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Right Strophoid
In geometry, a strophoid is a curve generated from a given curve and points (the fixed point) and (the pole) as follows: Let be a variable line passing through and intersecting at . Now let and be the two points on whose distance from is the same as the distance from to (i.e. ). The locus of such points and is then the strophoid of with respect to the pole and fixed point . Note that and are at right angles in this construction. In the special case where is a line, lies on , and is not on , then the curve is called an oblique strophoid. If, in addition, is perpendicular to then the curve is called a right strophoid, or simply ''strophoid'' by some authors. The right strophoid is also called the logocyclic curve or foliate. Equations Polar coordinates Let the curve be given by r = f(\theta), where the origin is taken to be . Let be the point . If K = (r \cos\theta,\ r \sin\theta) is a point on the curve the distance from to is :d = \sqrt = \sqrt. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crunode
In mathematics, a crunode (archaic; from Latin ''crux'' "cross" + ''node'') or node of an algebraic curve is a type of singular point at which the 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''. In the case of a smooth real plane curve , a point is a crunode provided that both first partial derivatives vanish \frac = \frac = 0 and the Hessian determinant is negative: \frac \frac - \left(\frac\right)^2 < 0. See also * * Acnode * * ...[...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] |
Circular Points At Infinity
In projective geometry, the circular points at infinity (also called cyclic points or isotropic points) are two special points at infinity in the complex projective plane that are contained in the complexification of every real circle. Coordinates A point of the complex projective plane may be described in terms of homogeneous coordinates, being a triple of complex numbers , where two triples describe the same point of the plane when the coordinates of one triple are the same as those of the other aside from being multiplied by the same nonzero factor. In this system, the points at infinity may be chosen as those whose ''z''-coordinate is zero. The two circular points at infinity are two of these, usually taken to be those with homogeneous coordinates : and . Trilinear coordinates Let ''A''. ''B''. ''C'' be the measures of the vertex angles of the reference triangle ABC. Then the trilinear coordinates of the circular points at infinity in the plane of the reference triangle ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptote
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity. The word asymptote is derived from the Greek ἀσύμπτωτος (''asumptōtos'') which means "not falling together", from ἀ priv. + σύν "together" + πτωτ-ός "fallen". The term was introduced by Apollonius of Perga in his work on conic sections, but in contrast to its modern meaning, he used it to mean any line that does not intersect the given curve. There are three kinds of asymptotes: ''horizontal'', ''vertical'' and ''oblique''. For curves given by the graph of a function , horizontal asymptotes are horizontal lines that the graph of the function approaches as ''x'' tends to Vertical asymptotes are vertical lines near which the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Folium Of Descartes
In geometry, the folium of Descartes (; named for René Descartes) is an algebraic curve defined by the implicit equation x^3+y^3-3a\cdot xy=0. dy/dx=(x^2-ay)/(ax-y^2), \,dx/dy=(ax-y^2)/(x^2-ay). History The curve was first proposed and studied by René Descartes in 1638. Its claim to fame lies in an incident in the development of calculus. Descartes challenged Pierre de Fermat to find the tangent line to the curve at an arbitrary point, since Fermat had recently discovered a method for finding tangent lines. Fermat solved the problem easily, something Descartes was unable to do. Since the invention of calculus, the slope of the tangent line can be found easily using implicit differentiation. Mayor Johan(nes) Hudde's second letter on maxima and minima (1658) mentions his calculation of the maximum width of the closed loop, part of ''Mathematical Exercitions'', 5 books (1656/57 Leyden) p. 498, by Frans van Schooten Jnr. Graphing the curve The folium of Descartes can be expr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Coordinate System
In geometry, a Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of real numbers called ''coordinates'', which are the positive and negative numbers, 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 (mathematics), origin'' and has as coordinates. The axes direction (geometry), 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 di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gerade Strophoide Rechts
In physics, a parity transformation (also called parity inversion) is the flip in the sign of ''one'' spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a point reflection): \mathbf: \beginx\\y\\z\end \mapsto \begin-x\\-y\\-z\end. It can also be thought of as a test for chirality of a physical phenomenon, in that a parity inversion transforms a phenomenon into its mirror image. All fundamental interactions of elementary particles, with the exception of the weak interaction, are symmetric under parity transformation. As established by the Wu experiment conducted at the US National Bureau of Standards by Chinese-American scientist Chien-Shiung Wu, the weak interaction is chiral and thus provides a means for probing chirality in physics. In her experiment, Wu took advantage of the controlling role of weak interactions in radioactive decay of atomic isotopes to establish the chirality of the weak force. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sectrix Of Maclaurin
In geometry, a sectrix of Maclaurin is defined as the curve swept out by the point of intersection of two lines which are each revolving at constant rates about different points called poles. Equivalently, a sectrix of Maclaurin can be defined as a curve whose equation in biangular coordinates is linear. The name is derived from the trisectrix of Maclaurin (named for Colin Maclaurin), which is a prominent member of the family, and their sectrix property, which means they can be used to divide an angle into a given number of equal parts. There are special cases known as arachnida or araneidans because of their spider-like shape, and Plateau curves after Joseph Plateau who studied them. Equations in polar coordinates We are given two lines rotating about two poles P and P_1. By translation and rotation we may assume P = (0,0) and P_1 = (a, 0). At time t, the line rotating about P has angle \theta = \kappa t + \alpha and the line rotating about P_1 has angle \theta_1 = \kappa_1 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] |
Law Of Sines
In trigonometry, the law of sines (sometimes called the sine formula or sine rule) is a mathematical equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, \frac \,=\, \frac \,=\, \frac \,=\, 2R, where , and are the lengths of the sides of a triangle, and , and are the opposite angles (see figure 2), while is the radius of the triangle's circumcircle. When the last part of the equation is not used, the law is sometimes stated using the Multiplicative inverse, reciprocals; \frac \,=\, \frac \,=\, \frac. The law of sines can be used to compute the remaining sides of a triangle when two angles and a side are known—a technique known as triangulation. It can also be used when two sides and one of the non-enclosed angles are known. In some such cases, the triangle is not uniquely determined by this data (called the ''ambiguous case'') and the technique gives two possible values for the enclosed angle. The law of sines is on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sectrix Of Maclaurin
In geometry, a sectrix of Maclaurin is defined as the curve swept out by the point of intersection of two lines which are each revolving at constant rates about different points called poles. Equivalently, a sectrix of Maclaurin can be defined as a curve whose equation in biangular coordinates is linear. The name is derived from the trisectrix of Maclaurin (named for Colin Maclaurin), which is a prominent member of the family, and their sectrix property, which means they can be used to divide an angle into a given number of equal parts. There are special cases known as arachnida or araneidans because of their spider-like shape, and Plateau curves after Joseph Plateau who studied them. Equations in polar coordinates We are given two lines rotating about two poles P and P_1. By translation and rotation we may assume P = (0,0) and P_1 = (a, 0). At time t, the line rotating about P has angle \theta = \kappa t + \alpha and the line rotating about P_1 has angle \theta_1 = \kappa_1 t + ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
