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Osculating Plane
In mathematics, particularly in differential geometry, an osculating plane is a plane in a Euclidean space or affine space which meets a submanifold at a point in such a way as to have a second order of contact at the point. The word ''osculate'' is ; an osculating plane is thus a plane which "kisses" a submanifold. The osculating plane in the geometry of Euclidean space curves can be described in terms of the Frenet-Serret formulas as the linear span of the tangent and normal vectors. See also * Normal plane (geometry) * Osculating circle An osculating circle is a circle that best approximates the curvature of a curve at a specific point. It is tangent to the curve at that point and has the same curvature as the curve at that point. The osculating circle provides a way to unders ... * References {{Reflist Differential geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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Differential Geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as classical antiquity, antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Nikolai Lobachevsky, Lobachevsky. The simplest examples of smooth spaces are the Differential geometry of curves, plane and space curves and Differential geometry of surfaces, surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Plane (mathematics)
In mathematics, a plane is a two-dimensional space or flat surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. When working exclusively in two-dimensional Euclidean space, the definite article is used, so ''the'' Euclidean plane refers to the whole space. Several notions of a plane may be defined. The Euclidean plane follows Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ..., and in particular the parallel postulate. A projective plane may be constructed by adding "points at infinity" where two otherwise parallel lines would intersect, so that every pair of lines intersects in exactly one point. The elliptic plane may be further defined by adding a metr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces'' of any positive integer dimension ''n'', which are called Euclidean ''n''-spaces when one wants to specify their dimension. For ''n'' equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of '' proving'' all properties of the space as theorems, by starting from a few fundamental properties, called '' postulates'', which either were considered as evid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Affine Space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments. Affine space is the setting for affine geometry. As in Euclidean space, the fundamental objects in an affine space are called '' points'', which can be thought of as locations in the space without any size or shape: zero-dimensional. Through any pair of points an infinite straight line can be drawn, a one-dimensional set of points; through any three points that are not collinear, a two-dimensional plane can be drawn; and, in general, through points in general position, a -dimensional flat or affine subspace can be drawn. Affine space is characterized by a notion of pairs of parallel lines that lie within the same plane but never meet each-other (non-parallel lines wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Submanifold
In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S \rightarrow M satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required. Different authors often have different definitions. Formal definition In the following we assume all manifolds are differentiable manifolds of class C^r for a fixed r\geq 1, and all morphisms are differentiable of class C^r. Immersed submanifolds An immersed submanifold of a manifold M is the image S of an immersion map f: N\rightarrow M; in general this image will not be a submanifold as a subset, and an immersion map need not even be injective (one-to-one) – it can have self-intersections. More narrowly, one can require that the map f: N\rightarrow M be an injection (one-to-one), in which we call it an injective immersion, and define an immersed submanifold to be the image subset S together with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Point (geometry)
In geometry, a point is an abstract idealization of an exact position, without size, in physical space, or its generalization to other kinds of mathematical spaces. As zero-dimensional objects, points are usually taken to be the fundamental indivisible elements comprising the space, of which one-dimensional curves, two-dimensional surfaces, and higher-dimensional objects consist. In classical Euclidean geometry, a point is a primitive notion, defined as "that which has no part". Points and other primitive notions are not defined in terms of other concepts, but only by certain formal properties, called axioms, that they must satisfy; for example, ''"there is exactly one straight line that passes through two distinct points"''. As physical diagrams, geometric figures are made with tools such as a compass, scriber, or pen, whose pointed tip can mark a small dot or prick a small hole representing a point, or can be drawn across a surface to represent a curve. A po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Contact (mathematics)
In mathematics, two functions have a contact of order if, at a point , they have the same value and their first derivatives are equal. This is an equivalence relation, whose equivalence classes are generally called jets. The point of osculation is also called the double cusp. Contact is a geometric notion; it can be defined algebraically as a valuation. One speaks also of curves and geometric objects having -th order contact at a point: this is also called ''osculation'' (i.e. kissing), generalising the property of being tangent. (Here the derivatives are considered with respect to arc length.) An osculating curve from a given family of curves is a curve that has the highest possible order of contact with a given curve at a given point; for instance a tangent line is an osculating curve from the family of lines, and has first-order contact with the given curve; an osculating circle is an osculating curve from the family of circles, and has second-order contact (same tang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Linear Span
In mathematics, the linear span (also called the linear hull or just span) of a set S of elements of a vector space V is the smallest linear subspace of V that contains S. It is the set of all finite linear combinations of the elements of , and the intersection of all linear subspaces that contain S. It is often denoted pp. 29-30, §§ 2.5, 2.8 or \langle S \rangle. For example, in geometry, two linearly independent vectors span a plane. To express that a vector space is a linear span of a subset , one commonly uses one of the following phrases: spans ; is a spanning set of ; is spanned or generated by ; is a generator set or a generating set of . Spans can be generalized to many mathematical structures, in which case, the smallest substructure containing S is generally called the substructure ''generated'' by S. Definition Given a vector space over a field , the span of a set of vectors (not necessarily finite) is defined to be the intersection of all subsp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Normal Plane (geometry)
In geometry, a normal plane is any plane containing the normal vector of a surface at a particular point. The normal plane also refers to the plane that is perpendicular to the tangent vector of a space curve; (this plane also contains the normal vector) see Frenet–Serret formulas. Normal Section The normal section of a surface at a particular point is the curve produced by the intersection of that surface with a normal plane. The curvature of the normal section is called the ''normal curvature''. If the surface is bow or cylinder shaped, the maximum and the minimum of these curvatures are the ''principal curvatures''. If the surface is saddle shaped the maxima of both sides are the principal curvatures. The product of the principal curvatures is the Gaussian curvature of the surface (negative for saddle shaped surfaces). The mean of the principal curvatures is the ''mean curvature'' of the surface; if (and only if) the mean curvature is zero, the surface is called a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |