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Spiral
In mathematics, a spiral is a curve which emanates from a point, moving further away as it revolves around the point. It is a subtype of whorled patterns, a broad group that also includes concentric objects. Two-dimensional A two-dimensional, or plane, spiral may be easily described using polar coordinates, where the radius r is a monotonic continuous function of angle \varphi: * r=r(\varphi)\; . The circle would be regarded as a degenerate case (the function not being strictly monotonic, but rather constant). In ''x-y-coordinates'' the curve has the parametric representation: * x=r(\varphi)\cos\varphi \ ,\qquad y=r(\varphi)\sin\varphi\; . Examples Some of the most important sorts of two-dimensional spirals include: * The Archimedean spiral: r=a \varphi * The hyperbolic spiral: r = a/ \varphi * Fermat's spiral: r= a\varphi^ * The lituus: r = a\varphi^ * The logarithmic spiral: r=ae^ * The Cornu spiral or ''clothoid'' * The Fibonacci spiral and golden spiral * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithmic Spiral
A logarithmic spiral, equiangular spiral, or growth spiral is a self-similarity, self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). More than a century later, the curve was discussed by René Descartes, Descartes (1638), and later extensively investigated by Jacob Bernoulli, who called it ''Spira mirabilis'', "the marvelous spiral". The logarithmic spiral is distinct from the Archimedean spiral in that the distances between the turnings of a logarithmic spiral increase in a geometric progression, whereas for an Archimedean spiral these distances are constant. Definition In polar coordinates (r, \varphi) the logarithmic spiral can be written as r = ae^,\quad \varphi \in \R, or \varphi = \frac \ln \frac, with e (mathematical constant), e being the base of natural logarithms, and a > 0, k\ne 0 being real constants. In Cartesian coordinates The logarithmi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Spiral
A hyperbolic spiral is a type of spiral with a Pitch angle of a spiral, pitch angle that increases with distance from its center, unlike the constant angles of logarithmic spirals or decreasing angles of Archimedean spirals. As this curve widens, it approaches an asymptotic line. It can be found in the view up a spiral staircase and the starting arrangement of certain footraces, and is used to model spiral galaxy, spiral galaxies and Volute, architectural volutes. As a plane curve, a hyperbolic spiral can be described in polar coordinates (r,\varphi) by the equation r=\frac, for an arbitrary choice of the scale factor a. Because of the Multiplicative inverse, reciprocal relation between r and \varphi it is also called a reciprocal spiral. The same relation between Cartesian coordinates would describe a hyperbola, and the hyperbolic spiral was first discovered by applying the equation of a hyperbola to polar coordinates. Hyperbolic spirals can also be generated as the inverse cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archimedean Spiral
The Archimedean spiral (also known as Archimedes' spiral, the arithmetic spiral) is a spiral named after the 3rd-century BC Ancient Greece, Greek mathematician Archimedes. The term ''Archimedean spiral'' is sometimes used to refer to the more general class of spirals of this type (see below), in contrast to ''Archimedes' spiral'' (the specific arithmetic spiral of Archimedes). It is the locus (mathematics), locus corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line that rotates with constant angular velocity. Equivalently, in Polar coordinate system, polar coordinates it can be described by the equation r = b\cdot\theta with real number . Changing the parameter controls the distance between loops. From the above equation, it can thus be stated: position of the particle from point of start is proportional to angle as time elapses. Archimedes described such a spiral in his book ''On Spirals''. Conon of Samos was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cornu Spiral
An Euler spiral is a curve whose curvature changes linearly with its curve length (the curvature of a circular curve is equal to the reciprocal of the radius). This curve is also referred to as a clothoid or Cornu spiral.Levien, Raph"The Euler spiral: a mathematical history."Rapp. tech (2008). The behavior of Fresnel integrals can be illustrated by an Euler spiral, a connection first made by Marie Alfred Cornu in 1874. Euler's spiral is a type of superspiral that has the property of a monotonic curvature function. The Euler spiral has applications to diffraction computations. They are also widely used in railway and highway engineering to design transition curves between straight and curved sections of railways or roads. A similar application is also found in photonic integrated circuits. The principle of linear variation of the curvature of the transition curve between a tangent and a circular curve defines the geometry of the Euler spiral: *Its curvature begins with zero at t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Golden Spiral
In geometry, a golden spiral is a logarithmic spiral whose growth factor is , the golden ratio. That is, a golden spiral gets wider (or further from its origin) by a factor of for every quarter Turn (angle), turn it makes. Approximations of the golden spiral There are several comparable spirals that approximate, but do not exactly equal, a golden spiral. For example, a golden spiral can be approximated by first starting with a rectangle for which the ratio between its length and width is the golden ratio. This rectangle can then be partitioned into a square and a similarity (geometry), similar rectangle and this rectangle can then be split in the same way. After continuing this process for an arbitrary number of steps, the result will be an almost complete partitioning of the rectangle into squares. The corners of these squares can be connected by quarter-circles. The result, though not a true logarithmic spiral, closely approximates a golden spiral. Another approximation i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat's Spiral
A Fermat's spiral or parabolic spiral is a plane curve with the property that the area between any two consecutive full turns around the spiral is invariant. As a result, the distance between turns grows in inverse proportion to their distance from the spiral center, contrasting with the Archimedean spiral (for which this distance is invariant) and the logarithmic spiral (for which the distance between turns is proportional to the distance from the center). Fermat spirals are named after Pierre de Fermat. Their applications include curvature continuous blending of curves, modeling phyllotaxis, plant growth and the shapes of certain spiral galaxy, spiral galaxies, and the design of variable capacitors, solar power reflector arrays, and cyclotrons. Coordinate representation Polar The representation of the Fermat spiral in polar coordinates is given by the equation r=\pm a\sqrt for . The parameter a is a scaling factor affecting the size of the spiral but not its shape. The tw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spiral Of Theodorus
In geometry, the spiral of Theodorus (also called the square root spiral, Pythagorean spiral, or Pythagoras's snail) is a spiral composed of right triangles, placed edge-to-edge. It was named after Theodorus of Cyrene. Construction The spiral is started with an isosceles right triangle, with each leg having unit length. Another right triangle (which is the ''only'' automedian right triangle) is formed, with one leg being the hypotenuse of the prior right triangle (with length the square root of 2) and the other leg having length of 1; the length of the hypotenuse of this second right triangle is the square root of 3. The process then repeats; the nth triangle in the sequence is a right triangle with the side lengths \sqrt and 1, and with hypotenuse \sqrt. For example, the 16th triangle has sides measuring 4=\sqrt, 1 and hypotenuse of \sqrt. History and uses Although all of Theodorus' work has been lost, Plato put Theodorus into his dialogue '' Theaetetus'', which tells of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polar Coordinates
In mathematics, the polar coordinate system specifies a given point (mathematics), point in a plane (mathematics), plane by using a distance and an angle as its two coordinate system, coordinates. These are *the point's distance from a reference point called the ''pole'', and *the point's direction from the pole relative to the direction of the ''polar axis'', a ray (geometry), ray drawn from the pole. The distance from the pole is called the ''radial coordinate'', ''radial distance'' or simply ''radius'', and the angle is called the ''angular coordinate'', ''polar angle'', or ''azimuth''. The pole is analogous to the origin in a Cartesian coordinate system. Polar coordinates are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point in a plane, such as spirals. Planar physical systems with bodies moving around a central point, or phenomena originating from a central point, are often simpler and more in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polar Coordinate System
In mathematics, the polar coordinate system specifies a given point in a plane by using a distance and an angle as its two coordinates. These are *the point's distance from a reference point called the ''pole'', and *the point's direction from the pole relative to the direction of the ''polar axis'', a ray drawn from the pole. The distance from the pole is called the ''radial coordinate'', ''radial distance'' or simply ''radius'', and the angle is called the ''angular coordinate'', ''polar angle'', or ''azimuth''. The pole is analogous to the origin in a Cartesian coordinate system. Polar coordinates are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point in a plane, such as spirals. Planar physical systems with bodies moving around a central point, or phenomena originating from a central point, are often simpler and more intuitive to model using polar coordinates. The polar coordinate system i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lituus (mathematics)
300px, Branch for positive The lituus spiral () is a spiral in which the angle is inversely proportional to the square of the radius . This spiral, which has two branches depending on the sign of , is asymptotic to the axis. Its points of inflexion are at : (\theta, r) = \left(\tfrac12, \pm\sqrt\right). The curve was named for the ancient Roman lituus by Roger Cotes in a collection of papers entitled ''Harmonia Mensurarum'' (1722), which was published six years after his death. Coordinate representations Polar coordinates The representations of the lituus spiral in polar coordinates is given by the equation : r = \frac, where and . Cartesian coordinates The lituus spiral with the polar coordinates can be converted to Cartesian coordinates like any other spiral with the relationships and . With this conversion we get the parametric representations of the curve: : \begin x &= \frac \cos\theta, \\ y &= \frac \sin\theta. \\ \end These equations can in turn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whorl
A whorl ( or ) is an individual circle, oval, volution or equivalent in a whorled pattern, which consists of a spiral or multiple concentric objects (including circles, ovals and arcs). In nature File:Photograph and axial plane floral diagram of Friesodielsia desmoides.jpg, Botanical whorls: sepals, petals, leaves, or branches radiating from a single point (photo of flower of Friesodielsia desmoides, family Annonaceae, juxtaposed with diagram of axial cross-section) File:Anisus septegyrus1pl.jpg, Mollusc whorls: Each complete 360° turn in the spiral growth of the shell of the mollusc Anisus septemgyratus, family Planorbidae. File:Baby hairy head DSCN2483.jpg, A hair whorl is a patch of hair growing in a circular direction around a visible center point. File:Fingerprint Whorl.jpg, In a fingerprint, a whorl is each ridge arranged circularly around a central point on the finger. File:Bovine Bone Sample and 430 times Magnification.jpg, In histopathologic architecture, a wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
