail eggs, are used as a gourmet ingredient in Western countries. Eggs are a common everyday food in many parts of Asia, such as China and Thailand, with Asian production providing 59 percent of the world total in 2013.
The largest bird eggs, from ostriches, tend to be used only as special luxury food. Gull eggs are considered a delicacy in England, as well as in some Scandinavian countries in Norway. In some African countries, guineafowl eggs are seen in marketplaces in the spring of each year. Pheasant eggs and emu eggs are edible, but less available, sometimes they are obtainable from farmers, poulterers, or luxury grocery stores. In many countries, wild bird eggs are protected by laws which prohibit the collecting or selling of them, or permit collection only during specific periods of the year. In 2013, world production of chicken eggs was 68.3 million tonnes. The largest four producers were China at 24.8 million of this total, the United States at 5.6 million, India at 3.8 million, Japan at 2.5 million. A typical large egg factory ships a million dozen eggs per week. For the month of January 2019, the United States produced 9.41 billion eggs, with 8.2 billion for table consumption and 1.2 billion for raising chicks.
Americans are projected to each consume 279 eggs in 2019, the highest since 1973, but less than the 405 eggs eaten per person in 1945. During production, eggs are candled to check their quality; the size of its air cell is determined, the examination reveals whether the egg was fertilized and thereby contains an embryo. Depending on local regulations, eggs may be washed before being placed in egg boxes, although washing may shorten their length of freshness; the shape of an egg resembles a prolate spheroid with one end larger than the other and has cylindrical symmetry along the long axis. An egg is surrounded by a hard shell. Thin membranes exist inside the shell; the egg yolk is suspended in the egg white by two spiral bands of tissue called the chalazae. The larger end of the egg contains an air cell that forms when the contents of the egg cool down and contract after it is laid. Chicken eggs are graded according to the size of this air cell, measured during candling. A fresh egg has a small air cell and receives a grade of AA.
As the size of the air cell increases and the quality of the egg decreases, the grade moves from AA to A to B. This provides a way of t
Egg
The egg is the organic vessel containing the zygote in which an embryo develops until it can survive on its own. An egg results from fertilization of an egg cell. Most arthropods and mollusks lay eggs, although some, such as scorpions do not. Reptile eggs, bird eggs, monotreme eggs are laid out of water, are surrounded by a protective shell, either flexible or inflexible. Eggs laid on land or in nests are kept within a warm and favorable temperature range while the embryo grows; when the embryo is adequately developed it hatches, i.e. breaks out of the egg's shell. Some embryos have a temporary egg tooth they use to pip, or break the eggshell or covering; the largest recorded egg is from a whale shark, was 30 cm × 14 cm × 9 cm in size. Whale shark eggs hatch within the mother. At 1.5 kg and up to 17.8 cm × 14 cm, the ostrich egg is the largest egg of any living bird, though the extinct elephant bird and some dinosaurs laid larger eggs. The bee hummingbird produces the smallest known bird egg; some eggs laid by reptiles and most fish, amphibians and other invertebrates can be smaller.
Reproductive structures similar to the egg in other kingdoms are termed "spores," or in spermatophytes "seeds," or in gametophytes "egg cells". Several major groups of animals have distinguishable eggs; the most common reproductive strategy for fish is known as oviparity, in which the female lays undeveloped eggs that are externally fertilized by a male. Large numbers of eggs are laid at one time and the eggs are left to develop without parental care; when the larvae hatch from the egg, they carry the remains of the yolk in a yolk sac which continues to nourish the larvae for a few days as they learn how to swim. Once the yolk is consumed, there is a critical point after which they must learn how to hunt and feed or they will die. A few fish, notably the rays and most sharks use ovoviviparity in which the eggs are fertilized and develop internally; however the larvae still grow inside the egg consuming the egg's yolk and without any direct nourishment from the mother. The mother gives birth to mature young.
In certain instances, the physically most developed offspring will devour its smaller siblings for further nutrition while still within the mother's body. This is known as intrauterine cannibalism. In certain scenarios, some fish such as the hammerhead shark and reef shark are viviparous, with the egg being fertilized and developed internally, but with the mother providing direct nourishment; the eggs of fish and amphibians are jellylike. Cartilagenous fish eggs are fertilized internally and exhibit a wide variety of both internal and external embryonic development. Most fish species spawn eggs that are fertilized externally with the male inseminating the eggs after the female lays them; these eggs would dry out in the air. Air-breathing amphibians lay their eggs in water, or in protective foam as with the Coast foam-nest treefrog, Chiromantis xerampelina. Bird eggs are incubated for a time that varies according to the species. Average clutch sizes range from one to about 17; some birds lay eggs when not fertilized.
The default color of vertebrate eggs is the white of the calcium carbonate from which the shells are made, but some birds passerines, produce colored eggs. The pigment biliverdin and its zinc chelate give a green or blue ground color, protoporphyrin produces reds and browns as a ground color or as spotting. Non-passerines have white eggs, except in some ground-nesting groups such as the Charadriiformes and nightjars, where camouflage is necessary, some parasitic cuckoos which have to match the passerine host's egg. Most passerines, in contrast, lay colored eggs if there is no need of cryptic colors; however some have suggested that the protoporphyrin markings on passerine eggs act to reduce brittleness by acting as a solid state lubricant. If there is insufficient calcium available in the local soil, the egg shell may be thin in a circle around the broad end. Protoporphyrin speckling compensates for this, increases inversely to the amount of calcium in the soil. For the same reason eggs in a clutch are more spotted than early ones as the female's store of calcium is depleted.
The color of individual eggs is genetically influenced, appears to be inherited through the mother only, suggesting that the gene responsible for pigmentation is on the sex determining W chromosome. It used to be thought that color was applied to the shell before laying, but this research shows that coloration is an integral part of the development of the shell, with the same protein responsible for depositing calcium carbonate, or protoporphyrins when there is a lack of that mineral. In species such as the common guillemot, which nest in large groups, each female's eggs have different markings, making it easier for females to identify their own eggs on the crowded cliff ledges on which they breed. Bird eggshells are diverse. For example: cormorant eggs are rough and chalky tinamou eggs are shiny duck eggs are oily and waterproof cassowary eggs are pittedTiny pores in bird eggshells allow the embryo to breathe; the domestic
Vesica piscis
The vesica piscis is a type of lens, a mathematical shape formed by the intersection of two disks with the same radius, intersecting in such a way that the center of each disk lies on the perimeter of the other. In Latin, "vesica piscis" means "bladder of a fish", reflecting the shape's resemblance to the conjoined dual air bladders found in most fish. In Italian, the shape's name is mandorla; this figure appears in the first proposition of Euclid's Elements, where it forms the first step in constructing an equilateral triangle using a compass and straightedge. The triangle has as its vertices the two disk centers and one of the two sharp corners of the vesica piscis. Mathematically, the vesica piscis is a special case of a lens, the shape formed by the intersection of two disks; the mathematical ratio of the height of the vesica piscis to the width across its center is the square root of 3, or 1.7320508.... The ratios 265:153 = 1.7320261... and 1351:780 = 1.7320513... are two of a series of approximations to this value, each with the property that no better approximation can be obtained with smaller whole numbers.
Archimedes of Syracuse, in his On the Measurement of the Circle, uses these ratios as upper and lower bounds: 1351 780 > 3 > 265 153. The area of the vesica piscis is formed by two equilateral triangles and four equal circular segments. In the drawing one triangle and one segment appear in blue. One triangle and one segment form a sector of one sixth of the circle; the area of the sector is then: 1 6 π r 2. Since the side of the equilateral triangle has length r, its area is 3 4 r 2; the area of the segment is the difference between those two areas: 1 6 π r 2 − 3 4 r 2 By summing the areas of two triangles and four segments, we obtain the area of the vesica piscis: 1 6 r 2 ≈ 1.2284 r 2 The two circles of the vesica piscis, or three circles forming in pairs three vesicae, are used in Venn diagrams. Arcs of the same three circles can be used to form the triquetra symbol, the Reuleaux triangle. In Christian art, some aureolas are in the shape of a vertically oriented vesica piscis, the seals of ecclesiastical organizations can be enclosed within a vertically oriented vesica piscis.
The icthys symbol incorporates the vesica piscis shape. Ecclesiastical heraldry of the Catholic Church appeared first in nearly all vesica-shaped; the cover of the Chalice Well in Glastonbury depicts a stylized version of the vesica piscis design. The vesica piscis has been used as a symbol within Freemasonry, most notably in the shapes of the collars worn by officiants of the Masonic rituals, it was considered the proper shape for the enclosure of the seals of Masonic lodges. The vesica piscis is used as proportioning system in architecture, in particular Gothic architecture; the system was illustrated in Cesare Cesariano's Vitruvius, which he called "the rule of the German architects". The vesica piscis is a leitmotif of architect Carlo Scarpa and is used as a “viewing device” in Tomba Brion in San Vito d'Altivole, Italy. Flower of Life, a figure based upon this principle Villarceau circles, a pair of congruent circles derived from a torus that, are not centered on each other's perimeter Weisstein, Eric W. "Vesica Piscis".
MathWorld
