The conchoid of de Sluze is the cubic curve firstconstructed by René de Sluze in 1662. It is given by the implicit equation(1)or the polar equation(2)This can be written in parametric form as(3)(4)The conchoid of de Sluze has a singular point at the origin which is a crunode for , a cusp for , and an acnode for .It has curvature and tangentialangle(5)(6)The curve has a loop if , in which case the loop is swept out by . The area of the loop is(7)
"The" trifolium is the three-lobed folium with , i.e., the 3-petalled rose. (Lawrence 1972 defines a trifolium as a folium with , but this more general definition is not so commonly used.)The trifolium with lobe along the negative -axis has polar equation(1)and Cartesian equation(2)The Cartesian equation can also be written in the alternative form(3)The two mirror images of the trifolium together have Cartesian equation(4)The area of the trifolium is given by(5)(6)(7)Rather surprisingly, this means that the area of the trifolium (left figure) is exactly one quarter of the area of the circumscribed circle, and even more surprisingly that the combined area of two mirror image trifoliums (middle figure) is identical to the area of the circle lying outside the curve (right figure).The arc length of the trifolium is(8)(9)(OEIS A093728), where is a complete elliptic integral of the second kind.The arc length function, curvature,and..
The swastika curve is Cundy and Rollett's (1989, p. 71) name for the quartic plane curve with Cartesian equationand polar equation
The Ochoa curve is the elliptic curvegiven in Weierstrass form asThe complete set of 23 integer solutions (where solutions of the form are counted as a single solution) to this equation consists of , (, 4520), (, 13356), (, 14616), (, 10656), (91, 8172), (227, 4228), (247, 3528), (271, 2592), (455, 200), (499, 3276), (523, 4356), (530, 4660), (599, 7576), (751, 14112), (1003, 25956), (1862, 75778), (3511, 204552), (5287, 381528), (23527, 3607272), (64507, 16382772), (100102, 31670478), and (1657891, 2134685628) (OEIS A141144 and A141145; Stroeker and de Weger 1994).
A cubic curve invented by Diocles in about 180 BC in connection with his attempt to duplicate the cube by geometrical methods. The name "cissoid" first appears in the work of Geminus about 100 years later. Fermat and Roberval constructed the tangent in 1634. From a given point there are either one or three tangents to the cissoid.Given an origin and a point on the curve, let be the point where the extension of the line intersects the line and be the intersection of the circle of radius and center with the extension of . Then the cissoid of Diocles is the curve which satisfies .The cissoid of Diocles is the roulette of a parabola vertex of a parabola rolling on an equal parabola. Newton gave a method of drawing the cissoid of Diocles using two line segments of equal length at right angles. If they are moved so that one line always passes through a fixed point and the end of the other line segment slides along a straight line, then the midpoint of the sliding..
An elliptic curve of the form for an integer. This equation has a finite number of solutions in integers for all nonzero . If is a solution, it therefore follows that is as well.Uspensky and Heaslet (1939) give elementary solutions for , , and 2, and then give , , , and 1 as exercises. Euler found that the only integer solutions to the particular case (a special case of Catalan's conjecture) are , , and . This can be proved using Skolem's method, using the Thue equation , using 2-descent to show that the elliptic curve has rank 0, and so on. It is given as exercise 6b in Uspensky and Heaslet (1939, p. 413), and proofs published by Wakulicz (1957), Mordell (1969, p. 126), Sierpiński and Schinzel (1988, pp. 75-80), and Metsaenkylae (2003).Solutions of the Mordell curve with are summarized in the table below for small .solutions123456none7none8910Values of such that the Mordell curve has no integer solutions are given by 6, 7, 11, 13,..
A plane curve discovered by Maclaurin but first studied in detail by Cayley. The name Cayley's sextic is due to R. C. Archibald, who attempted to classify curves in a paper published in Strasbourg in 1900 (MacTutor Archive). Cayley's sextic is given in polar coordinates by(1)The Cartesian equation is(2)Parametric equations can be given by(3)(4)for . In this parametrization, the loop corresponds to .The area enclosed by the outer boundary is(5)(6)(OEIS A118308), and by the inner loop is(7)(8)(OEIS A118309), and the arclength of the entire curve is(9)The arc length, curvature,and tangential angle are given by(10)(11)(12)
There are two incompatible definitions of the squircle.The first defines the squircle as the quartic plane curve which is special case of the superellipse with and , namely(1)illustrated above. This curve as arc length(2)(3)(OEIS A186642), where is a Meijer G-function (M. Trott, pers. comm., Oct. 21, 2011), encloses area(4)and has area moment of inertia tensor(5)The second definition of the squircle was given by Fernandez Guasti (1992), but apparently not dubbed with the name "squircle" until later (Fernández Guasti et al. 2005). This curve has quartic Cartesian equation(6)with squareness parameter , where corresponds to a circle with radius and to a square of side length . This curve is actually semialgebraic, as it must be restricted to to exclude other branches. This squircle encloses area(7)where is an elliptic integral of the second kind, which can be verified reduces to for and for .Both versions..
The Maltese cross curve is the cubic algebraiccurve with Cartesian equation(1)and polar equation(2)(Cundy and Rollett 1989, p. 71), so named for its resemblance to the Maltesecross.It has curvature and tangentialangle given by(3)(4)
The Cassini ovals are a family of quartic curves, also called Cassini ellipses, described by a point such that the product of its distances from two fixed points a distance apart is a constant . The shape of the curve depends on . If , the curve is a single loop with an oval (left figure above) or dog bone (second figure) shape. The case produces a lemniscate (third figure). If , then the curve consists of two loops (right figure). Cassini ovals are anallagmatic curves.A series of ovals for values of to 1.5 are illustrated above.The curve was first investigated by Cassini in 1680 when he was studying the relative motions of the Earth and the Sun. Cassini believed that the Sun traveled around the Earth on one of these ovals, with the Earth at one focus of the oval.The Cassini ovals are defined in two-center bipolarcoordinates by the equation(1)with the origin at a focus. Even more incredible curves are produced by the locus of a point the product of whose distances..
An algebraic curve of degree six.Examples include the astroid, atriphtaloid, Cayley's sextic, cornoid, cycloid of Ceva, dumbbell curve, ellipse evolute, epicycloid, Freeth's nephroid, heart curve (first), limaçon evolute, nephroid, quadrifolium, scarabaeus curve, and Talbot's curve.
The links curve is the quartic curve given by theCartesian equation(1)The area enclosed by the outer envelope is(2)and the area of the inner loop is(3)The origin of the curve is a tacnode.
The "Cartesian ovals," sometimes also known as the Cartesian curve or oval of Descartes, are the quartic curve consisting of two ovals. They were first studied by Descartes in 1637 and by Newton while classifying cubic curves. It is the locus of a point whose distances from two foci and in two-center bipolar coordinates satisfy(1)where are positive integers, is a positive real, and and are the distances from and (Lockwood 1967, p. 188).Cartesian ovals are anallagmatic curves. Unlikethe Cartesian ovals, these curves possess three foci.In Cartesian coordinates, the Cartesianovals can be written(2)Moving the quantity involving to the right-hand side, squaring both sides, simplifying, and rearranging gives(3)Once again squaring both sides gives(4)Defining(5)(6)gives the slightly simpler form(7)which corresponds to the form given by Lawrence (1972, p. 157) in the case and .If , the oval becomes a central conic.If..
The Scarabaeus curve is a sextic curve given by the equationand by the polar equationwhere .
The lemniscate, also called the lemniscate of Bernoulli, is a polar curve whose most common form is the locus of points the product of whose distances from two fixed points (called the foci) a distance away is the constant . This gives the Cartesian equation(1)where both sides of the equation have been squared. Expanding and simplifying then gives(2)Jakob Bernoulli published an article in Acta Eruditorum in 1694 in which he called this curve the lemniscus (Latin for "a pendant ribbon"). Bernoulli was not aware that the curve he was describing was a special case of Cassini ovals which had been described by Cassini in 1680. The general properties of the lemniscate were discovered by G. Fagnano in 1750 (MacTutor Archive). Gauss's and Euler's investigations of the arc length of the curve led to later work on elliptic functions.The most general form of the lemniscate is a toric sectionof a torus(3)cut by a plane . Plugging in and rearranging..
There are two curves known as the butterfly curve.The first is the sextic planecurve given by the implicit equation(1)(Cundy and Rollett 1989, p. 72; left figure). The total area of both wings is then given by(2)(3)(4)(OEIS A118292). The arclength is(5)(OEIS A118811).The second is the curve with polar equation(6)which has the corresponding parametric equations(7)(8)(Bourke, Fay 1989, Fay 1997, Kantel-Chaos-Team, Wassenaar; right figure).
By analogy with the squircle, a term first apparently used by Fernández Guasti et al. (2005), the term "rectellipse" (used here for the first time) is a natural generalization to the case of unequal vertical and horizontal dimensions.The first definition of the rectellipse is the quartic plane curve which is special case of the superellipse with , namely(1)illustrated above. This curve encloses area(2)and has area moment of inertia tensor(3)The second definition of the rectellipse was given, though not explicitly named, by Fernandez Guasti (1992). This curve has quartic Cartesian equation(4)with squareness parameter , where corresponds to an ellipse with semiaxes and and to a rectangle the side lengths and . This curve is actually semialgebraic, as it must be restricted to and to exclude other branches. This rectellipse encloses area(5)where is an elliptic integral of the second kind, which can be verified reduces..
The keratoid cusp is quintic algebraiccurve defined by(1)It has a ramphoid cusp at the origin, horizontal tangents at and , and a vertical tangent at .The curvature is given implicitly by(2)The loop has area(3)and arc length(4)
The only known classically known algebraic curve of curve genus that has an explicit parametrization in terms of standard special functions (Burnside 1893, Brezhnev 2001). This equation is given by(1)The closed portion of the curve has area(2)(3)where is a gamma function.The closed portion of this curve has a parametrization in terms of the Weierstrasselliptic function given by(4)(5)where(6)the half-periods are given by and ranges over complex values (Brezhnev 2001).
A hyperbola (plural "hyperbolas"; Gray 1997, p. 45) is a conic section defined as the locus of all points in the plane the difference of whose distances and from two fixed points (the foci and ) separated by a distance is a given positive constant ,(1)(Hilbert and Cohn-Vossen 1999, p. 3). Letting fall on the left -intercept requires that(2)so the constant is given by , i.e., the distance between the -intercepts (left figure above). The hyperbola has the important property that a ray originating at a focus reflects in such a way that the outgoing path lies along the line from the other focus through the point of intersection (right figure above).The special case of the rectangular hyperbola, corresponding to a hyperbola with eccentricity , was first studied by Menaechmus. Euclid and Aristaeus wrote about the general hyperbola, but only studied one branch of it. The hyperbola was given its present name by Apollonius, who was..
A quartic curve with implicit equation(1)or(2)and . In parametric form,(3)(4)The curvature is(5)
A general plane quartic curve is a curve of the form(1)Examples include the ampersand curve, bean curve, bicorn, bicuspid curve, bifoliate, bifolium, bitangent-rich curve, bow, bullet nose, butterfly curve, capricornoid, cardioid, Cartesian ovals, Cassini ovals, conchoid of Nicomedes, cruciform, deltoid, devil's curve, Dürer's conchoid, eight curve, fish curve, folium, hippopede, Kampyle of Eudoxus, Klein quartic, knot curve, lemniscate, limaçon, links curve, pear-shaped curve, piriform curve, swastika curve, trefoil curve, and trifolium.The incidence relations of the 28 bitangents of the general quartic curve can be put into a one-to-one correspondence with the vertices of a particular polytope in seven-dimensional space (Coxeter 1928, Du Val 1933). This fact is essentially similar to the discovery by Schoute (1910) that the 27 Solomon's seal lines on a cubic surface can be connected with a polytope in six-dimensional..
There are a number of mathematical curves that produced heart shapes, some of which are illustrated above. The first curve is a rotated cardioid (whose name means "heart-shaped") given by the polar equation(1)The second is obtained by taking the cross section of the heart surface and relabeling the -coordinates as , giving the order-6 algebraic equation(2)The third curve is given by the parametric equations(3)(4)where (H. Dascanio, pers. comm., June 21, 2003). The fourth curve is given by(5)(P. Kuriscak, pers. comm., Feb. 12, 2006). Each half of this heart curve is a portion of an algebraic curve of order 6. And the fifth curve is the polar curve(6)due to an anonymous source and obtained from the log files of Wolfram|Alpha in early February 2010.Each half of this heart curve is a portion of an algebraic curve of order 12, so the entire curve is a portion of an algebraic curve of order 24.A sixth heart curve can be defined..
The quartic with implicit equation(1)The bow has vertical tangents at and horizontal tangents at .Its curvature is implicitly given by(2)The area enclosed by the two loops is given by(3)(4)(OEIS A118321).The portion of the curve bounding the two loops has approximate perimeter(5)(OEIS A118322).
The quadrifolium is the 4-petalled rose having . It has polar equation(1)and Cartesian equation(2)The area of the quadrifolium is(3)(4)(5)Rather surprisingly, this means that the area inside the curve is equal to that of its complement within the curve's circumcircle.The arc length is(6)(7)(OEIS A138500), where is a complete elliptic integral of the second kind.The arc length function, curvature,and tangential angle are(8)(9)(10)where is an elliptic integral of the second kind and is the floor function.
Given a general quadratic curve(1)the quantity is known as the discriminant, where(2)and is invariant under rotation. Using the coefficients from quadratic equations for a rotation by an angle ,(3)(4)(5)(6)(7)(8)Now let(9)(10)(11)(12)and use(13)(14)to rewrite the primed variables(15)(16)(17)(18)From (16) and (18), it follows that(19)Combining with (17) yields, for an arbitrary (20)(21)(22)(23)which is therefore invariant under rotation. This invariant therefore provides a useful shortcut to determining the shape represented by a quadratic curve. Choosing to make (see quadratic equation), the curve takes on the form(24)Completing the square and defining new variablesgives(25)Without loss of generality, take the sign of to be positive. The discriminant is(26)Now, if , then and both have the same sign, and the equation has the general form of an ellipse (if and are positive). If , then and have opposite signs, and the equation..
An ellipse is a curve that is the locus of all points in the plane the sum of whose distances and from two fixed points and (the foci) separated by a distance of is a given positive constant (Hilbert and Cohn-Vossen 1999, p. 2). This results in the two-center bipolar coordinate equation(1)where is the semimajor axis and the origin of the coordinate system is at one of the foci. The corresponding parameter is known as the semiminor axis.The ellipse is a conic section and a Lissajouscurve.An ellipse can be specified in the Wolfram Language using Circle[x, y, a, b].If the endpoints of a segment are moved along two intersecting lines, a fixed point on the segment (or on the line that prolongs it) describes an arc of an ellipse. This is known as the trammel construction of an ellipse (Eves 1965, p. 177).It is possible to construct elliptical gears that rotate smoothly against one another (Brown 1871, pp. 14-15; Reuleaux and Kennedy 1876,..
The quartic curve given by the implicit equation(1)so-named because of its resemblance to a tooth.The bicuspid curve has cusps at and .The horizontal tangents are located at , and the vertical tangents at , , and , where is a polynomial root.The bicuspid with has approximate area(2)and approximate perimeter(3)
The general bivariate quadratic curve can be written(1)Define the following quantities:(2)(3)(4)(5)Then the quadratics are classified into the types summarized in the following table (Beyer 1987). The real (nondegenerate) quadratics (the ellipse and its special case the circle, hyperbola, and parabola) correspond to the curves which can be created by the intersection of a plane with a (two-nappes) cone, and are therefore known as conic sections.curvecoincident lines000ellipse (imaginary)ellipse (real)hyperbolaintersecting lines (imaginary)0intersecting lines (real)0parabola0parallel lines (imaginary)00parallel lines (real)00It is always possible to eliminate the cross term by a suitable rotation of the axes. To see this, consider rotation by an arbitrary angle . The rotation matrix is(6)(7)so(8)(9)(10)(11)(12)Plugging these into (◇) and grouping terms gives(13)Comparing the coefficients with (◇)..
A curve also known as the Gerono lemniscate. It is given by Cartesiancoordinates(1)polar coordinates,(2)and parametric equations(3)(4)It has vertical tangents at and horizontal tangents at .Setting , , and in the equation of the eight surface (i.e., scaling by half and relabeling the -axis as the -axis) gives the eight curve.The area of the curve is(5)The curvature and tangentialangle are(6)(7)The arc length of the entire curve is given by(8)(9)(10)(11)(12)(13)(OEIS A118178), where is a complete elliptic integral of the first kind, is a complete elliptic integral of the second kind, and is a complete elliptic integral of the third kind, all with elliptic modulus (D. W. Cantrell, pers. comm., Apr. 22, 2006). The arc length also has a surprising connection to 1-dimensional random walks via(14)where(15)(16)(17)and is a regularized hypergeometric function, the first few terms of which for , 1, ... are 1, 0, 4, 6, 36,..
The bicorn, sometimes also called the "cocked hat curve" (Cundy and Rollett 1989, p. 72), is the name of a collection of quartic curves studied by Sylvester in 1864 and Cayley in 1867 (MacTutor Archive). The bicorn is given by the parametric equations(1)(2)(Lawrence 1972, p. 147) and Cartesian equation(3)(Lawrence 1972, p. 147; Cundy and Rollett 1989, p. 72; Mactutor, with the final corrected to be squared instead of to the first power).The bicorn has cusps at .The area enclosed by the curve is(4)The curvature and tangentialangle are given by(5)for . There does not seem to be a simple closed form for the arc length of the curve, but its numerical value is approximately given by .
A curve obtained by fitting polynomials to each ordinate of an ordered sequence of points. The above plots show polynomial curves where the order of the fitting polynomial varies from to , where is the number of points.Polynomial curves have several undesirable features, including a nonintuitive variation of fitting curve with varying coefficients, and numerical instability for high orders. Splines such as the Bézier curve are therefore used more commonly.
The dumbbell curve is the sextic curve(1)It has area(2)and approximate arc length(3)For the parametrization(4)(5)the curvature is(6)
The quartic curve given by the implicit equation(1)It has horizontal tangents at and vertical tangents at and .The area enclosed by the curve is given by(2)(3)(4)(OEIS A193505), the geometric centroid of the interior by(5)(6)and the area moment of inertia tensor ofthe interior by(7)(8)(9)(E. Weisstein, Feb. 3-5, 2018).The perimeter is given approximately by(10)(OEIS A193506).
The devil's curve was studied by G. Cramer in 1750 and Lacroix in 1810 (MacTutor Archive). It appeared in Nouvelles Annales in 1858. The Cartesian equation is(1)equivalent to(2)the polar equation is(3)and the parametric equations are(4)(5)The curve illustrated above corresponds to parameters and .It has a crunode at the origin.For , the cental hourglass is horizontal, for , it is vertical, and as it passes through , the curve changes to a circle.A special case of the Devil's curve is the so-called "electric motor curve":(6)(Cundy and Rollett 1989).
A 4-cusped hypocycloid which is sometimes also called a tetracuspid, cubocycloid, or paracycle. The parametric equations of the astroid can be obtained by plugging in or into the equations for a general hypocycloid, giving parametric equations(1)(2)(3)(4)(5)(6)for .The polar equation can be obtained by computing(7)and plugging in to to obtain(8)for .In Cartesian coordinates,(9)A generalization of the curve to(10)gives "squashed" astroids, which are a special case of the superellipse corresponding to parameter .In pedal coordinates with the pedalpoint at the center, the equation is(11)and the Cesàro equation is(12)A further generalization to an equation of the form(13)is known as a superellipse.The arc length, curvature,and tangential angle are(14)(15)(16)where the formula for holds for .The perimeter of the entire astroid can be computedfrom the general hypocycloid formula(17)with ,(18)For a squashed..
A parabola (plural "parabolas"; Gray 1997, p. 45) is the set of all points in the plane equidistant from a given line (the conic section directrix) and a given point not on the line (the focus). The focal parameter (i.e., the distance between the directrix and focus) is therefore given by , where is the distance from the vertex to the directrix or focus. The surface of revolution obtained by rotating a parabola about its axis of symmetry is called a paraboloid.The parabola was studied by Menaechmus in an attempt to achieve cube duplication. Menaechmus solved the problem by finding the intersection of the two parabolas and . Euclid wrote about the parabola, and it was given its present name by Apollonius. Pascal considered the parabola as a projection of a circle, and Galileo showed that projectiles falling under uniform gravity follow parabolic paths. Gregory and Newton considered the catacaustic properties of a parabola that..
A cubic curve is an algebraic curve of curve order 3. An algebraic curve over a field is an equation , where is a polynomial in and with coefficients in , and the degree of is the maximum degree of each of its terms (monomials).Examples include the cissoid of Diocles, conchoid of de Sluze, folium of Descartes, Maclaurin trisectrix, Maltese cross curve, right strophoid, semicubical parabola, serpentine curve, Tschirnhausen cubic, and witch of Agnesi, as well as elliptic curves such as the Mordell curve and Ochoa curve.Newton showed that all cubics can be generated by the projection of the five divergent cubic parabolas. Newton's classification of cubic curves appeared in the chapter "Curves" in Lexicon Technicum by John Harris published in London in 1710. Newton also classified all cubics into 72 types, missing six of them. In addition, he showed that any cubic can be obtained by a suitable projection of the elliptic curve(1)where the..
The ampersand curve is the name given by Cundy and Rowlett (1989, p. 72) tothe quartic curve with implicit equation(1)Although it is not mentioned by Cundy and Rowlett, this curve is significant because it is the original example (after subtracting a small positive constant ) of a quartic curve having 28 real bitangents constructed by Plücker (Plücker 1839, Gray 1982).It has crunodes at , , and .The horizontal asymptotes are at , , and . The vertical asymptotes are at and The polar equation is given by solving the quadraticequation(2)The area enclosed by the ampersand is given approximately by(3)(OEIS A101801) and the perimeterapproximately by(4)(OEIS A101802).
A plane quartic curve also called the cross curve or policeman on point duty curve (Cundy and Rollett 1989). It is given by the implicit equation(1)which is equivalent to(2)and(3)In parametric form,(4)(5)The curvature is given by(6)(7)which, in the special case , reduces to(8)
Algebraic geometry is the study of geometries that come from algebra, in particular, from rings. In classical algebraic geometry, the algebra is the ring of polynomials, and the geometry is the set of zeros of polynomials, called an algebraic variety. For instance, the unit circle is the set of zeros of and is an algebraic variety, as are all of the conic sections.In the twentieth century, it was discovered that the basic ideas of classical algebraic geometry can be applied to any commutative ring with a unit, such as the integers. The geometry of such a ring is determined by its algebraic structure, in particular its prime ideals. Grothendieck defined schemes as the basic geometric objects, which have the same relationship to the geometry of a ring as a manifold to a coordinate chart. The language of category theory evolved at around the same time, largely in response to the needs of the increasing abstraction in algebraic geometry.As a consequence,..
The ophiuride is a cubic curve (left figure) given by the implicit equation(1)where , or by the polar equation(2)for . The curve is named base on its resemblance to a particular species of star-fish (right figure). Taking yields a cissoid of Diocles.Its curvature is(3)
The cornoid is the curve illustrated above given by the parametric equations(1)(2)where .It is a sextic algebraiccurve with equation(3)The arc length of the curve is given by(4)(5)(OEIS A141108), where is a complete elliptic integral of the first kind, is a complete elliptic integral of the second kind, is a complete elliptic integral of the third kind, and .The area of a single of the loops is(6)the area of the outer envelope is(7)and the area of the region enclosed is(8)(9)
An algebraic curve over a field is an equation , where is a polynomial in and with coefficients in . A nonsingular algebraic curve is an algebraic curve over which has no singular points over . A point on an algebraic curve is simply a solution of the equation of the curve. A -rational point is a point on the curve, where and are in the field .The following table lists the names of algebraic curves of a given degree.ordercurveexamples2quadratic curvecircle, ellipse, hyperbola, parabola3cubic curvecissoid of Diocles, conchoid of de Sluze, folium of Descartes, Maclaurin trisectrix, Maltese cross curve, Mordell curve, Ochoa curve, right strophoid, semicubical parabola, serpentine curve, Tschirnhausen cubic, witch of Agnesi4quartic curveampersand curve, bean curve, bicorn, bicuspid curve, bifoliate, bifolium, bitangent-rich curve, bow, bullet nose, butterfly curve, capricornoid, cardioid, Cartesian ovals, Cassini ovals, conchoid..