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List Of Curves
This is a list of Wikipedia articles about curves used in different fields: mathematics (including geometry, statistics, and applied mathematics), physics, engineering, economics, medicine, biology, psychology, ecology, etc. Mathematics (Geometry) Algebraic curves Rational curves Rational curves are subdivided according to the degree of the polynomial. =Degree 1= * Line =Degree 2= Plane curves of degree 2 are known as conics or conic sections and include *Circle **Unit circle *Ellipse *Parabola *Hyperbola **Unit hyperbola =Degree 3= Cubic plane curves include *Cubic parabola *Folium of Descartes *Cissoid of Diocles *Conchoid of de Sluze *Right strophoid *Semicubical parabola *Serpentine curve *Trident curve *Trisectrix of Maclaurin *Tschirnhausen cubic *Witch of Agnesi =Degree 4= Quartic plane curves include *Ampersand curve *Bean curve * Bicorn *Bow curve *Bullet-nose curve *Cartesian oval *Cruciform curve *Deltoid curve * Devil's curve *Hippopede *Kampyle of Eudoxus *Kapp ... [...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] |
Unit Circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. In topology, it is often denoted as because it is a one-dimensional unit -sphere. If is a point on the unit circle's circumference, then and are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, and satisfy the equation x^2 + y^2 = 1. Since for all , and since the reflection of any point on the unit circle about the - or -axis is also on the unit circle, the above equation holds for all points on the unit circle, not only those in the first quadrant. The interior of the unit circle is called the open unit disk, while the interior of the unit circle combined with the unit circle itself is called the closed unit disk. One may also use other notions of "dista ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trident Curve
In mathematics, a trident curve (also trident of Newton or parabola of Descartes) is any member of the family of curves that have the formula: :xy+ax^3+bx^2+cx=d Trident curves are cubic plane curves with an ordinary double point in the real projective plane at ''x'' = 0, ''y'' = 1, ''z'' = 0; if we substitute ''x'' = and ''y'' = into the equation of the trident curve, we get :ax^3+bx^2z+cxz^2+xz = dz^3, which has an ordinary double point at the origin. Trident curves are therefore rational plane algebraic curves of genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ... zero. References * External links * Algebraic curves {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Serpentine Curve
A serpentine curve is a curve whose equation is of the form :x^2y+a^2y-abx=0, \quad ab > 0. Equivalently, it has a parametric representation :x=a\cot(t), y=b\sin (t)\cos(t), or functional representation :y=\frac. The curve has an inflection point at the origin. It has local extrema at x = \pm a, with a maximum value of y=b/2 and a minimum value of y=-b/2. History Serpentine curves were studied by L'Hôpital and Huygens, and named and classified by Newton. Visual appearance External links MathWorld – Serpentine Equation {{geometry-stub Plane curves ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semicubical Parabola
In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve that has an implicit equation of the form : y^2 - a^2 x^3 = 0 (with ) in some Cartesian coordinate system. Solving for leads to the ''explicit form'' : y = \pm a x^, which imply that every real point satisfies . The exponent explains the term ''semicubical parabola''. (A parabola can be described by the equation .) Solving the implicit equation for yields a second ''explicit form'' :x = \left(\frac\right)^. The parametric equation : \quad x = t^2, \quad y = a t^3 can also be deduced from the implicit equation by putting t = \frac. . The semicubical parabolas have a cuspidal singularity; hence the name of ''cuspidal cubic''. The arc length of the curve was calculated by the English mathematician William Neile and published in 1657 (see section History). Properties of semicubical parabolas Similarity Any semicubical parabola (t^2,at^3) is similar to the ''semicubical unit parabola'' ... [...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. The ... [...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 *, right stroph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cissoid Of Diocles
In geometry, the cissoid of Diocles (; named for Diocles) is a cubic plane curve notable for the property that it can be used to construct two mean proportionals to a given ratio. In particular, it can be used to double a cube. It can be defined as the cissoid of a circle and a line tangent to it with respect to the point on the circle opposite to the point of tangency. In fact, the curve family of cissoids is named for this example and some authors refer to it simply as ''the'' cissoid. It has a single cusp at the pole, and is symmetric about the diameter of the circle which is the line of tangency of the cusp. The line is an asymptote. It is a member of the conchoid of de Sluze family of curves and in form it resembles a tractrix. Construction and equations Let the radius of be . By translation and rotation, we may take to be the origin and the center of the circle to be (''a'', 0), so is . Then the polar equations of and are: :\begin & r=2a\sec\theta \\ & r=2a\cos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Folium Of Descartes
In geometry, the folium of Descartes (; named for René Decartes) is an algebraic curve defined by the implicit equation :x^3 + y^3 - 3 a x y = 0. 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. Graphing the curve The folium of Descartes can be expressed in polar coordinates as :r = \frac, which is plotted on the left. This is equivalent to r = \frac. Another technique is to write y = px and solve for x and y in terms of p. This yields the rational parametric equations: x = ,\, y = . We can see that the parameter is rela ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Parabola
In arithmetic and algebra, the cube of a number is its third power, that is, the result of multiplying three instances of together. The cube of a number or any other mathematical expression is denoted by a superscript 3, for example or . The cube is also the number multiplied by its square: :. The ''cube function'' is the function (often denoted ) that maps a number to its cube. It is an odd function, as :. The volume of a geometric cube is the cube of its side length, giving rise to the name. The inverse operation that consists of finding a number whose cube is is called extracting the cube root of . It determines the side of the cube of a given volume. It is also raised to the one-third power. The graph of the cube function is known as the cubic parabola. Because the cube function is an odd function, this curve has a center of symmetry at the origin, but no axis of symmetry. In integers A cube number, or a perfect cube, or sometimes just a cube, is a number wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Plane Curve
In mathematics, a cubic plane curve is a plane algebraic curve defined by a cubic equation : applied to homogeneous coordinates for the projective plane; or the inhomogeneous version for the affine space determined by setting in such an equation. Here is a non-zero linear combination of the third-degree monomials : These are ten in number; therefore the cubic curves form a projective space of dimension 9, over any given field . Each point imposes a single linear condition on , if we ask that pass through . Therefore, we can find some cubic curve through any nine given points, which may be degenerate, and may not be unique, but will be unique and non-degenerate if the points are in general position; compare to two points determining a line and how five points determine a conic. If two cubics pass through a given set of nine points, then in fact a pencil of cubics does, and the points satisfy additional properties; see Cayley–Bacharach theorem. A cubic curve may ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Hyperbola
In geometry, the unit hyperbola is the set of points (''x'',''y'') in the Cartesian plane that satisfy the implicit equation x^2 - y^2 = 1 . In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an ''alternative radial length'' : r = \sqrt . Whereas the unit circle surrounds its center, the unit hyperbola requires the ''conjugate hyperbola'' y^2 - x^2 = 1 to complement it in the plane. This pair of hyperbolas share the asymptotes ''y'' = ''x'' and ''y'' = −''x''. When the conjugate of the unit hyperbola is in use, the alternative radial length is r = \sqrt . The unit hyperbola is a special case of the rectangular hyperbola, with a particular orientation, location, and scale. As such, its eccentricity equals \sqrt. The unit hyperbola finds applications where the circle must be replaced with the hyperbola for purposes of analytic geometry. A prominent instance is the depiction of spacetime as a pseudo-Euclidean space. There the asymptotes of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
