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Monkey Saddle
In mathematics, the monkey saddle is the surface defined by the equation : z = x^3 - 3xy^2, \, or in cylindrical coordinates :z = \rho^3 \cos(3\varphi). It belongs to the class of saddle surfaces, and its name derives from the observation that a saddle used by a monkey would require two depressions for its legs and one for its tail. The point on the monkey saddle corresponds to a degenerate critical point of the function at . The monkey saddle has an isolated umbilical point with zero Gaussian curvature at the origin, while the curvature is strictly negative at all other points. One can relate the rectangular and cylindrical equations using complex numbers x+iy = r e^: : z = x^3 - 3xy^2 = \operatorname x+iy)^3= \operatorname ^3 e^= r^3\cos(3\varphi). By replacing 3 in the cylindrical equation with any integer one can create a saddle with depressions. Another orientation of the monkey saddle is the ''Smelt petal'' defined by x+y+z+xyz=0, so that the ''z-''axis of the mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monkey Saddle Surface (Shaded)
Monkey is a common name that may refer to most mammals of the infraorder Simiiformes, also known as simians. Traditionally, all animals in the group now known as simians are counted as monkeys except the apes. Thus monkeys, in that sense, constitute an incomplete paraphyletic grouping; alternatively, if apes (Hominoidea) are included, ''monkeys'' and ''simians'' are synonyms. In 1812, Étienne Geoffroy grouped the apes and the Cercopithecidae group of monkeys together and established the name Catarrhini, "Old World monkeys" ("''singes de l'Ancien Monde''" in French). The extant sister of the Catarrhini in the monkey ("singes") group is the Platyrrhini (New World monkeys). Some nine million years before the divergence between the Cercopithecidae and the apes, the Platyrrhini emerged within "monkeys" by migration to South America likely by ocean. Apes are thus deep in the tree of extant and extinct monkeys, and any of the apes is distinctly closer related to the Cercopithecida ... [...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] |
Surface (mathematics)
In mathematics, a surface is a mathematical model of the common concept of a surface. It is a generalization of a plane, but, unlike a plane, it may be curved; this is analogous to a curve generalizing a straight line. There are several more precise definitions, depending on the context and the mathematical tools that are used for the study. The simplest mathematical surfaces are planes and spheres in the Euclidean 3-space. The exact definition of a surface may depend on the context. Typically, in algebraic geometry, a surface may cross itself (and may have other singularities), while, in topology and differential geometry, it may not. A surface is a topological space of dimension two; this means that a moving point on a surface may move in two directions (it has two degrees of freedom). In other words, around almost every point, there is a '' coordinate patch'' on which a two-dimensional coordinate system is defined. For example, the surface of the Earth resembles ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cylindrical Coordinate System
A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions around a main axis (a chosen directed line) and an auxiliary axis (a reference ray). The three cylindrical coordinates are: the point perpendicular distance from the main axis; the point signed distance ''z'' along the main axis from a chosen origin; and the plane angle of the point projection on a reference plane (passing through the origin and perpendicular to the main axis) The main axis is variously called the ''cylindrical'' or ''longitudinal'' axis. The auxiliary axis is called the ''polar axis'', which lies in the reference plane, starting at the origin, and pointing in the reference direction. Other directions perpendicular to the longitudinal axis are called ''radial lines''. The distance from the axis may be called the ''radial distance'' or ''radius'', while the angular coordinate is sometimes referred to as the ''angular position'' or as the ''azimuth''. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saddle Surface
In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function. An example of a saddle point is when there is a critical point with a relative minimum along one axial direction (between peaks) and a relative maximum along the crossing axis. However, a saddle point need not be in this form. For example, the function f(x,y) = x^2 + y^3 has a critical point at (0, 0) that is a saddle point since it is neither a relative maximum nor relative minimum, but it does not have a relative maximum or relative minimum in the y-direction. The name derives from the fact that the prototypical example in two dimensions is a surface that ''curves up'' in one direction, and ''curves down'' in a different direction, resembling a riding saddle. In terms of contour lines, a saddle point in two dimensions gives rise to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saddle
A saddle is a supportive structure for a rider of an animal, fastened to an animal's back by a girth. The most common type is equestrian. However, specialized saddles have been created for oxen, camels and other animals. It is not known precisely when riders first began to use some sort of padding or protection, but a blanket attached by some form of surcingle or girth was probably the first "saddle", followed later by more elaborate padded designs. The solid saddle tree was a later invention, and though early stirrup designs predated the invention of the solid tree, the paired stirrup, which attached to the tree, was the last element of the saddle to reach the basic form that is still used today. Present-day saddles come in a wide variety of styles, each designed for a specific equestrianism discipline, and require careful fit to both the rider and the horse. Proper saddle care can extend the useful life of a saddle, often for decades. The saddle was a crucial step in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monkey
Monkey is a common name that may refer to most mammals of the infraorder Simiiformes, also known as simians. Traditionally, all animals in the group now known as simians are counted as monkeys except the apes. Thus monkeys, in that sense, constitute an incomplete paraphyletic grouping; alternatively, if apes (Hominoidea) are included, ''monkeys'' and ''simians'' are synonyms. In 1812, Étienne Geoffroy grouped the apes and the Cercopithecidae group of monkeys together and established the name Catarrhini, "Old World monkeys" ("''singes de l'Ancien Monde''" in French). The extant sister of the Catarrhini in the monkey ("singes") group is the Platyrrhini (New World monkeys). Some nine million years before the divergence between the Cercopithecidae and the apes, the Platyrrhini emerged within "monkeys" by migration to South America likely by ocean. Apes are thus deep in the tree of extant and extinct monkeys, and any of the apes is distinctly closer related to the Cercopith ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Critical Point (mathematics)
In mathematics, a critical point is the argument of a function where the function derivative is zero (or undefined, as specified below). The value of the function at a critical point is a . More specifically, when dealing with functions of a real variable, a critical point is a point in the domain of the function where the function derivative is equal to zero (also known as a ''stationary point'') or where the function is not differentiable. Similarly, when dealing with complex variables, a critical point is a point in the function's domain where its derivative is equal to zero (or the function is not ''holomorphic''). Likewise, for a function of several real variables, a critical point is a value in its domain where the gradient norm is equal to zero (or undefined). This sort of definition extends to differentiable maps between and a critical point being, in this case, a point where the rank of the Jacobian matrix is not maximal. It extends further to differentiable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Umbilical Point
In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical. At such points the normal curvatures in all directions are equal, hence, both principal curvatures are equal, and every tangent vector is a ''principal direction''. The name "umbilic" comes from the Latin ''umbilicus'' (navel). Umbilic points generally occur as isolated points in the elliptical region of the surface; that is, where the Gaussian curvature is positive. The sphere is the only surface with non-zero curvature where every point is umbilic. A flat umbilic is an umbilic with zero Gaussian curvature. The monkey saddle is an example of a surface with a flat umbilic and on the plane every point is a flat umbilic. A closed surface topologically equivalent to a torus may or may not have zero umbilics, but every closed surface of nonzero Euler characteristic, embedded smoothly into Euclidean space, has at least one umbilic. A famous c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a smooth Surface (topology), surface in three-dimensional space at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. For example, a sphere of radius has Gaussian curvature everywhere, and a flat plane and a cylinder have Gaussian curvature zero everywhere. The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus. Gaussian curvature is an ''intrinsic'' measure of curvature, meaning that it could in principle be measured by a 2-dimensional being living entirely within the surface, because it depends only on distances that are measured “within” or along the surface, not on the way it is isometrically embedding, embedded in Euclidean space. This is the content of the ''Theorema Egregium''. Gaussian curvature is named after Carl Friedrich Gauss, who published the ''Theorema Egregium'' in 1827. Informal definit ... [...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] |
Shape Petal
A shape is a graphical representation of an object's form or its external boundary, outline, or external surface. It is distinct from other object properties, such as color, texture, or material type. In geometry, ''shape'' excludes information about the object's position, size, orientation and chirality. A ''figure'' is a representation including both shape and size (as in, e.g., figure of the Earth). A plane shape or plane figure is constrained to lie on a '' plane'', in contrast to ''solid'' 3D shapes. A two-dimensional shape or two-dimensional figure (also: 2D shape or 2D figure) may lie on a more general curved ''surface'' (a two-dimensional space). Classification of simple shapes Some simple shapes can be put into broad categories. For instance, polygons are classified according to their number of edges as triangles, quadrilaterals, pentagons, etc. Each of these is divided into smaller categories; triangles can be equilateral, isosceles, obtuse, acute, scalene, et ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
