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Conchoid Of Nicomedes
In geometry, a conchoid is a curve derived from a fixed point , another curve, and a length . It was invented by the ancient Greek mathematician Nicomedes. Description For every line through that intersects the given curve at the two points on the line which are from are on the conchoid. The conchoid is, therefore, the cissoid of the given curve and a circle of radius and center . They are called ''conchoids'' because the shape of their outer branches resembles conch shells. The simplest expression uses polar coordinates with at the origin. If :r=\alpha(\theta) expresses the given curve, then :r=\alpha(\theta)\pm d expresses the conchoid. If the curve is a line, then the conchoid is the ''conchoid of Nicomedes''. For instance, if the curve is the line , then the line's polar form is and therefore the conchoid can be expressed parametrically as :x=a \pm d \cos \theta,\, y=a \tan \theta \pm d \sin \theta. A limaçon is a conchoid with a circle as the given cur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conchoid Of Nicomedes
In geometry, a conchoid is a curve derived from a fixed point , another curve, and a length . It was invented by the ancient Greek mathematician Nicomedes. Description For every line through that intersects the given curve at the two points on the line which are from are on the conchoid. The conchoid is, therefore, the cissoid of the given curve and a circle of radius and center . They are called ''conchoids'' because the shape of their outer branches resembles conch shells. The simplest expression uses polar coordinates with at the origin. If :r=\alpha(\theta) expresses the given curve, then :r=\alpha(\theta)\pm d expresses the conchoid. If the curve is a line, then the conchoid is the ''conchoid of Nicomedes''. For instance, if the curve is the line , then the line's polar form is and therefore the conchoid can be expressed parametrically as :x=a \pm d \cos \theta,\, y=a \tan \theta \pm d \sin \theta. A limaçon is a conchoid with a circle as the given cur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nicomedes , 94–74 BC
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Nicomedes may refer to: *Nicomedes (mathematician), ancient Greek mathematician who discovered the conchoid *Nicomedes of Sparta, regent during the youth of King Pleistoanax, commanded the Spartan army at the Battle of Tanagra (457 BC) *Saint Nicomedes, Martyr of unknown era, whose feast is observed 15 September Four kings of Bithynia in Anatolia, 3rd–1st century BC: *Nicomedes I of Bithynia, ruled 278–255 BC *Nicomedes II of Bithynia, 149–127 BC *Nicomedes III of Bithynia, 127–94 BC *Nicomedes IV of Bithynia Nicomedes IV Philopator ( grc-gre, Νικομήδης Φιλοπάτωρ) was the king of Bithynia from c. 94 BC to 74 BC. (''numbered as III. not IV.'') He was the first son and successor of Nicomedes III of Bithynia. Life Memnon of Heraclea wro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geometry), point. This is the definition that appeared more than 2000 years ago in Euclid's Elements, Euclid's ''Elements'': "The [curved] line is […] the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which […] will leave from its imaginary moving some vestige in length, exempt of any width." This definition of a curve has been formalized in modern mathematics as: ''A curve is the image (mathematics), image of an interval (mathematics), interval to a topological space by a continuous function''. In some contexts, the function that defines the curve is called a ''parametrization'', and the curve is a parametric curve. In this artic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nicomedes (mathematician)
Nicomedes (; grc-gre, Νικομήδης; c. 280 – c. 210 BC) was an ancient Greek mathematician. Life and work Almost nothing is known about Nicomedes' life apart from references in his works. Studies have stated that Nicomedes was born in about 280 BC and died in about 210 BC. It is known that he lived around the time of Eratosthenes or after, because he criticized Eratosthenes' method of doubling the cube. It is also known that Apollonius of Perga called a curve of his creation a "sister of the conchoid", suggesting that he was naming it after Nicomedes' already famous curve. Consequently, it is believed that Nicomedes lived after Eratosthenes and before Apollonius of Perga. Like many geometers of the time, Nicomedes was engaged in trying to solve the problems of doubling the cube and trisecting the angle, both problems we now understand to be impossible using the tools of classical geometry. In the course of his investigations, Nicomedes created the conchoid of Nicom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cissoid
In geometry, a cissoid (() is a plane curve generated from two given curves , and a point (the pole). Let be a variable line passing through and intersecting at and at . Let be the point on so that \overline = \overline. (There are actually two such points but is chosen so that is in the same direction from as is from .) Then the locus of such points is defined to be the cissoid of the curves , relative to . Slightly different but essentially equivalent definitions are used by different authors. For example, may be defined to be the point so that \overline = \overline + \overline. This is equivalent to the other definition if is replaced by its reflection through . Or may be defined as the midpoint of and ; this produces the curve generated by the previous curve scaled by a factor of 1/2. Equations If and are given in polar coordinates by r=f_1(\theta) and r=f_2(\theta) respectively, then the equation r=f_2(\theta)-f_1(\theta) describes the cissoid of and r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conch
Conch () is a common name of a number of different medium-to-large-sized sea snails. Conch shells typically have a high spire and a noticeable siphonal canal (in other words, the shell comes to a noticeable point at both ends). In North America, a conch is often identified as a queen conch, indigenous to the waters of the Gulf of Mexico and Caribbean. Queen conches are valued for seafood and are also used as fish bait. The group of conches that are sometimes referred to as "true conches" are marine gastropod molluscs in the family Strombidae, specifically in the genus ''Strombus'' and other closely related genera. For example, ''Lobatus gigas'', the queen conch, and ''Laevistrombus canarium'', the dog conch, are true conches. Many other species are also often called "conch", but are not at all closely related to the family Strombidae, including ''Melongena'' species (family Melongenidae) and the horse conch ''Triplofusus papillosus'' (family Fasciolariidae). Species comm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line (mathematics)
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segment in everyday life, which has two points to denote its ends. Lines can be referred by two points that lay on it (e.g., \overleftrightarrow) or by a single letter (e.g., \ell). Euclid described a line as "breadthless length" which "lies evenly with respect to the points on itself"; he introduced several postulates as basic unprovable properties from which he constructed all of geometry, which is now called Euclidean geometry to avoid confusion with other geometries which have been introduced since the end of the 19th century (such as non-Euclidean, projective and affine geometry). In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parametric Equation
In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, in which case the equations are collectively called a parametric representation or parameterization (alternatively spelled as parametrisation) of the object. For example, the equations :\begin x &= \cos t \\ y &= \sin t \end form a parametric representation of the unit circle, where ''t'' is the parameter: A point (''x'', ''y'') is on the unit circle if and only if there is a value of ''t'' such that these two equations generate that point. Sometimes the parametric equations for the individual scalar output variables are combined into a single parametric equation in vectors: :(x, y)=(\cos t, \sin t). Parametric representations are generally nonunique (see the "Examples in two dimensions" section belo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limaçon
In geometry, a limaçon or limacon , also known as a limaçon of Pascal or Pascal's Snail, is defined as a roulette curve formed by the path of a point fixed to a circle when that circle rolls around the outside of a circle of equal radius. It can also be defined as the roulette formed when a circle rolls around a circle with half its radius so that the smaller circle is inside the larger circle. Thus, they belong to the family of curves called centered trochoids; more specifically, they are epitrochoids. The cardioid is the special case in which the point generating the roulette lies on the rolling circle; the resulting curve has a cusp. Depending on the position of the point generating the curve, it may have inner and outer loops (giving the family its name), it may be heart-shaped, or it may be oval. A limaçon is a bicircular rational plane algebraic curve of degree 4. History The earliest formal research on limaçons is generally attributed to Étienne Pascal, father ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conchoid Of De Sluze
In algebraic geometry, the conchoids of de Sluze are a family of plane curves studied in 1662 by Walloon mathematician René François Walter, baron de Sluze.. The curves are defined by the polar equation :r=\sec\theta+a\cos\theta \,. In cartesian coordinates, the curves satisfy the implicit equation :(x-1)(x^2+y^2)=ax^2 \, except that for the implicit form has an acnode not present in polar form. They are rational, circular, cubic plane curves. These expressions have an asymptote (for ). The point most distant from the asymptote is . is a crunode for . The area between the curve and the asymptote is, for , :, a, (1+a/4)\pi \, while for , the area is :\left(1-\frac a2\right)\sqrt-a\left(2+\frac a2\right)\arcsin\frac1. If , the curve will have a loop. The area of the loop is :\left(2+\frac a2\right)a\arccos\frac1 + \left(1-\frac a2\right)\sqrt. Four of the family have names of their own: *, line (asymptote to the rest of the family) *, cissoid of Diocles *, right stroph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conchoid Of Dürer
In geometry, the conchoid of Dürer, also called Dürer's shell curve, is a plane, algebraic curve, named after Albrecht Dürer and introduced in 1525. It is not a true conchoid. Construction Suppose two perpendicular lines are given, with intersection point ''O''. For concreteness we may assume that these are the coordinate axes and that ''O'' is the origin, that is (0, 0). Let points and move on the axes in such a way that , a constant. On the line , extended as necessary, mark points and at a fixed distance from . The locus of the points and is Dürer's conchoid. Equation The equation of the conchoid in Cartesian form is :::2y^2(x^2+y^2) - 2by^2(x+y) + (b^2-3a^2)y^2 - a^2x^2 + 2a^2b(x+y) + a^2(a^2-b^2) = 0 . In parametric form the equation is given by :\begin x &= \frac + a \cos(t),\\ y &= a \sin(t), \end where the parameter is measured in radians. Properties The curve has two components, asymptotic to the lines y = \pm a / \sqrt2. Each component is a ration ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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