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Curve Sketching
In geometry, curve sketching (or curve tracing) are techniques for producing a rough idea of overall shape of a plane curve given its equation, without computing the large numbers of points required for a detailed plot. It is an application of the theory of curves to find their main features. Basic techniques The following are usually easy to carry out and give important clues as to the shape of a curve: *Determine the ''x'' and ''y'' intercepts of the curve. The ''x'' intercepts are found by setting ''y'' equal to 0 in the equation of the curve and solving for ''x''. Similarly, the ''y'' intercepts are found by setting ''x'' equal to 0 in the equation of the curve and solving for ''y''. *Determine the symmetry of the curve. If the exponent of ''x'' is always even in the equation of the curve then the ''y''-axis is an axis of symmetry for the curve. Similarly, if the exponent of ''y'' is always even in the equation of the curve then the ''x''-axis is an axis of symmetry for the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean Paul De Gua De Malves
Jean Paul de Gua de Malves (1713, Malves-en-Minervois (Aude) – June 2, 1785, Paris) was a French mathematician who published in 1740 a work on analytical geometry in which he applied it, without the aid of differential calculus, to find the tangents, asymptotes, and various Mathematical singularity, singular points of an algebraic curve. He further showed how singular points and isolated loops were affected by conical projection. He gave the proof of Descartes's rule of signs which is to be found in most modern works. It is not clear whether Descartes ever proved it strictly, and Isaac Newton, Newton seems to have regarded it as obvious. De Gua de Malves was acquainted with many of the French philosophes during the last decades of the Ancien Régime. He was an early, short-lived, participant, then editor (later replaced by Diderot) of the project that ended up as the ''Encyclopédie''. Condorcet claimed that it was in fact the de Gua who recruited Diderot to the project, though ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boundary Tracing
Boundary tracing, also known as contour tracing, of a binary digital region can be thought of as a segmentation technique that identifies the boundary pixels of the digital region. Boundary tracing is an important first step in the analysis of that region. Boundary is a topological notion. However, a digital image is no topological space. Therefore, it is impossible to define the notion of a boundary in a digital image mathematically exactly. Most publications about tracing the boundary of a subset S of a digital image I describe algorithms which find a set of pixels belonging to S and having in their direct neighborhood pixels belonging both to S and to its complement I - S. According to this definition the boundary of a subset S is different from the boundary of the complement I – S which is a topological paradox. To define the boundary correctly it is necessary to introduce a topological space corresponding to the given digital image. Such space can be a two-dimensional abst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marching Cubes
Marching cubes is a computer graphics algorithm, published in the 1987 SIGGRAPH proceedings by Lorensen and Cline, for extracting a polygonal mesh of an isosurface from a three-dimensional discrete scalar field (the elements of which are sometimes called voxels). The applications of this algorithm are mainly concerned with medical visualizations such as CT and MRI scan data images, and special effects or 3-D modelling with what is usually called metaballs or other metasurfaces. The marching cubes algorithm is meant to be used for 3-D; the 2-D version of this algorithm is called the marching squares algorithm. History The algorithm was developed by William E. Lorensen (1946-2019) and Harvey E. Cline as a result of their research for General Electric. At General Electric they worked on a way to efficiently visualize data from CT and MRI devices. The premise of the algorithm is to divide the input volume into a discrete set of cubes. By assuming linear reconstruction filte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Continuation
Numerical continuation is a method of computing approximate solutions of a system of parameterized nonlinear equations, :F(\mathbf u,\lambda) = 0. The ''parameter'' \lambda is usually a real scalar, and the ''solution'' \mathbf u an ''n''-vector. For a fixed ''parameter value'' \lambda, F(\ast,\lambda) maps Euclidean n-space into itself. Often the original mapping F is from a Banach space into itself, and the Euclidean n-space is a finite-dimensional Banach space. A steady state, or fixed point, of a parameterized family of flows or maps are of this form, and by discretizing trajectories of a flow or iterating a map, periodic orbits and heteroclinic orbits can also be posed as a solution of F=0. Other forms In some nonlinear systems, parameters are explicit. In others they are implicit, and the system of nonlinear equations is written :F(\mathbf u) = 0 where \mathbf u is an ''n''-vector, and its image F(\mathbf u) is an ''n-1'' vector. This formulation, without an e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parent Function
In mathematics, a parent function is the simplest function of a family of functions that preserves the definition (or shape) of the entire family. For example, for the family of quadratic functions having the general form : y = ax^2 + bx + c\,, the simplest function is : y = x^2. This is therefore the parent function of the family of quadratic equations. For linear and quadratic functions, the graph of any function can be obtained from the graph of the parent function by simple translations and stretches parallel to the axes. For example, the graph of ''y'' = ''x'' − 4''x'' + 7 can be obtained from the graph of ''y'' = ''x'' by translating +2 units along the X axis and +3 units along Y axis. This is because the equation can also be written as ''y'' − 3 = (''x'' − 2). For many trigonometric functions, the parent function is usually a basic sin(''x''), cos(''x''), or tan(''x''). For example, the graph of ''y ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation can be restricted to the affine algebraic plane curve of equation . These two operations are each inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered. More generally, an algebraic curve is an algebraic variety of dimension one. Equivalently, an algebraic curve is an algebraic variety that is birationally equivalent to an algebraic plane curve. If the curve is contained in an affine space or a projective space, one can take a projection for such a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locus (mathematics)
In geometry, a locus (plural: ''loci'') (Latin word for "place", "location") is a set of all points (commonly, a line, a line segment, a curve or a surface), whose location satisfies or is determined by one or more specified conditions.. In other words, the set of the points that satisfy some property is often called the ''locus of a point'' satisfying this property. The use of the singular in this formulation is a witness that, until the end of the 19th century, mathematicians did not consider infinite sets. Instead of viewing lines and curves as sets of points, they viewed them as places where a point may be ''located'' or may move. History and philosophy Until the beginning of the 20th century, a geometrical shape (for example a curve) was not considered as an infinite set of points; rather, it was considered as an entity on which a point may be located or on which it moves. Thus a circle in the Euclidean plane was defined as the ''locus'' of a point that is at a given dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geometry), point. This is the definition that appeared more than 2000 years ago in Euclid's Elements, Euclid's ''Elements'': "The [curved] line is […] the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which […] will leave from its imaginary moving some vestige in length, exempt of any width." This definition of a curve has been formalized in modern mathematics as: ''A curve is the image (mathematics), image of an interval (mathematics), interval to a topological space by a continuous function''. In some contexts, the function that defines the curve is called a ''parametrization'', and the curve is a parametric curve. In this artic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fluid Dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and hydrodynamics (the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation. Fluid dynamics offers a systematic structure—which underlies these practical disciplines—that embraces empirical and semi-empirical laws derived from flow measurement and used to solve practical problems. The solution to a fluid dynamics problem typically involves the calculation of various properties of the fluid, such as flow velocity, pressure, density, and temperature, as functions of space and time. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Streamlines, Streaklines, And Pathlines
Streamlines, streaklines and pathlines are field lines in a fluid flow. They differ only when the flow changes with time, that is, when the flow is not steady. Considering a velocity vector field in three-dimensional space in the framework of continuum mechanics, we have that: * Streamlines are a family of curves whose tangent vectors constitute the velocity vector field of the flow. These show the direction in which a massless fluid element will travel at any point in time. * Streaklines are the loci of points of all the fluid particles that have passed continuously through a particular spatial point in the past. Dye steadily injected into the fluid at a fixed point extends along a streakline. * Pathlines are the trajectories that individual fluid particles follow. These can be thought of as "recording" the path of a fluid element in the flow over a certain period. The direction the path takes will be determined by the streamlines of the fluid at each moment in time. * T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset. Convex hulls of open sets are open, and convex hulls of compact sets are compact. Every compact convex set is the convex hull of its extreme points. The convex hull operator is an example of a closure operator, and every antimatroid can be represented by applying this closure operator to finite sets of points. The algorithmic problems of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces, and its dual problem of intersecting half-spaces, are fundamental problems of com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
