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Homothetic Transformation
In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point ''S'' called its ''center'' and a nonzero number ''k'' called its ''ratio'', which sends point X to a point X' by the rule : \overrightarrow=k\overrightarrow for a fixed number k\ne 0. Using position vectors: :\mathbf x'=\mathbf s + k(\mathbf x -\mathbf s). In case of S=O (Origin): :\mathbf x'=k\mathbf x, which is a uniform scaling and shows the meaning of special choices for k: :for k=1 one gets the ''identity'' mapping, :for k=-1 one gets the ''reflection'' at the center, For 1/k one gets the ''inverse'' mapping defined by k. In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if k>0) or reverse (if k<0) the direction of all vectors. Together with the translations, all homotheties of an affine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matryoshka Doll
Matryoshka dolls ( ; rus, матрёшка, p=mɐˈtrʲɵʂkə, a=Ru-матрёшка.ogg), also known as stacking dolls, nesting dolls, Russian tea dolls, or Russian dolls, are a set of wooden dolls of decreasing size placed one inside another. The name ''matryoshka'', mainly known as "little matron", is a diminutive form of ''Matryosha'' (), in turn a diminutive of the Russian female first name ''Matryona'' (). A set of matryoshkas consists of a wooden figure, which separates at the middle, top from bottom, to reveal a smaller figure of the same sort inside, which has, in turn, another figure inside of it, and so on. The first Russian nested doll set was made in 1890 by wood turning craftsman and wood carver Vasily Zvyozdochkin from a design by Sergey Malyutin, who was a folk crafts painter at Abramtsevo. Traditionally the outer layer is a woman, dressed in a sarafan, a long and shapeless traditional Russian peasant jumper dress. The figures inside may be of any ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points, including points at infinity, can be represented using finite coordinates. Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts. Homogeneous coordinates have a range of applications, including computer graphics and 3D computer vision, where they allow affine transformations and, in general, projective transformations to be easily represented by a matrix. If homogeneous coordinates of a point are multiplied by a non-zero scalar then the resulting coordinates represent the same point. Since homogeneous coordinates are also given to points at infinity, the number of coordinates required to allow this extension is one mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Point Reflection
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invariant under point reflection through its center, it is said to possess central symmetry or to be centrally symmetric. Point reflection can be classified as an affine transformation. Namely, it is an isometric involutive affine transformation, which has exactly one fixed point, which is the point of inversion. It is equivalent to a homothetic transformation with scale factor equal to −1. The point of inversion is also called homothetic center. Terminology The term ''reflection'' is loose, and considered by some an abuse of language, with ''inversion'' preferred; however, ''point reflection'' is widely used. Such maps are involutions, meaning that they have order 2 – they are their own inverse: applying them twice yields the identi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. These three axioms hold for number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compass (drawing Tool)
A compass, more accurately known as a pair of compasses, is a technical drawing instrument that can be used for inscribing circles or arcs. As dividers, it can also be used as a tool to mark out distances, in particular, on maps. Compasses can be used for mathematics, drafting, navigation and other purposes. Prior to computerization, compasses and other tools for manual drafting were often packaged as a set with interchangeable parts. By the mid-twentieth century, circle templates supplemented the use of compasses. Today those facilities are more often provided by computer-aided design programs, so the physical tools serve mainly a didactic purpose in teaching geometry, technical drawing, etc. Construction and parts Compasses are usually made of metal or plastic, and consist of two "legs" connected by a hinge which can be adjusted to allow changing of the radius of the circle drawn. Typically one leg has a spike at its end for anchoring, and the other leg holds a drawing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pantograph
A pantograph (, from their original use for copying writing) is a mechanical linkage connected in a manner based on parallelograms so that the movement of one pen, in tracing an image, produces identical movements in a second pen. If a line drawing is traced by the first point, an identical, enlarged, or miniaturized copy will be drawn by a pen fixed to the other. Using the same principle, different kinds of pantographs are used for other forms of duplication in areas such as sculpting, minting, engraving, and milling. Because of the shape of the original device, a pantograph also refers to a kind of structure that can compress or extend like an accordion, forming a characteristic rhomboidal pattern. This can be found in extension arms for wall-mounted mirrors, temporary fences, pantographic knives, scissor lifts, and other scissor mechanisms such as the pantograph used on electric locomotives and trams. History The ancient Greek engineer Hero of Alexandria described ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pantograph01
A pantograph (, from their original use for copying writing) is a mechanical linkage connected in a manner based on parallelograms so that the movement of one pen, in tracing an image, produces identical movements in a second pen. If a line drawing is traced by the first point, an identical, enlarged, or miniaturized copy will be drawn by a pen fixed to the other. Using the same principle, different kinds of pantographs are used for other forms of duplication in areas such as sculpting, minting, engraving, and milling. Because of the shape of the original device, a pantograph also refers to a kind of structure that can compress or extend like an accordion, forming a characteristic rhomboidal pattern. This can be found in extension arms for wall-mounted mirrors, temporary fences, pantographic knives, scissor lifts, and other scissor mechanisms such as the pantograph used on electric locomotives and trams. History The ancient Greek engineer Hero of Alexandria described ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pantograph Animation
A pantograph (, from their original use for copying writing) is a mechanical linkage connected in a manner based on parallelograms so that the movement of one pen, in tracing an image, produces identical movements in a second pen. If a line drawing is traced by the first point, an identical, enlarged, or miniaturized copy will be drawn by a pen fixed to the other. Using the same principle, different kinds of pantographs are used for other forms of duplication in areas such as sculpting, minting, engraving, and milling. Because of the shape of the original device, a pantograph also refers to a kind of structure that can compress or extend like an accordion, forming a characteristic rhomboidal pattern. This can be found in extension arms for wall-mounted mirrors, temporary fences, pantographic knives, scissor lifts, and other scissor mechanisms such as the pantograph used on electric locomotives and trams. History The ancient Greek engineer Hero of Alexandria described ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
