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Quincuncial Map
The Peirce quincuncial projection is the conformal map projection from the sphere to an unfolded square dihedron, developed by Charles Sanders Peirce in 1879. Each octant projects onto an isosceles right triangle, and these are arranged into a square. The name ''quincuncial'' refers to this arrangement: the north pole at the center and quarters of the south pole in the corners form a quincunx pattern like the pips on the ''five'' face of a traditional die. The projection has the distinctive property that it forms a seamless square tiling of the plane, conformal except at four singular points along the equator. Typically the projection is square and oriented such that the north pole lies at the center, but an oblique aspect in a rectangle was proposed by Émile Guyou in 1887, and a transverse aspect was proposed by Oscar S. Adams in 1925. The projection has seen use in digital photography for portraying spherical panoramas. History The maturation of complex analysis led to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peirce Quincuncial Projection SW 20W
Peirce may refer to: * Charles Sanders Peirce (1839–1914), American philosopher, founder of pragmatism Schools * Peirce College, Philadelphia, formerly known as Peirce College of Business, Peirce Junior College and Peirce School of Business Administration * Peirce School (also known as Old Peirce School), West Newton, Massachusetts * Helen C. Peirce School of International Studies, an elementary school in Chicago Others * Peirce (crater), a lunar crater * Peirce (given name), including a list of people with the given name * Peirce (surname), including a list of people with the surname See also * * * Pierce (other) * Peirse (other) {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Map
In mathematics, a conformal map is a function (mathematics), function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in U if it preserves angles between directed curves through u_0, as well as preserving orientation. Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or curvature. The conformal property may be described in terms of the Jacobian matrix and determinant, Jacobian derivative matrix of a coordinate transformation. The transformation is conformal whenever the Jacobian at each point is a positive scalar times a rotation matrix (Orthogonal matrix, orthogonal with determinant one). Some authors define conformality to include orientation-reversing mappings whose Jacobians can be written as any scalar times any orthogonal matrix. For mappings in two dimensions, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lemniscate Elliptic Functions
In mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied by Giulio Carlo de' Toschi di Fagnano, Giulio Fagnano in 1718 and later by Leonhard Euler and Carl Friedrich Gauss, among others. The lemniscate sine and lemniscate cosine functions, usually written with the symbols and (sometimes the symbols and or and are used instead), are analogous to the trigonometric functions sine and cosine. While the trigonometric sine relates the arc length to the chord length in a unit-diameter circle x^2+y^2 = x, the lemniscate sine relates the arc length to the chord length of a lemniscate \bigl(x^2+y^2\bigr)^2=x^2-y^2. The lemniscate functions have periods related to a number called the lemniscate constant, the ratio of a lemniscate's perimeter to its diameter. This number is a Quartic plane curve, quartic analog of the (Conic section, quadratic) , pi, ratio of perimeter to diameter of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lemniscatic
In algebraic geometry, a lemniscate ( or ) is any of several figure-eight or -shaped curves. The word comes from the Latin , meaning "decorated with ribbons", from the Greek (), meaning "ribbon",. or which alternatively may refer to the wool from which the ribbons were made. Curves that have been called a lemniscate include three quartic plane curves: the hippopede or lemniscate of Booth, the lemniscate of Bernoulli, and the lemniscate of Gerono. The hippopede was studied by Proclus (5th century), but the term "lemniscate" was not used until the work of Jacob Bernoulli in the late 17th century. History and examples Lemniscate of Booth The consideration of curves with a figure-eight shape can be traced back to Proclus, a Greek Neoplatonist philosopher and mathematician who lived in the 5th century AD. Proclus considered the cross-sections of a torus by a plane parallel to the axis of the torus. As he observed, for most such sections the cross section consists of either one o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Integral
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in connection with the problem of finding the arc length of an ellipse. Modern mathematics defines an "elliptic integral" as any function which can be expressed in the form f(x) = \int_^ R \, dt, where is a rational function of its two arguments, is a polynomial of degree 3 or 4 with no repeated roots, and is a constant. In general, integrals in this form cannot be expressed in terms of elementary functions. Exceptions to this general rule are when has repeated roots, or when contains no odd powers of or if the integral is pseudo-elliptic. However, with the appropriate reduction formula, every elliptic integral can be brought into a form that involves integrals over rational functions and the three Legendre canonical forms, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Elliptic Functions
In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum, as well as in the design of electronic elliptic filters. While trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation \operatorname for \sin. The Jacobi elliptic functions are used more often in practical problems than the Weierstrass elliptic functions as they do not require notions of complex analysis to be defined and/or understood. They were introduced by . Carl Friedrich Gauss had already studied special Jacobi elliptic functions in 1797, the lemniscate elliptic functions in particular, but his work was published much later. Overview There are twelve Jacobi elliptic functions denoted by \operatorna ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stereographic Projection
In mathematics, a stereographic projection is a perspective transform, perspective projection of the sphere, through a specific point (geometry), point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (the ''projection plane'') perpendicular to the diameter through the point. It is a smooth function, smooth, bijection, bijective function (mathematics), function from the entire sphere except the center of projection to the entire plane. It maps circle of a sphere, circles on the sphere to generalised circle, circles or lines on the plane, and is conformal map, conformal, meaning that it preserves angles at which curves meet and thus Local property, locally approximately preserves similarity (geometry), shapes. It is neither isometry, isometric (distance preserving) nor Equiareal map, equiareal (area preserving). The stereographic projection gives a way to representation (mathematics), represent a sphere by a plane. The metric tensor, metric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Map Projection
In cartography, a map projection is any of a broad set of Transformation (function) , transformations employed to represent the curved two-dimensional Surface (mathematics), surface of a globe on a Plane (mathematics), plane. In a map projection, coordinates, often expressed as latitude and longitude, of locations from the surface of the globe are transformed to coordinates on a plane. Projection is a necessary step in creating a two-dimensional map and is one of the essential elements of cartography. All projections of a sphere on a plane necessarily distort the surface in some way. Depending on the purpose of the map, some distortions are acceptable and others are not; therefore, different map projections exist in order to preserve some properties of the sphere-like body at the expense of other properties. The study of map projections is primarily about the characterization of their distortions. There is no limit to the number of possible map projections. More generally, proje ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Northern Hemisphere
The Northern Hemisphere is the half of Earth that is north of the equator. For other planets in the Solar System, north is defined by humans as being in the same celestial sphere, celestial hemisphere relative to the invariable plane of the Solar System as Earth's North Pole. Due to Earth's axial tilt of 23.439281°, there is a seasonal variation in the lengths of the day and night. There is also a seasonal variation in temperatures, which lags the variation in day and night. Conventionally, winter in the Northern Hemisphere is taken as the period from the December solstice (typically December 21 UTC) to the March equinox (typically March 20 UTC), while summer is taken as the period from the June solstice through to the September equinox (typically on 23 September UTC). The dates vary each year due to the difference between the calendar year and the Year#Astronomical years, astronomical year. Within the Northern Hemisphere, oceanic currents can change the weather patterns that aff ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwarz–Christoffel Mapping
In complex analysis, a Schwarz–Christoffel mapping is a conformal map of the upper half-plane or the complex unit disk onto the interior of a simple polygon. Such a map is guaranteed to exist by the Riemann mapping theorem (stated by Bernhard Riemann in 1851); the Schwarz–Christoffel formula provides an explicit construction. They were introduced independently by Elwin Christoffel in 1867 and Hermann Schwarz in 1869. Schwarz–Christoffel mappings are used in potential theory and some of its applications, including minimal surfaces, hyperbolic art, and fluid dynamics. Definition Consider a polygon in the complex plane. The Riemann mapping theorem implies that there is a biholomorphic mapping ''f'' from the upper half-plane : \ to the interior of the polygon. The function ''f'' maps the real axis to the edges of the polygon. If the polygon has interior angles \alpha,\beta,\gamma, \ldots, then this mapping is given by : f(\zeta) = \int^\zeta \frac \,\mathrmw where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
