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List Of Curves Topics
This is an alphabetical index of articles related to curves used in mathematics. * Acnode * Algebraic curve * Arc * Asymptote * Asymptotic curve * Barbier's theorem * Bézier curve * Bézout's theorem * Birch and Swinnerton-Dyer conjecture * Bitangent * Bitangents of a quartic * Cartesian coordinate system * Caustic * Cesàro equation * Chord (geometry) * Cissoid * Circumference * Closed timelike curve * concavity * Conchoid (mathematics) * Confocal * Contact (mathematics) * Contour line * Crunode * Cubic Hermite curve * Curvature * Curve orientation * Curve fitting ** Curve-fitting compaction * Curve of constant width * Curve of pursuit * Curves in differential geometry * Cusp * Cyclogon * De Boor algorithm * Differential geometry of curves * Eccentricity (mathematics) * Elliptic curve cryptography * Envelope (mathematics) * Fenchel's theorem * Genus (mathematics) * Geodesic * Geometric genus * Great-circle distance * Harmonograph * Hedgehog (curve)br>* Hilbert's sixteenth ... [...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] |
Cissoid
In geometry, a cissoid (() is a plane curve generated from two given curves , and a point (the pole). Let be a variable line passing through and intersecting at and at . Let be the point on so that \overline = \overline. (There are actually two such points but is chosen so that is in the same direction from as is from .) Then the locus of such points is defined to be the cissoid of the curves , relative to . Slightly different but essentially equivalent definitions are used by different authors. For example, may be defined to be the point so that \overline = \overline + \overline. This is equivalent to the other definition if is replaced by its reflection through . Or may be defined as the midpoint of and ; this produces the curve generated by the previous curve scaled by a factor of 1/2. Equations If and are given in polar coordinates by r=f_1(\theta) and r=f_2(\theta) respectively, then the equation r=f_2(\theta)-f_1(\theta) describes the cissoid of and r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curve-fitting Compaction
Curve-fitting compaction is data compaction accomplished by replacing data to be stored or transmitted with an analytical expression. Examples of curve-fitting compaction consisting of discretization and then interpolation are: * Breaking of a continuous curve into a series of straight line segments and specifying the slope, intercept, and range for each segment * Using a mathematical expression, such as a polynomial or a trigonometric function In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ..., and a single point on the corresponding curve instead of storing or transmitting the entire graphic curve or a series of points on it. References Curves Interpolation Data compression {{compu-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curve Fitting
Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is required, or smoothing, in which a "smooth" function is constructed that approximately fits the data. A related topic is regression analysis, which focuses more on questions of statistical inference such as how much uncertainty is present in a curve that is fit to data observed with random errors. Fitted curves can be used as an aid for data visualization, to infer values of a function where no data are available, and to summarize the relationships among two or more variables. Extrapolation refers to the use of a fitted curve beyond the range of the observed data, and is subject to a degree of uncertainty since it may reflect the method used to construct the curve as much as it reflects the observed data. For linear-algebraic analysis o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curve Orientation
In mathematics, an orientation of a curve is the choice of one of the two possible directions for travelling on the curve. For example, for Cartesian coordinates, the -axis is traditionally oriented toward the right, and the -axis is upward oriented. In the case of a planar simple closed curve (that is, a curve in the plane whose starting point is also the end point and which has no other self-intersections), the curve is said to be positively oriented or counterclockwise oriented, if one always has the curve interior to the left (and consequently, the curve exterior to the right), when traveling on it. Otherwise, that is if left and right are exchanged, the curve is negatively oriented or clockwise oriented. This definition relies on the fact that every simple closed curve admits a well-defined interior, which follows from the Jordan curve theorem. The inner loop of a beltway road in a country where people drive on the right side of the road is an example of a negatively orie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature ''at a point'' of a differentiable curve is the curvature of its osculating circle, that is the circle that best approximates the curve near this point. The curvature of a straight line is zero. In contrast to the tangent, which is a vector quantity, the curvature at a point is typically a scalar quantity, that is, it is expressed by a single real number. For surfaces (and, more generally for higher-dimensional manifolds), that are embedded in a Euclidean space, the concept of curvature is more complex, as it depends on the choice of a direction on the surface or man ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Hermite Curve
In numerical analysis, a cubic Hermite spline or cubic Hermite interpolator is a spline where each piece is a third-degree polynomial specified in Hermite form, that is, by its values and first derivatives at the end points of the corresponding domain interval. Cubic Hermite splines are typically used for interpolation of numeric data specified at given argument values x_1,x_2,\ldots,x_n, to obtain a continuous function. The data should consist of the desired function value and derivative at each x_k. (If only the values are provided, the derivatives must be estimated from them.) The Hermite formula is applied to each interval (x_k, x_) separately. The resulting spline will be continuous and will have continuous first derivative. Cubic polynomial splines can be specified in other ways, the Bezier cubic being the most common. However, these two methods provide the same set of splines, and data can be easily converted between the Bézier and Hermite forms; so the names are of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crunode
In mathematics, a crunode (archaic) or node is a point where a curve intersects itself so that both branches of the curve have distinct tangent lines at the point of intersection. A crunode is also known as an ''ordinary double point''. For a plane curve, defined as the locus of points , where is a smooth function of variables and ranging over the real numbers, a crunode of the curve is a singularity of the function , where both partial derivatives \tfrac and \tfrac vanish. Further the Hessian matrix of second derivatives will have both positive and negative eigenvalues. See also *Singular point of a curve *Acnode *Cusp *Tacnode In classical algebraic geometry, a tacnode (also called a point of osculation or double cusp). is a kind of singular point of a curve. It is defined as a point where two (or more) osculating circles to the curve at that point are tangent I ... * Saddle point References Curves Algebraic curves {{differential-geometry-stub es: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contour Line
A contour line (also isoline, isopleth, or isarithm) of a 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 plane section of the 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 the function is always perpendicular to the contour lines. When the lines are close together the magnitude of the grad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contact (mathematics)
In mathematics, two functions have a contact of order ''k'' if, at a point ''P'', they have the same value and ''k'' equal derivatives. This is an equivalence relation, whose equivalence classes are generally called jets. The point of osculation is also called the double cusp. Contact is a geometric notion; it can be defined algebraically as a valuation. One speaks also of curves and geometric objects having ''k''-th order contact at a point: this is also called ''osculation'' (i.e. kissing), generalising the property of being tangent. (Here the derivatives are considered with respect to arc length.) An osculating curve from a given family of curves is a curve that has the highest possible order of contact with a given curve at a given point; for instance a tangent line is an osculating curve from the family of lines, and has first-order contact with the given curve; an osculating circle is an osculating curve from the family of circles, and has second-order contact (same tan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Confocal
In geometry, confocal means having the same foci: confocal conic sections. * For an optical cavity consisting of two mirrors, confocal means that they share their foci. If they are identical mirrors, their radius of curvature, ''R''mirror, equals ''L'', where ''L'' is the distance between the mirrors. * In conic sections, it is said of two ellipses, two hyperbolas, or an ellipse and a hyperbola which share both foci with each other. If an ellipse and a hyperbola are confocal, they are perpendicular to each other. * In optics, it means that one focus or image point of one lens is the same as one focus of the next lens. See also *Confocal laser scanning microscopy *Confocal microscopy Confocal microscopy, most frequently confocal laser scanning microscopy (CLSM) or laser confocal scanning microscopy (LCSM), is an optical imaging technique for increasing optical resolution and contrast of a micrograph by means of using a sp ... * {{set index article, mathematics Elementary geo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conchoid (mathematics)
In geometry, a conchoid is a curve derived from a fixed point , another curve, and a length . It was invented by the ancient Greek mathematician Nicomedes. Description For every line through that intersects the given curve at the two points on the line which are from are on the conchoid. The conchoid is, therefore, the cissoid of the given curve and a circle of radius and center . They are called ''conchoids'' because the shape of their outer branches resembles conch shells. The simplest expression uses polar coordinates with at the origin. If :r=\alpha(\theta) expresses the given curve, then :r=\alpha(\theta)\pm d expresses the conchoid. If the curve is a line, then the conchoid is the ''conchoid of Nicomedes''. For instance, if the curve is the line , then the line's polar form is and therefore the conchoid can be expressed parametrically as :x=a \pm d \cos \theta,\, y=a \tan \theta \pm d \sin \theta. A limaçon is a conchoid with a circle as the given curve. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
