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Reparameterization
In mathematics, and more specifically in geometry, parametrization (or parameterization; also parameterisation, parametrisation) is the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, defined by an implicit equation. The inverse process is called implicitization. "To parameterize" by itself means "to express in terms of parameters". Parametrization is a mathematical process consisting of expressing the state of a system, process or model as a function of some independent quantities called parameters. The state of the system is generally determined by a finite set of coordinates, and the parametrization thus consists of one function of several real variables for each coordinate. The number of parameters is the number of degrees of freedom of the system. For example, the position of a point that moves on a curve in three-dimensional space is determined by the time needed to reach the point when starting from a fixe ... [...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] |
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] |
CMYK
The CMYK color model (also known as process color, or four color) is a subtractive color model, based on the CMY color model, used in color printing, and is also used to describe the printing process itself. The abbreviation ''CMYK'' refers to the four ink plates used: cyan, magenta, yellow, and key (most often black). The CMYK model works by partially or entirely masking colors on a lighter, usually white, background. The ink reduces the light that would otherwise be reflected. Such a model is called ''subtractive'', as inks ''subtract'' some colors from white light; in the CMY model, white light minus red leaves cyan, white light minus green leaves magenta, and white light minus blue leaves yellow. In additive color models, such as RGB, white is the ''additive'' combination of all primary colored lights, and black is the absence of light. In the CMYK model, it is the opposite: white is the natural color of the paper or other background, and black results from a ful ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trichromacy
Trichromacy or trichromatism is the possession of three independent channels for conveying color information, derived from the three different types of cone cells in the eye. Organisms with trichromacy are called trichromats. The normal explanation of trichromacy is that the organism's retina contains three types of color receptors (called cone cells in vertebrates) with different absorption spectra. In actuality, the number of such receptor types may be greater than three, since different types may be active at different light intensities. In vertebrates with three types of cone cells, at low light intensities the rod cells may contribute to color vision. Humans and other animals that are trichromats Humans and some other mammals have evolved trichromacy based partly on pigments inherited from early vertebrates. In fish and birds, for example, four pigments are used for vision. These extra cone receptor visual pigments detect energy of other wavelengths, sometimes includi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coordinate System
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are not interchangeable; they are commonly distinguished by their position in an ordered tuple, or by a label, such as in "the ''x''-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and ''vice versa''; this is the basis of analytic geometry. Common coordinate systems Number line The simplest example of a coordinate system is the identification of points on a line with real numbers using the '' number line''. In this system, an arbitrary point ''O'' (the ''origin'') is chosen on a given line. The coordinate o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Coordinate
In mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinates. These are * the radial distance along the line connecting the point to a fixed point called the origin; * the polar angle between this radial line and a given ''polar axis''; and * the azimuthal angle , which is the angle of rotation of the radial line around the polar axis. (See graphic regarding the "physics convention".) Once the radius is fixed, the three coordinates (''r'', ''θ'', ''φ''), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates. The plane passing through the origin and perpendicular to the polar axis (where the polar angle is a right angle) is called the ''reference plane'' (sometimes '' fundamental plane''). Terminology The radial distance from the fixed point of origin is also called the ''radius'', or ''radial line'', or ''radial coordi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signed Distance Function
In mathematics and its applications, the signed distance function or signed distance field (SDF) is the orthogonal distance of a given point ''x'' to the boundary of a set Ω in a metric space (such as the surface of a geometric shape), with the sign determined by whether or not ''x'' is in the interior of Ω. The function has positive values at points ''x'' inside Ω, it decreases in value as ''x'' approaches the boundary of Ω where the signed distance function is zero, and it takes negative values outside of Ω. However, the alternative convention is also sometimes taken instead (i.e., negative inside Ω and positive outside). The concept also sometimes goes by the name oriented distance function/field. Definition Let be a subset of a metric space with metric , and \partial\Omega be its boundary. The distance between a point of and the subset \partial\Omega of is defined as usual as : d(x, \partial \Omega) = \inf_d(x, y), where \inf denotes the infimum. The ''sign ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Azimuth
An azimuth (; from ) is the horizontal angle from a cardinal direction, most commonly north, in a local or observer-centric spherical coordinate system. Mathematically, the relative position vector from an observer ( origin) to a point of interest is projected perpendicularly onto a reference plane (the horizontal plane); the angle between the projected vector and a reference vector on the reference plane is called the azimuth. When used as a celestial coordinate, the azimuth is the horizontal direction of a star or other astronomical object in the sky. The star is the point of interest, the reference plane is the local area (e.g. a circular area with a 5 km radius at sea level) around an observer on Earth's surface, and the reference vector points to true north. The azimuth is the angle between the north vector and the star's vector on the horizontal plane. Azimuth is usually measured in degrees (°), in the positive range 0° to 360° or in the signed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radial Distance (geometry)
In mathematics, the polar coordinate system specifies a given point in a plane by using a distance and an angle as its two coordinates. These are *the point's distance from a reference point called the ''pole'', and *the point's direction from the pole relative to the direction of the ''polar axis'', a ray drawn from the pole. The distance from the pole is called the ''radial coordinate'', ''radial distance'' or simply ''radius'', and the angle is called the ''angular coordinate'', ''polar angle'', or ''azimuth''. The pole is analogous to the origin in a Cartesian coordinate system. Polar coordinates are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point in a plane, such as spirals. Planar physical systems with bodies moving around a central point, or phenomena originating from a central point, are often simpler and more intuitive to model using polar coordinates. The polar coordinate system is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cylindrical Polar Coordinates
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 r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Coordinate
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular oriented lines, called '' coordinate lines'', ''coordinate axes'' or just ''axes'' (plural of ''axis'') of the system. The point where the axes meet is called the '' origin'' and has as coordinates. The axes directions represent an orthogonal basis. The combination of origin and basis forms a coordinate frame called the Cartesian frame. Similarly, the position of any point in three-dimensional space can be specified by three ''Cartesian coordinates'', which are the signed distances from the point to three mutually perpendicular planes. More generally, Cartesian coordinates specify the point in an -dimensional Euclidean space for any dimension . These coordinates are the signed distances from the point to mutually perpendicular fixed h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Three-dimensional
In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (''coordinates'') are required to determine the position (geometry), position of a point (geometry), point. Most commonly, it is the three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which models physical space. More general three-dimensional spaces are called ''3-manifolds''. The term may also refer colloquially to a subset of space, a ''three-dimensional region'' (or 3D domain (mathematical analysis), domain), a ''solid figure''. Technically, a tuple of Real number, numbers can be understood as the Cartesian coordinates of a location in a -dimensional Euclidean space. The set of these -tuples is commonly denoted \R^n, and can be identified to the pair formed by a -dimensional Euclidean space and a Cartesian coordinate system. When , this space is called the three-dimensional Euclidean space (or simply "E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
