Analytic geometry

In classical mathematics, analytic geometry known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineering, in aviation, space science, spaceflight, it is the foundation of most modern fields of geometry, including algebraic, differential and computational geometry. The Cartesian coordinate system is applied to manipulate equations for planes, straight lines, squares in two and sometimes in three dimensions. Geometrically, one studies the Euclidean plane and Euclidean space; as taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometrical shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations. That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom.

The Greek mathematician Menaechmus solved problems and proved theorems by using a method that had a strong resemblance to the use of coordinates and it has sometimes been maintained that he had introduced analytic geometry. Apollonius of Perga, in On Determinate Section, dealt with problems in a manner that may be called an analytic geometry of one dimension. Apollonius in the Conics further developed a method, so similar to analytic geometry that his work is sometimes thought to have anticipated the work of Descartes by some 1800 years, his application of reference lines, a diameter and a tangent is no different from our modern use of a coordinate frame, where the distances measured along the diameter from the point of tangency are the abscissas, the segments parallel to the tangent and intercepted between the axis and the curve are the ordinates. He further developed relations between the abscissas and the corresponding ordinates that are equivalent to rhetorical equations of curves. However, although Apollonius came close to developing analytic geometry, he did not manage to do so since he did not take into account negative magnitudes and in every case the coordinate system was superimposed upon a given curve a posteriori instead of a priori.

That is, equations were determined by curves. Coordinates and equations were subsidiary notions applied to a specific geometric situation; the 11th-century Persian mathematician Omar Khayyám saw a strong relationship between geometry and algebra and was moving in the right direction when he helped close the gap between numerical and geometric algebra with his geometric solution of the general cubic equations, but the decisive step came with Descartes. Omar Khayyam is credited with identifying the foundations of algebraic geometry, his book Treatise on Demonstrations of Problems of Algebra, which laid down the principles of algebra, is part of the body of Persian mathematics, transmitted to Europe; because of his thoroughgoing geometrical approach to algebraic equations, Khayyam can be considered a precursor to Descartes in the invention of analytic geometry. Analytic geometry was independently invented by René Descartes and Pierre de Fermat, although Descartes is sometimes given sole credit.

Cartesian geometry, the alternative term used for analytic geometry, is named after Descartes. Descartes made significant progress with the methods in an essay titled La Geometrie, one of the three accompanying essays published in 1637 together with his Discourse on the Method for Rightly Directing One's Reason and Searching for Truth in the Sciences referred to as Discourse on Method; this work, written in his native French tongue, its philosophical principles, provided a foundation for calculus in Europe. The work was not well received, due, in part, to the many gaps in arguments and complicated equations. Only after the translation into Latin and the addition of commentary by van Schooten in 1649 did Descartes's masterpiece receive due recognition. Pierre de Fermat pioneered the development of analytic geometry. Although not published in his lifetime, a manuscript form of Ad locos planos et solidos isagoge was circulating in Paris in 1637, just prior to the publication of Descartes' Discourse.

Written and well received, the Introduction laid the groundwork for analytical geometry. The key difference between Fermat's and Descartes' treatments is a matter of viewpoint: Fermat always started with an algebraic equation and described the geometric curve that satisfied it, whereas Descartes started with geometric curves and produced their equations as one of several properties of the curves; as a consequence of this approach, Descartes had to deal with more complicated equations and he had to develop the methods to work with polynomial equations of higher degree. It was Leonhard Euler who first applied the coordinate method in a systematic study of space curves and surfaces. In analytic geometry, the plane is given a coordinate system, by which every point has a pair of real number coordinates. Euclidean space is given coordinates where every point has three coordinates; the value of the coordinates depends on the choice of the initial point of origin. There are a variety of coordinate systems used, but the most common are the following: The most common coordinate system to use is the Cartesian coordinate system, where each point has an x-coordinate representing its horizontal position, a y-coordinate representing its vertica

Bayraktar Mini UAV

Bayraktar Mini UAV is a Miniature UAV produced by Turkish company Baykar. With the concept of short range day and night aerial reconnaissance and surveillance applications, system design activities started within 2004. Initial prototype Bayraktar A has been developed in 2005, following successful autonomous flight demonstrations, Baykar has been awarded with a contract to start serial production. After hundreds of trials and feed backs, system was subject to major modifications; as a result, Bayraktar B Mini UAV Systems fielded and became operational in 2007 to serve for Turkish Armed Forces. Due to its success in the region, the system was purchased by the Qatar Armed Forces in 2012. Bayraktar-B is a hand-launched, portable UAV system, designed to operate under harsh geographic and meteorological conditions. Bayraktar-B is fielded with small army units, as of 2012 have recorded more than 50,000 flight hours. System offers a complete autonomy with protective features with a high rank of reliability and easiness for the operators, which makes it a valuable technological asset.

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Captivity, or being held captive, is a state wherein humans or other animals are confined to a particular space and prevented from leaving or moving freely. An example in humans is imprisonment. Prisoners of war are held in captivity by a government hostile to their own. Animals are held in captivity in zoos, as pets and as livestock. Captivity is the state of being imprisoned or confined; the word derives from the late Middle English captivitas, the Latin captivus and capere, meaning to seize or take, the root of the English word, "capture". In humans, captivity may include arrest and detention as a function of law enforcement and a civilian correctional system, detention of combatants in a time of war, as well as human trafficking, slave taking, other forms of involuntary confinement, forced relocation, servitude. In non-human animals, captivity may include confinement for the purpose of food production or labor, such as that done on a farm, confinement for the purpose of human recreation or education, such as that done at a zoo or aquarium, or confinement for the purpose of keeping domesticated pets, such as that done with animals such as the house cat or the dog.

In relation to non-living objects, captivity may describe the state of having control, whether that be control of one person over an object, such as "capturing a piece" in the game of chess, the control of a group over an area, such as the "capture" of a fort or city during a time of war, or control exercised by one object over another, such as one celestial body being "captured" by the gravitational pull of another, or a "captive balloon", tethered to the ground by a rope or string. In a philosophical sense, captivity may refer not to confinement or lack of individual freedom, but to the nature of a relationship between the captive and the captor, characterized by a lack of self-direction and autonomy. "Although the paradigm case of captivity is a free person, held against her will by another, the existence of captive children and animals makes it clear that the denial of autonomy as it is understood is not a condition for captivity". In some instances, the captivity of the subject is clear, as with an animal kept in a cage at a zoo.

However, circumstances exist. For example, it has been noted that it is hard to say whether members of a rhinoceros family kept in a thousand-acre enclosure within their normal area of habitation, for purposes of insuring their preservation, are in captivity. Captivity may be employed in more abstract or figurative senses, such as to captivate, meaning to subdue through charm, or to capture such as an artist attempting to "capture a mood", or "capture a scene". Humans are held captive under the authority of their own government for a number of different reasons. Under certain circumstances, a person suspected of committing a crime is subject to detention for a period of time while awaiting trial for that crime. In some cases, a person may be detained and released without being charged with criminal wrongdoing. Persons convicted of a crime may sent to a prison. According to the Institute for Criminal Policy Research at the University of London, as of 2016 an estimated 10.35 million people were imprisoned worldwide.

Throughout human history, the practice of "captive taking" during war was practiced. Those taken from the defeated group, most women and children, would be enslaved, sold into slavery to others, forced to marry members of the victorious group, or held in permanent sexual captivity; the first Roman gladiators, for example, were prisoners of war. The taking of captives may have been a byproduct, but was often a primary goal of conducting raids and warfare in small scale societies. According to some estimates of ancient societies, war captives and slaves may have at various points comprised as much as 20% of Roman Italy, 33% of Greece, 70% of Korea, 20% of some Islamic states, 40% of tropical American societies, as much as half of some African societies; the practice of conducting raids for captive taking extended in some forms until modern times, for example, piracy in the Mediterranean Sea and the taking of captives to be sold as slaves continued until the 19th century, when it culminated in the Barbary Wars.

Over time, nations found it to be in their interests to agree to international standards regarding the treatment of captured soldiers. The 1648 Peace of Westphalia, which ended the Thirty Years' War, established that prisoners of war should be released without ransom at the end of hostilities and that they should be allowed to return to their homelands. Chapter II of the Annex to the 1907 Hague Convention IV – The Laws and Customs of War on Land covered the treatment of prisoners of war in detail; these provisions were further expanded in the 1929 Geneva Convention on the Prisoners of War and were revised in the Third Geneva Convention in 1949. Article 4 of the Third Geneva Convention protects captured military personnel, some guerrilla fighters, certain civilians, it applies from the moment a prisoner is captured until she is released or repatriated. One of the main provisions of the convention makes it illegal to torture prisoners and states that a prisoner can only be required to give their name, date of birth and service number.

In some wars, such as the First World War, the conditions of captivity were separated between camps for prisoners of war, those for civilian internment. Some wars have seen mass wartime imprisonment. In addition to enemy military personnel, the Nazi regime imprisoned large numbers of private citizens based on their ethnicity, culture, or political views, as part of the regime's efforts to impose a vision of et