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Normal Vector
In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at the point. A normal vector may have length one (a unit vector) or its length may represent the curvature of the object (a '' curvature vector''); its algebraic sign may indicate sides (interior or exterior). In three dimensions, a surface normal, or simply normal, to a surface at point P is a vector perpendicular to the tangent plane of the surface at P. The word "normal" is also used as an adjective: a line ''normal'' to a plane, the ''normal'' component of a force, the normal vector, etc. The concept of normality generalizes to orthogonality ( right angles). The concept has been generalized to differentiable manifolds of arbitrary dimension embedded in a Euclidean space. The normal vector space or normal space of a manifold a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Normal Vectors2
Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Normal'' (2009 film), an adaptation of Anthony Neilson's 1991 play ''Normal: The Düsseldorf Ripper'' * ''Normal!'', a 2011 Algerian film * ''The Normals'' (film), a 2012 American comedy film * "Normal" (''New Girl''), an episode of the TV series Mathematics * Normal (geometry), an object such as a line or vector that is perpendicular to a given object * Normal basis (of a Galois extension), used heavily in cryptography * Normal bundle * Normal cone, of a subscheme in algebraic geometry * Normal coordinates, in differential geometry, local coordinates obtained from the exponential map (Riemannian geometry) * Normal distribution, the Gaussian continuous probability distribution * Normal equations, describing the solution of the linear least sq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Force
A force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. The concept of force makes the everyday notion of pushing or pulling mathematically precise. Because the Magnitude (mathematics), magnitude and Direction (geometry, geography), direction of a force are both important, force is a Euclidean vector, vector quantity. The SI unit of force is the newton (unit), newton (N), and force is often represented by the symbol . Force plays an important role in classical mechanics. The concept of force is central to all three of Newton's laws of motion. Types of forces often encountered in classical mechanics include Elasticity (physics), elastic, frictional, Normal force, contact or "normal" forces, and gravity, gravitational. The rotational version of force is torque, which produces angular acceleration, changes in the rotational speed of an object. In an extended body, each part often applies forces on the adjacent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Foot Of A Perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can be defined between two lines (or two line segments), between a line and a plane, and between two planes. Perpendicularity is one particular instance of the more general mathematical concept of ''orthogonality''; perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the word "perpendicular" is sometimes used to describe much more complicated geometric orthogonality conditions, such as that between a surface and its ''normal vector''. Definitions A line is said to be perpendicular to another line if the two lines intersect at a right angle. Explicitly, a first line is perpendicular to a second line if (1) the two lines meet; and (2) at the point of intersection the straight angle on one side ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Phong Shading
In 3D computer graphics, Phong shading, Phong interpolation, or normal-vector interpolation shading is an interpolation technique for surface shading invented by computer graphics pioneer Bui Tuong Phong. Phong shading interpolates surface normals across rasterized polygons and computes pixel colors based on the interpolated normals and a reflection model. ''Phong shading'' may also refer to the specific combination of Phong interpolation and the Phong reflection model. History Phong shading and the Phong reflection model were developed at the University of Utah by Bui Tuong Phong, who published them in his 1973 Ph.D. dissertation and a 1975 paper.Bui Tuong Phong, "Illumination for Computer Generated Pictures," Comm. ACM, Vol 18(6):311-317, June 1975. Phong's methods were considered radical at the time of their introduction, but have since become the de facto baseline shading method for many rendering applications. Phong's methods have proven popular due to their generall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Vertex (geometry)
In geometry, a vertex (in plural form: vertices or vertexes) is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. Definition Of an angle The ''vertex'' of an angle is the point where two rays begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments, and lines that result in two straight "sides" meeting at one place. :(3 vols.): (vol. 1), (vol. 2), (vol. 3). Of a polytope A vertex is a corner point of a polygon, polyhedron, or other higher-dimensional polytope, formed by the intersection of edges, faces or facets of the object. In a polygon, a vertex is called " convex" if the internal angle of the polygon (i.e., the angle formed by the two edges at the vertex with the polygon inside the angle) is less than π radians (180°, two right angl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Flat Shading
Shading refers to the depiction of depth perception in 3D models (within the field of 3D computer graphics) or illustrations (in visual art) by varying the level of darkness. Shading tries to approximate local behavior of light on the object's surface and is not to be confused with techniques of adding shadows, such as shadow mapping or shadow volumes, which fall under global behavior of light. In drawing Shading is used traditionally in drawing for depicting a range of darkness by applying media more densely or with a darker shade for darker areas, and less densely or with a lighter shade for lighter areas. Light patterns, such as objects having light and shaded areas, help when creating the illusion of depth on paper. There are various techniques of shading, including cross hatching, where perpendicular lines of varying closeness are drawn in a grid pattern to shade an area. The closer the lines are together, the darker the area appears. Likewise, the farther apart the li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Light Source
Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 terahertz, between the infrared (with longer wavelengths) and the ultraviolet (with shorter wavelengths). In physics, the term "light" may refer more broadly to electromagnetic radiation of any wavelength, whether visible or not. In this sense, gamma rays, X-rays, microwaves and radio waves are also light. The primary properties of light are intensity, propagation direction, frequency or wavelength spectrum and polarization. Its speed in a vacuum, 299 792 458 metres a second (m/s), is one of the fundamental constants of nature. Like all types of electromagnetic radiation, visible light propagates by massless elementary particles called photons that represents the quanta of electromagnetic field, and can be analyzed as both waves and particle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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3D Computer Graphics
3D computer graphics, or “3D graphics,” sometimes called CGI, 3D-CGI or three-dimensional computer graphics are graphics that use a three-dimensional representation of geometric data (often Cartesian) that is stored in the computer for the purposes of performing calculations and rendering digital images, usually 2D images but sometimes 3D images. The resulting images may be stored for viewing later (possibly as an animation) or displayed in real time. 3D computer graphics, contrary to what the name suggests, are most often displayed on two-dimensional displays. Unlike 3D film and similar techniques, the result is two-dimensional, without visual depth. More often, 3D graphics are being displayed on 3D displays, like in virtual reality systems. 3D graphics stand in contrast to 2D computer graphics which typically use completely different methods and formats for creation and rendering. 3D computer graphics rely on many of the same algorithms as 2D computer vector gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Differential Geometry Of Surfaces
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: ''extrinsically'', relating to their embedding in Euclidean space and ''intrinsically'', reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space. Surfaces naturally arise as graphs of functions of a pair of variables, and sometimes appear in parametric form or as loci associated to space curves. An important role in their study has been played by Lie groups (in the spirit of the Erlangen program), namely the symmetry groups of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Differential Geometry Of Curves
Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus. Many specific curves have been thoroughly investigated using the synthetic approach. Differential geometry takes another path: curves are represented in a parametrized form, and their geometric properties and various quantities associated with them, such as the curvature and the arc length, are expressed via derivatives and integrals using vector calculus. One of the most important tools used to analyze a curve is the Frenet frame, a moving frame that provides a coordinate system at each point of the curve that is "best adapted" to the curve near that point. The theory of curves is much simpler and narrower in scope than the theory of surfaces and its higher-dimensional generalizations because a regular curve in a Euclidean space has no intrinsic geometry. Any regular curve may be parametrized by t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Tangent Space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a manifold at a point can be viewed as the space of possible velocities for a particle moving on the manifold. Informal description In differential geometry, one can attach to every point x of a differentiable manifold a ''tangent space''—a real vector space that intuitively contains the possible directions in which one can tangentially pass through x . The elements of the tangent space at x are called the '' tangent vectors'' at x . This is a generalization of the notion of a vector, based at a given initial point, in a Euclidean space. The dimension of the tangent space at every point of a connected manifold is the same as that of the manifold itself. For example, if the given manifold is a 2 - sphere, then one can pi ... [...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, that is, 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, including the three-dimensional space and the ''Euclidean plane'' (dimension two). 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 evident (for example, there is exactly one straight line passing through two points), or seemed impossible to prove (paral ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |