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Isophote
In geometry, an isophote is a curve on an illuminated surface that connects points of equal brightness. One supposes that the illumination is done by parallel light and the brightness is measured by the following scalar product: :b(P)= \vec n(P)\cdot \vec v=\cos\varphi where is the unit normal vector of the surface at point and the unit vector of the light's direction. If , i.e. the light is perpendicular to the surface normal, then point is a point of the surface silhouette observed in direction Brightness 1 means that the light vector is perpendicular to the surface. A plane has no isophotes, because every point has the same brightness. In astronomy, an isophote is a curve on a photo connecting points of equal brightness. J. Binney, M. Merrifield: ''Galactic Astronomy'', Princeton University Press, 1998, , p. 178. Application and example In computer-aided design, isophotes are used for checking optically the smoothness of surface connections. For a surface (implicit o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Continuity
In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives (''differentiability class)'' it has over its Domain of a function, domain. A function of class C^k is a function of smoothness at least ; that is, a function of class C^k is a function that has a th derivative that is continuous in its domain. A function of class C^\infty or C^\infty-function (pronounced C-infinity function) is an infinitely differentiable function, that is, a function that has derivatives of all Order of derivation, orders (this implies that all these derivatives are continuous). Generally, the term smooth function refers to a C^-function. However, it may also mean "sufficiently differentiable" for the problem under consideration. Differentiability classes Differentiability class is a classification of functions according to the properties of their derivatives. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contour Line
A contour line (also isoline, isopleth, isoquant or isarithm) of a Function of several real variables, function of two variables is a curve along which the function has a constant value, so that the curve joins points of equal value. It is a cross-section (geometry)#Definition, plane section of the graph of a function of two variables, three-dimensional graph of the function f(x,y) parallel to the (x,y)-plane. More generally, a contour line for a function of two variables is a curve connecting points where the function has the same particular value. In cartography, a contour line (often just called a "contour") joins points of equal elevation (height) above a given level, such as mean sea level. A contour map is a map illustrated with contour lines, for example a topographic map, which thus shows valleys and hills, and the steepness or gentleness of slopes. The contour interval of a contour map is the difference in elevation between successive contour lines. The gradient of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Implicit Curve
In mathematics, an implicit curve is a plane curve defined by an implicit equation relating two coordinate variables, commonly ''x'' and ''y''. For example, the unit circle is defined by the implicit equation x^2+y^2=1. In general, every implicit curve is defined by an equation of the form : F(x,y)=0 for some function ''F'' of two variables. Hence an implicit curve can be considered as the set of zeros of a function of two variables. ''Implicit'' means that the equation is not expressed as a solution for either ''x'' in terms of ''y'' or vice versa. If F(x,y) is a polynomial in two variables, the corresponding curve is called an ''algebraic curve'', and specific methods are available for studying it. Plane curves can be represented in Cartesian coordinates (''x'', ''y'' coordinates) by any of three methods, one of which is the implicit equation given above. The graph of a function is usually described by an equation y=f(x) in which the functional form is explicitly stated; this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parametric Surface
A parametric surface is a surface in the Euclidean space \R^3 which is defined by a parametric equation with two parameters Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence theorem, are frequently given in a parametric form. The curvature and arc length of curves on the surface, surface area, differential geometric invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization. Examples * The simplest type of parametric surfaces is given by the graphs of functions of two variables: z = f(x,y), \quad \mathbf r(x,y) = (x, y, f(x,y)). * A rational surface is a surface that admits parameterizations by a rational function. A rational surface is an algebraic surface. Given an algebraic surface, it is commonly easier to decide if it i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygon
In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon's '' vertices'' or ''corners''. An ''n''-gon is a polygon with ''n'' sides; for example, a triangle is a 3-gon. A simple polygon is one which does not intersect itself. More precisely, the only allowed intersections among the line segments that make up the polygon are the shared endpoints of consecutive segments in the polygonal chain. A simple polygon is the boundary of a region of the plane that is called a ''solid polygon''. The interior of a solid polygon is its ''body'', also known as a ''polygonal region'' or ''polygonal area''. In contexts where one is concerned only with simple and solid polygons, a ''polygon'' may refer only to a simple polygon or to a solid polygon. A polygonal chain may cross over itself, creating star polyg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intersection Curve
In geometry, an intersection curve is a curve that is common to two geometric objects. In the simplest case, the intersection of two non-parallel planes in Euclidean 3-space is a line. In general, an intersection curve consists of the common points of two ''transversally'' intersecting surfaces, meaning that at any common point the surface normals are not parallel. This restriction excludes cases where the surfaces are touching or have surface parts in common. The analytic determination of the intersection curve of two surfaces is easy only in simple cases; for example: a) the intersection of two planes, b) plane section of a quadric (sphere, cylinder, cone, etc.), c) intersection of two quadrics in special cases. For the general case, literature provides algorithms, in order to calculate points of the intersection curve of two surfaces. Intersection line of two planes Given: two planes \varepsilon_i: \quad \vec n_i\cdot\vec x=d_i, \quad i=1,2, \quad \vec n_1,\vec n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear System
In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists since most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems. Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one. In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a lin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Implicit Surface
In mathematics, an implicit surface is a Surface (geometry), surface in Euclidean space defined by an equation : F(x,y,z)=0. An ''implicit surface'' is the set of Zero of a function, zeros of a Function of several real variables, function of three variables. ''Implicit function, Implicit'' means that the equation is not solved for or or . The graph of a function is usually described by an equation z=f(x,y) and is called an ''explicit'' representation. The third essential description of a surface is the ''Parametric equation, parametric'' one: (x(s,t),y(s,t), z(s,t)), where the -, - and -coordinates of surface points are represented by three functions x(s,t)\, , y(s,t)\, , z(s,t) depending on common parameters s,t. Generally the change of representations is simple only when the explicit representation z=f(x,y) is given: z-f(x,y)=0 (implicit), (s,t,f(s,t)) (parametric). ''Examples'': #The Plane (geometry), plane x+2y-3z+1=0. #The Sphere (geometry), sphere x^2+y^2+z^2-4=0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer-aided Design
Computer-aided design (CAD) is the use of computers (or ) to aid in the creation, modification, analysis, or optimization of a design. This software is used to increase the productivity of the designer, improve the quality of design, improve communications through documentation, and to create a database for manufacturing. Designs made through CAD software help protect products and inventions when used in patent applications. CAD output is often in the form of electronic files for print, machining, or other manufacturing operations. The terms computer-aided drafting (CAD) and computer-aided design and drafting (CADD) are also used. Its use in designing electronic systems is known as ''electronic design automation'' (''EDA''). In mechanical design it is known as ''mechanical design automation'' (''MDA''), which includes the process of creating a technical drawing with the use of computer software. CAD software for mechanical design uses either vector-based graphics to depict t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
