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Inverse Curve
In inversive geometry, an inverse curve of a given curve is the result of applying an inverse operation to . Specifically, with respect to a fixed circle with center and radius the inverse of a point is the point for which lies on the ray and . The inverse of the curve is then the locus of as runs over . The point in this construction is called the center of inversion, the circle the circle of inversion, and the radius of inversion. An inversion applied twice is the identity transformation, so the inverse of an inverse curve with respect to the same circle is the original curve. Points on the circle of inversion are fixed by the inversion, so its inverse is itself. Equations The inverse of the point with respect to the unit circle is where :X = \frac,\qquad Y=\frac, or equivalently :x = \frac,\qquad y=\frac. So the inverse of the curve determined by with respect to the unit circle is :f\left(\frac, \frac\right)=0. It is clear from this that inverting an algeb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Curves Parabola Cardioid
Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse, the inverse of a number that, when added to the original number, yields zero * Compositional inverse, a function that "reverses" another function * Inverse element * Inverse function, a function that "reverses" another function **Generalized inverse, a matrix that has some properties of the inverse matrix but not necessarily all of them * Multiplicative inverse (reciprocal), a number which when multiplied by a given number yields the multiplicative identity, 1 ** Inverse matrix of an Invertible matrix Other uses * Invert level, the base interior level of a pipe, trench or tunnel * Inverse (website), ''Inverse'' (website), an online magazine * An outdated term for an LGBT person; see Sexual inversion (sexology) See also * Inversion (other) * Inverter (other) * Opp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lituus (mathematics)
300px, Branch for positive The lituus spiral () is a spiral in which the angle is inversely proportional to the square of the radius . This spiral, which has two branches depending on the sign of , is asymptotic to the axis. Its points of inflexion are at : (\theta, r) = \left(\tfrac12, \pm\sqrt\right). The curve was named for the ancient Roman lituus by Roger Cotes in a collection of papers entitled ''Harmonia Mensurarum'' (1722), which was published six years after his death. Coordinate representations Polar coordinates The representations of the lituus spiral in polar coordinates is given by the equation : r = \frac, where and . Cartesian coordinates The lituus spiral with the polar coordinates can be converted to Cartesian coordinates like any other spiral with the relationships and . With this conversion we get the parametric representations of the curve: : \begin x &= \frac \cos\theta, \\ y &= \frac \sin\theta. \\ \end These equations can in turn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oval Of Cassini
In geometry, a Cassini oval is a quartic plane curve defined as the locus (mathematics), locus of points in the plane (geometry), plane such that the Product (mathematics), product of the distances to two fixed points (Focus (geometry), foci) is constant. This may be contrasted with an ellipse, for which the ''sum'' of the distances is constant, rather than the product. Cassini ovals are the special case of polynomial lemniscates when the polynomial used has degree of a polynomial, degree 2. Cassini ovals are named after the astronomer Giovanni Domenico Cassini who studied them in the late 17th century. Cassini believed that a planet orbiting around another body traveled on one of these ovals, with the body it orbited around at one focus of the oval. Other names include Cassinian ovals, Cassinian curves and ovals of Cassini. Formal definition A Cassini oval is a set of points, such that for any point P of the set, the ''product'' of the distances , PP_1, ,\, , PP_2, to two fix ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is called the radius. The length of a line segment connecting two points on the circle and passing through the centre is called the diameter. A circle bounds a region of the plane called a Disk (mathematics), disc. The circle has been known since before the beginning of recorded history. Natural circles are common, such as the full moon or a slice of round fruit. The circle is the basis for the wheel, which, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus. Terminology * Annulus (mathematics), Annulus: a ring-shaped object, the region bounded by two concentric circles. * Circular arc, Arc: any Connected ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hippopede
In geometry, a hippopede () is a plane curve determined by an equation of the form :(x^2+y^2)^2=cx^2+dy^2, where it is assumed that and since the remaining cases either reduce to a single point or can be put into the given form with a rotation. Hippopedes are circular algebraic curve, bicircular, Rational number, rational, algebraic curves of Degree of a polynomial, degree 4 and symmetric with respect to both the and axes. Special cases When ''d'' > 0 the curve has an oval form and is often known as an oval of Booth, and when the curve resembles a sideways figure eight, or lemniscate, and is often known as a lemniscate of Booth, after 19th-century mathematician James Booth (mathematician), James Booth who studied them. Hippopedes were also investigated by Proclus (for whom they are sometimes called Hippopedes of Proclus) and Eudoxus of Cnidus, Eudoxus. For , the hippopede corresponds to the lemniscate of Bernoulli. Definition as spiric sections Hippopedes can be defined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Trisectrix Of Maclaurin
In algebraic geometry, the trisectrix of Maclaurin is a cubic plane curve notable for its trisectrix property, meaning it can be used to trisect an angle. It can be defined as locus of the point of intersection of two lines, each rotating at a uniform rate about separate points, so that the ratio of the rates of rotation is 1:3 and the lines initially coincide with the line between the two points. A generalization of this construction is called a sectrix of Maclaurin. The curve is named after Colin Maclaurin who investigated the curve in 1742. Equations Let two lines rotate about the points P = (0,0) and P_1 = (a, 0) so that when the line rotating about P has angle \theta with the ''x'' axis, the rotating about P_1 has angle 3\theta. Let Q be the point of intersection, then the angle formed by the lines at Q is 2\theta. By the law of sines, : = \! so the equation in polar coordinates is (up to translation and rotation) :r= a \frac = \frac = (4 \cos \theta - \sec \theta).\! Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conchoid Of De Sluze
In algebraic geometry, the conchoids of de Sluze are a family of plane curves studied in 1662 by Walloon mathematician René François Walter, baron de Sluze. The curves are defined by the polar equation :r=\sec\theta+a\cos\theta \,. In cartesian coordinates, the curves satisfy the implicit equation :(x-1)(x^2+y^2)=ax^2 \, except that for the implicit form has an acnode not present in polar form. They are rational, circular, cubic plane curves. These expressions have an asymptote (for ). The point most distant from the asymptote is . is a crunode for . The area between the curve and the asymptote is, for , :, a, (1+a/4)\pi \, while for , the area is :\left(1-\frac a2\right)\sqrt-a\left(2+\frac a2\right)\arcsin\frac1. If , the curve will have a loop. The area of the loop is :\left(2+\frac a2\right)a\arccos\frac1 + \left(1-\frac a2\right)\sqrt. Four of the family have names of their own: *, line (asymptote to the rest of the family) *, cissoid of Diocles In geometr ... [...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] |
Cardioid
In geometry, a cardioid () is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal spiral, and an inverse curve of the parabola with the focus as the center of inversion. A cardioid can also be defined as the set of points of reflections of a fixed point on a circle through all tangents to the circle. The name was coined by Giovanni Salvemini in 1741 but the cardioid had been the subject of study decades beforehand.Yates Although named for its heart-like form, it is shaped more like the outline of the cross-section of a round apple without the stalk. A cardioid microphone exhibits an acoustic pickup pattern that, when graphed in two dimensions, resembles a cardioid (any 2d plane containing the 3d straight line of the microphone body). In three dimensions, the cardioid is shaped like an apple centred around the mic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acnode
An acnode is an isolated point in the solution set of a polynomial equation in two real variables. Equivalent terms are isolated point and hermit point. For example the equation :f(x,y)=y^2+x^2-x^3=0 has an acnode at the origin, because it is equivalent to :y^2 = x^2 (x-1) and x^2(x-1) is non-negative only when x ≥ 1 or x = 0. Thus, over the ''real'' numbers the equation has no solutions for x < 1 except for (0, 0). In contrast, over the complex numbers the origin is not isolated since square roots of negative real numbers exist. In fact, the complex solution set of a polynomial equation in two complex variables can never have an isolated point. An acnode is a critical point, or singularity, of the defining polynomial function, in the sense that both partial derivatives and vanish. Furth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limaçon
In geometry, a limaçon or limacon , also known as a limaçon of Pascal or Pascal's Snail, is defined as a roulette curve formed by the path of a point fixed to a circle when that circle rolls around the outside of a circle of equal radius. It can also be defined as the roulette formed when a circle rolls around a circle with half its radius so that the smaller circle is inside the larger circle. Thus, they belong to the family of curves called centered trochoids; more specifically, they are epitrochoids. The cardioid is the special case in which the point generating the roulette lies on the rolling circle; the resulting curve has a cusp. Depending on the position of the point generating the curve, it may have inner and outer loops (giving the family its name), it may be heart-shaped, or it may be oval. A limaçon is a bicircular rational plane algebraic curve of degree 4. History The earliest formal research on limaçons is generally attributed to Étienne Pascal, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
