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A plane curve given by the parametric equations(1)(2)The plots above show curves for values of from 0 to 7.The teardrop curve has area(3)

Fermat's spiral, also known as the parabolic spiral, is an Archimedean spiral with having polar equation(1)This curve was discussed by Fermat in 1636 (MacTutor Archive). For any given positive value of , there are two corresponding values of of opposite signs. The left plot above shows(2)only, while the right plot shows equation (1) in red and(3)in blue. Taking both signs, the resulting spiral is symmetrical about the origin.The curvature and arc lengthof the positive branch of Fermat's spiral are(4)(5)(6)where is a hypergeometric function and is an incomplete beta function.

The inverse curve of the epispiralwith inversion center at the origin and inversion radius is the rose

The epispiral is a plane curve with polar equationThere are sections if is odd and if is even.A slightly more symmetric version considers instead

The pedal curve of a logarithmicspiral with parametric equation(1)(2)for a pedal point at the pole is an identical logarithmicspiral(3)(4)so(5)

The inverse curve of the logarithmicspiralwith inversion center at the origin and inversion radius is the logarithmic spiral

For a logarithmic spiral given parametricallyas(1)(2)evolute is given by(3)(4)As first shown by Johann Bernoulli, the evolute of a logarithmic spiral is therefore another logarithmic spiral, having and ,In some cases, the evolute is identical to the original,as can be demonstrated by making the substitution to the new variable(5)Then the above equations become(6)(7)(8)(9)which are equivalent to the form of the original equation if(10)(11)(12)where only solutions with the minus sign in exist. Solving gives the values summarized in the following table.10.2744106319...20.1642700512...30.1218322508...40.0984064967...50.0832810611...60.0725974881...70.0645958183...80.0583494073...90.0533203211...100.0491732529...

A spiral that gives the solution to the central orbitproblem under a radial force law(1)where is a positive constant. There are three solution regimes,(2)where and are constants,(3)(4)and is the specific angular momentum (Whittaker 1944, p. 83). The case gives an epispiral, while leads to a hyperbolic spiral.

The catacaustic of a logarithmic spiral, where the origin is taken as the radiant point, is another logarithmic spiral. For an original spiral with parametric equations(1)(2)the catacaustic with radiant point at the originis(3)(4)

The logarithmic spiral is a spiral whose polarequation is given by(1)where is the distance from the origin, is the angle from the x-axis, and and are arbitrary constants. The logarithmic spiral is also known as the growth spiral, equiangular spiral, and spira mirabilis. It can be expressed parametrically as(2)(3)This spiral is related to Fibonacci numbers, the golden ratio, and the golden rectangle, and is sometimes called the golden spiral.The logarithmic spiral can be constructed from equally spaced rays by starting at a point along one ray, and drawing the perpendicular to a neighboring ray. As the number of rays approaches infinity, the sequence of segments approaches the smooth logarithmic spiral (Hilton et al. 1997, pp. 2-3).The logarithmic spiral was first studied by Descartes in 1638 and Jakob Bernoulli. Bernoulli was so fascinated by the spiral that he had one engraved on his tombstone (although the engraver did not draw..

The pedal curve of a sinusoidalspiralwith pedal point at the center is another sinusoidalspiral with polar equationA few examples are illustrated above.

The inverse curve of a sinusoidalspiralwith inversion center at the origin and inversion radius is another sinusoidal spiral

The most common statement known as Steiner's theorem (Casey 1893, p. 329) states that the Pascal lines of the hexagons 123456, 143652, and 163254 formed by interchanging the vertices at positions 2, 4, and 6 are concurrent (where the numbers denote the order in which the vertices of the hexagon are taken). The 20 points of concurrence so generated are known as Steiner points.Another theorem due to Steiner lets lines and join a variable point on a conic section to two fixed points on the same conic section. Then and are projectively related.A third "Steiner's theorem" states that if two opposite edges of a tetrahedron move on two fixed skew lines in any way whatsoever but remain fixed in length, then the volume of the tetrahedron remains constant (Altshiller-Court 1979, p. 87)...

The inner and outer spheres tangent internally to a cone and also to a plane intersecting the cone are called Dandelin spheres.The spheres can be used to show that the intersection of the plane with the cone is an ellipse. Let be a plane intersecting a right circular cone with vertex in the curve . Call the spheres tangent to the cone and the plane and , and the circles on which the spheres are tangent to the cone and . Pick a line along the cone which intersects at , at , and at . Call the points on the plane where the sphere are tangent and . Because intersecting tangents have the same length,(1)(2)Therefore,(3)which is a constant independent of , so is an ellipse with .

The term diamond is another word for a rhombus. The term is also used to denote a square tilted at a angle.The diamond shape is a special case of the superellipse with parameter , giving it implicit Cartesian equation(1)Since the diamond is a rhombus with diagonals and , it has inradius(2)(3)Writing as an algebraic curve gives the quartic curve(4)which is a diamond curve with the diamond edges extended to infinity.When considered as a polyomino, the diamond of order can be considered as the set of squares whose centers satisfy the inequality . There are then squares in the order- diamond, which is precisely the centered square number of order . For , 2, ..., the first few values are 1, 5, 13, 25, 41, 61, 85, 113, 145, ... (OEIS A001844).The diamond is also the name given to the unique 2-polyiamond...

If three conics pass through two given points and , then the lines joining the other two intersections of each pair of conics are concurrent at a point (Evelyn 1974, p. 15). The converse states that if two conics and meet at four points , , , and , and if and are chords of and , respectively, which meet on , then the six points lie on a conic. The dual of the theorem states that if three conics share two common tangents, then their remaining pairs of common tangents intersect at three collinear points.If the points and are taken as the points at infinity, then the theorem reduces to the theorem that radical lines of three circles are concurrent in a point known as the radical center (Evelyn 1974, p. 15).If two of the points and are taken as the points at infinity, then the theorem becomes that if two circles and pass through two points and on a conic , then the lines determined by the pair of intersections of each circle with the conic are parallel (Evelyn..

If two intersections of each pair of three conics , , and lie on a conic , then the lines joining the other two intersections of each pair are concurrent (Evelyn et al. 1974, pp. 23 and 25).The dual theorem states that if two common tangents of each pair of three conics touch a fourth conic, then the remaining common tangents of each pair intersect in three collinear points (Evelyn et al. 1974, pp. 24-25).

The chord through a focus parallel to the conic section directrix of a conic section is called the latus rectum, and half this length is called the semilatus rectum (Coxeter 1969). "Semilatus rectum" is a compound of the Latin semi-, meaning half, latus, meaning 'side,' and rectum, meaning 'straight.'For an ellipse, the semilatus rectum is the distance measured from a focus such that(1)where and are the apoapsis and periapsis, and is the ellipse's eccentricity. Plugging in for and then gives(2)so(3)For a parabola,(4)where is the distance between the focus and vertex (or directrix).

The conic sections are the nondegenerate curves generated by the intersections of a plane with one or two nappes of a cone. For a plane perpendicular to the axis of the cone, a circle is produced. For a plane that is not perpendicular to the axis and that intersects only a single nappe, the curve produced is either an ellipse or a parabola (Hilbert and Cohn-Vossen 1999, p. 8). The curve produced by a plane intersecting both nappes is a hyperbola (Hilbert and Cohn-Vossen 1999, pp. 8-9).The ellipse and hyperbolaare known as central conics.Because of this simple geometric interpretation, the conic sections were studied by the Greeks long before their application to inverse square law orbits was known. Apollonius wrote the classic ancient work on the subject entitled On Conics. Kepler was the first to notice that planetary orbits were ellipses, and Newton was then able to derive the shape of orbits mathematically using calculus, under..

The -roll mill curve is given by the equationwhere is a binomial coefficient.

A curve named after James Watt (1736-1819), the Scottish engineer who developed the steam engine (MacTutor Archive). The curve is produced by a linkage of rods connecting two wheels of equal diameter. Let the two wheels have radius and let their centers be located a distance apart. Further suppose that a rod of length is fixed at each end to the circumference of the two wheels. Let be the midpoint of the rod. Then Watt's curve is the locus of .The polar equation of Watt's curve is(1)The areas of one of the inner lenses, heart-shaped half-region, and entire enclosed region (which resembles a lemniscate are(2)(3)(4)If , then is a circle of radius with a figure of eight inside it.

The fish curve is a term coined in this work for the ellipse negative pedal curve with pedal point at the focus for the special case of the eccentricity . For an ellipse with parametric equations(1)(2)the corresponding fish curve has parametric equations(3)(4)The Cartesian equation is(5)which, when the origin is translated to the node, canbe written(6)(Lockwood 1957).The interior of the curve is not consistently oriented in the above parametrization, with the fish's head being on the left of the curve and the tail on the right as the curve is traversed. Treating the two pieces separately then gives the areas of the tail and head as(7)(8)giving an overall area for the fish as(9)(Lockwood 1957).The arc length of the curve is given by(10)(11)(12)(Lockwood 1957).The curvature and tangentialangle are given by(13)(14)where is the complex argument.The Tschirnhausen cubic, illustrated above,also resembles a fish, as does the trefoil curve...

The class of curve known as Dürer's conchoid appears in Dürer's work Instruction in Measurement with Compasses and Straight Edge (1525) and arose in investigations of perspective. Dürer constructed the curve by drawing lines and of length 16 units through and , where . The locus of and is the curve, although Dürer found only one of the two branches of the curve.The envelope of the lines and is a parabola, and the curve is therefore a glissette of a point on a line segment sliding between a parabola and one of its tangents.Dürer called the curve "muschellini," which means conchoid. However, it is not a true conchoid and so is sometimes called Dürer's shell curve. The Cartesian equation isThere are a number of interesting special cases. For , the curve becomes the line pair together with the circle . If , the curve becomes two coincident straight lines . If , the curve has a cusp at ...

The elastica formed by bent rods and considered in physics can be generalized to curves in a Riemannian manifold which are a critical point forwhere is the normal curvature of , is a real number, and is closed or satisfies some specified boundary condition. The curvature of an elastica must satisfywhere is the signed curvature of , is the Gaussian curvature of the oriented Riemannian surface along , is the second derivative of with respect to , and is a constant.

The envelope of the lines connecting corresponding points on the Jacobian curve and Steinerian curve. The Cayleyian curve of a net of curves of order has the same curve genus as the Jacobian curve and Steinerian curve and, in general, the class .

The bifolium is a folium with . The bifolium is a quartic curve and is given by the implicit equation is(1)and the polar equation(2)The bifolium has area(3)(4)(5)Its arc length is(6)(7)(OEIS A118307), where , , , and are elliptic integrals with(8)(9)The curvature is given by(10)(11)The bifolium is the pedal curve of the deltoid where the pedal point is the midpoint of one of the three curved sides.

The necessary and sufficient condition that an algebraic curve has an algebraic involute is that the arc length is a two-valued algebraic function of the coordinates of the extremities. Furthermore, this function is a root of a quadratic equation whose coefficients are rational functions of and .

The curve traced by a fixed point on a closed convex curve as that curve rolls without slipping along a second curve. The roulettes described by the foci of conics when rolled upon a line are sections of minimal surfaces (i.e., they yield minimal surfaces when revolved about the line) known as unduloids.A particularly interesting case of a roulette is a regular -gon rolling on a "road" composed of a sequence of truncated catenaries, as illustrated above. This motion is smooth in the sense that the geometric centroid follows a straight line, although in the case of the rolling equilateral triangle, a physical model would be impossible to construct because the vertices of the triangles would get "stuck" in the ruts (Wagon 2000). For the rolling square, the shape of the road is the catenary truncated at (Wagon 2000). For a regular -gon, the Cartesian equation of the corresponding catenary iswhereThe roulette consisting of a square..

The curve produced by fixed point on the circumference of a small circle of radius rolling around the inside of a large circle of radius . A hypocycloid is therefore a hypotrochoid with .To derive the equations of the hypocycloid, call the angle by which a point on the small circle rotates about its center , and the angle from the center of the large circle to that of the small circle . Then(1)so(2)Call . If , then the first point is at minimum radius, and the Cartesian parametric equations of the hypocycloid are(3)(4)(5)(6)If instead so the first point is at maximum radius (on the circle), then the equations of the hypocycloid are(7)(8)The curvature, arc length, and tangential angle of a hypocycloid are given by(9)(10)(11)An -cusped hypocycloid has . For an integer and with , the equations of the hypocycloid therefore become(12)(13)and the arc length and area are therefore(14)(15)A 2-cusped hypocycloid is a line segment (Steinhaus 1999, p. 145;..

The path traced out by a point on the edge of a circle of radius rolling on the outside of a circle of radius . An epicycloid is therefore an epitrochoid with . Epicycloids are given by the parametric equations(1)(2)A polar equation can be derived by computing(3)(4)so(5)But(6)so(7)(8)Note that is the parameter here, not the polar angle. The polar angle from the center is(9)To get cusps in the epicycloid, , because then rotations of bring the point on the edge back to its starting position.(10)(11)(12)(13)so(14)(15)An epicycloid with one cusp is called a cardioid, one with two cusps is called a nephroid, and one with five cusps is called a ranunculoid.Epicycloids can also be constructed by beginning with the diameter of a circle and offsetting one end by a series of steps of equal arc length along the circumference while at the same time offsetting the other end along the circumference by steps times as large. After traveling around the circle once,..

Guilloché patterns are spirograph-like curves that frame a curve within an inner and outer envelope curve. They are used on banknotes, securities, and passports worldwide for added security against counterfeiting. For currency, the precise techniques used by the governments of Russia, the United States, Brazil, the European Union, Madagascar, Egypt, and all other countries are likely quite different. The figures above show the same guilloche pattern plotted in polar and Cartesian coordinates generated by a series of nested additions and multiplications of sinusoids of various periods.Guilloché machines (alternately called geometric lathes, rose machines, engine-turners, and cycloidal engines) were first used for a watch casing dated 1624, and consist of myriad gears and settings that can produce many different patterns. Many goldsmiths, including Fabergè, employed guilloché machines.The..

A superellipse is a curve with Cartesian equation(1)first discussed in 1818 by Lamé. A superellipse may be described parametrically by(2)(3)The restriction to is sometimes made.Superellipses with are also known as Lamé curves or Lamé ovals, and the case with is sometimes known as the squircle. By analogy, the superellipse with and might be termed the rectellipse.A range of superellipses are shown above, with special cases , 1, and 2 illustrated right above. The following table summarizes a few special cases. Piet Hein used with a number of different ratios for various of his projects. For example, he used for Sergels Torg (Sergel's Square) in Stockholm, Sweden (Vestergaard), and for his table.curve(squashed) astroid1(squashed) diamond2ellipsePiet Hein's "superellipse"4rectellipseIf is a rational, then a superellipse is algebraic. However, for irrational , it is transcendental. For even integers..

A hypotrochoid generated by a fixed point on a circle rolling inside a fixed circle. The curves above correspond to values of , 0.2, ..., 1.0.Additional attractive designs such as the one above can also be made by superposing individual spirographs.The Season 1 episode "Counterfeit Reality" (2005) of the television crime drama NUMB3RS features spirographs when discussing Guilloché patterns.

The curve given by the polar equation(1)sometimes also written(2)where .The cardioid has Cartesian equation(3)and the parametric equations(4)(5)The cardioid is a degenerate case of the limaçon. It is also a 1-cusped epicycloid (with ) and is the catacaustic formed by rays originating at a point on the circumference of a circle and reflected by the circle.The cardioid has a cusp at the origin.The name cardioid was first used by de Castillon in Philosophical Transactions of the Royal Society in 1741. Its arc length was found by la Hire in 1708. There are exactly three parallel tangents to the cardioid with any given gradient. Also, the tangents at the ends of any chord through the cusp point are at right angles. The length of any chord through the cusp point is .The cardioid may also be generated as follows. Draw a circle and fix a point on it. Now draw a set of circles centered on the circumference of and passing through . The envelope of these circles..

A strophoid of a circle with the pole at the center of the circle and the fixed point on the circumference of the circle. Freeth (1878, pp. 130 and 228) described this and various other strophoids (MacTutor Archive).It has polar equation(1)The area enclosed by the outer boundary of the curve is(2)and the total arc length is(3)(4)(OEIS A138498), 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.If the line through parallel to the y-axis cuts the nephroid at , then angle is , so this curve can be used to construct a regular heptagon.The curvature and tangentialangle are given by(5)(6)where is the floor function.

The 2-cusped epicycloid is called a nephroid. The name nephroid means "kidney shaped" and was first used for the two-cusped epicycloid by Proctor in 1878 (MacTutor Archive).The nephroid is the catacaustic for rays originating at the cusp of a cardioid and reflected by it. In addition, Huygens showed in 1678 that the nephroid is the catacaustic of a circle when the light source is at infinity, an observation which he published in his Traité de la luminère in 1690 (MacTutor Archive). (Trott 2004, p. 17, mistakenly states that the catacaustic for parallel light falling on any concave mirror is a nephroid.) The shape of the "flat visor curve" produced by a pop-up card dubbed the "knight's visor" is half a nephroid (Jakus and O'Rourke 2012).Since the nephroid has cusps, , and the equation for in terms of the parameter is given by epicycloid equation(1)with ,(2)where(3)This can be written(4)The..

For some range of , the Mandelbrot set lemniscate in the iteration towards the Mandelbrot set is a pear-shaped curve. In Cartesian coordinates with a constant , the equation is given byThe plots above show the resulting curve for (left figure) and for a range of between 0 and 2 (right figure).

The "trefoil" curve is the name given by Cundy and Rollett (1989, p. 72) to the quartic plane curve given by the equation(1)As such, it is a simply a modification of the folium with and (2)obtained by dropping the coefficients 2.The area enclosed by the trefoil curve is(3)the geometric centroid of the enclosed region is(4)(5)and the area moment of inertia elements by(6)(7)(8)(E. Weisstein, Feb 3, 2018).

Given two curves and and a fixed point , let a line from cut at and at . Then the locus of a point such that is the cissoid. The word cissoid means "ivy shaped."curve 1curve 2polecissoidlineparallel lineany pointlinelinecirclecenter of circleconchoid of Nicomedescirclecircle tangent lineon circumferenceoblique cissoidcirclecircle tangent lineon circumference opposite tangentcissoid of Dioclescircleradial lineon circumferencestrophoidcircleconcentric circlecenter of circlescirclecirclesame circlelemniscate

A plane curve proposed by Descartes to challenge Fermat's extremum-finding techniques. In parametric form,(1)(2)The curve has a discontinuity at . The left wing is generated as runs from to 0, the loop as runs from 0 to , and the right wing as runs from to .In Cartesian coordinates,(3)(MacTutor Archive). The equation of the asymptote is(4)The curvature and tangentialangle of the folium of Descartes are(5)(6)where is the Heaviside step function.Converting the parametric equations to polar coordinates gives(7)(8)so the polar equation is(9)The area enclosed by the curve is(10)(11)(12)The arc length of the loop is given by(13)(14)

The folium (meaning leaf-shaped, referring to the lobes present in this curve), also known as Kepler's folium, is the curve with polar equation(1)Its Cartesian equation is(2)If , it is a single folium. If , it is a bifolium. If , it is a three-lobed curve sometimes called a trifolium. A modification of the case , is sometimes called the trefoil curve (Cundy and Rollett 1989, p. 72).The area of the folium is(3)The plots above show families of the folium for between 0 and 4.The simple folium is the pedal curve of the deltoidwhere the pedal point is one of the cusps.

A curve which has the shape of a petalled flower. This curve was named rhodonea by the Italian mathematician Guido Grandi between 1723 and 1728 because it resembles a rose (MacTutor Archive). The polar equation of the rose isorIf is odd, the rose is -petalled. If is even, the rose is -petalled.If is a rational number, then the curve closes at a polar angle of , where if is odd and if is even.If is irrational, then there are an infinite number of petals.The following table summarizes special names gives to roses with various values of .curve2quadrifolium3trifolium, paquerette de mélibée

A curve whose name means skull-like. It is given by the polar equationwhere , , , , and . The top of the curve corresponds to , while the bottom corresponds to .It has area given bywhere is an Appell hypergeometric function.

A curve whose name means "shell form." Let be a curve and a fixed point. Let and be points on a line from to meeting it at , where , with a given constant. For example, if is a circle and is on , then the conchoid is a limaçon, while in the special case that is the diameter of , then the conchoid is a cardioid. The equation for a parametrically represented curve with is(1)(2)

The inverse curve of the cochleoid(1)with inversion center at the origin and inversion radius is the quadratrix of Hippias.(2)(3)

The cochleoid, whose name means "snail-form" in Latin, was first considered by John Perks as referenced in Wallis et al. (1699). The cochleoid has also been called the oui-ja board curve (Beyer 1987, p. 215). The points of contact of parallel tangents to the cochleoid lie on a strophoid.Smith (1958, p. 327) gives historical references for the cochleoid, but corrections to the name and date mentioned as "discussed by J. Perk Phil. Trans. 1700" (actually John Perks, as mentioned in Wallis et al. 1699 and Pedersen 1963), the separateness of papers by Falkenburg (1844) and Benthem (1844), and the spelling of the latter's name should all be noted.In polar coordinates, the curve is given by(1)For the parametric form(2)(3)the curvature is(4)

The pedal curve of the parabolawith parametric equations(1)(2)with pedal point is(3)(4)On the conic section directrix, the pedal curve of a parabola is a strophoid (top left). On the foot of the conic section directrix, it is a right strophoid (top middle). On reflection of the focus in the conic section directrix, it is a Maclaurin trisectrix (top right). On the parabola vertex, it is a cissoid of Diocles (bottom left; Gray 1997, p. 119). On the focus, it is a straight line (bottom right; Hilbert and Cohn-Vossen 1999, pp. 26-27). On the symmetry axis for a parabola with , it is a conchoid of de Sluze (H. Smith, pers. comm., Aug. 4, 2004). The following table summarizes these special cases.pedal pointpedal curvedirectrixstrophoidfoot of directrixright strophoidreflection of focus in directrixMaclaurin trisectrixparabola vertexcissoid of Dioclesfocuslineaxis of a parabola with conchoid of de Sluze..

The lituus is an Archimedean spiral with , having polar equation(1)Lituus means a "crook," in the sense of a bishop's crosier. The lituus curve originated with Cotes in 1722. Maclaurin used the term lituus in his book Harmonia Mensurarum in 1722 (MacTutor Archive). The lituus is the locus of the point moving such that the area of a circular sector remains constant.The arc length, curvature,and tangential angle are given by(2)(3)(4)where the arc length is measured from .

A sinusoidal spiral is a curve of the form(1)with rational, which is not a true spiral. Sinusoidal spirals were first studied by Maclaurin. Special cases are given in the following table.curvehyperbolalineparabolaTschirnhausen cubicCayley's sexticcardioid1circle2lemniscateThe curvature and tangentialangle are(2)(3)

An Archimedean spiral with polarequation(1)The hyperbolic spiral, also called the inverse spiral (Whittaker 1944, p. 83), originated with Pierre Varignon in 1704 and was studied by Johann Bernoulli between 1710 and 1713, as well as by Cotes in 1722 (MacTutor Archive).It is also a special case of a Cotes' spiral, i.e.,the path followed by a particle in a central orbit with power law(2)when is a constant and is the specific angular momentum.The curvature and tangentialangle are given by(3)(4)

Archimedes' spiral is an Archimedean spiralwith polar equation(1)This spiral was studied by Conon, and later by Archimedes in On Spirals about225 BC. Archimedes was able to work out the lengths of various tangents to the spiral.The curvature of Archimedes' spiral is(2)and the arc length is(3)(4)This has the series expansion(5)(6)(OEIS A091154 and A002595), where is a Legendre polynomial.Archimedes' spiral can be used for compass and straightedge division of an angle into parts (including angle trisection) and can also be used for circle squaring. In addition, the curve can be used as a cam to convert uniform circular motion into uniform linear motion (Brown 1923; Steinhaus 1999, p. 137). The cam consists of one arch of the spiral above the x-axis together with its reflection in the x-axis. Rotating this with uniform angular velocity about its center will result in uniform linear motion of the point where it crosses the y-axis...

The inverse curve of the Archimedeanspiralwith inversion center at the origin and inversion radius is the Archimedean spiral

An Archimedean spiral is a spiral with polarequation(1)where is the radial distance, is the polar angle, and is a constant which determines how tightly the spiral is "wrapped."Values of corresponding to particular special named spirals are summarized in the following table, together with the colors with which they are depicted in the plot above.spiralcolorlituusredhyperbolic spiralorangeArchimedes' spiralgreen1Fermat's spiralblue2The curvature of an Archimedean spiral is given by(2)and the arc length for by(3)where is a hypergeometric function.If a fly crawls radially outward along a uniformly spinning disk, the curve it traces with respect to a reference frame in which the disk is at rest is an Archimedean spiral (Steinhaus 1999, p. 137). Furthermore, a heart-shaped frame composed of two arcs of an Archimedean spiral which is fixed to a rotating disk converts uniform rotational motion to uniform back-and-forth..

The radial curve of a unit circle from a radial point and parametric equations(1)(2)is another circle with parametricequations(3)(4)

The pedal curve of a unitcircle with parametric equation(1)(2)with pedal point is(3)(4)The pedal curve with respect to the center is thecircle itself (Gray 1997, pp. 119 and 124-135).If the pedal point is taken on the circumference (and in particular at the point ), the pedal curve is the cardioid(5)(6)and otherwise is a limaçon.

The orthotomic of the unitcircle represented by(1)(2)with a source at is(3)(4)

For a unit circle with parametricequations(1)(2)the negative pedal curve with respect to the pedal point is(3)(4)Therefore if the point is inside the circle (), the negative pedal is an ellipse, if , it is a single point, if the point is outside the circle (), the negative pedal is a hyperbola.

The pedal curve of circleinvolute(1)(2)with the center as the pedal point is the Archimedes'spiral(3)(4)

The involute of the circle was first studied by Huygens when he was considering clocks without pendula for use on ships at sea. He used the circle involute in his first pendulum clock in an attempt to force the pendulum to swing in the path of a cycloid. For a circle of radius ,(1)(2)the parametric equation of the involute is given by(3)(4)The arc length, curvature,and tangential angle are(5)(6)(7)The Cesàro equation is(8)

Consider a unit circle and a radiant point located at . There are four different regimes of caustics, illustrated above.For radiant point at , the catacaustic is the nephroid(1)(2)(Trott 2004, p. 17, mistakenly states that the catacaustic for parallel light falling on any concave mirror is a nephroid.)For radiant point a finite distance , the catacaustic is the curve(3)(4)which is apparently incorrectly described as a limaçonby Lawrence (1972, p. 207).For radiant point on the circumference of the circle (), the catacaustic is the cardioid(5)(6)with Cartesian equation(7)For radiant point inside the circle, the catacausticis a discontinuous two-part curve.If the radiant point is the origin, then the catacaustic degenerates to a single point at the origin since all rays reflect upon themselves back through the origin...

A circle is the set of points in a plane that are equidistant from a given point . The distance from the center is called the radius, and the point is called the center. Twice the radius is known as the diameter . The angle a circle subtends from its center is a full angle, equal to or radians.A circle has the maximum possible area for a given perimeter,and the minimum possible perimeter for a given area.The perimeter of a circle is called the circumference, and is given by(1)This can be computed using calculus using the formula for arc length in polar coordinates,(2)but since , this becomes simply(3)The circumference-to-diameter ratio for a circle is constant as the size of the circle is changed (as it must be since scaling a plane figure by a factor increases its perimeter by ), and also scales by . This ratio is denoted (pi), and has been proved transcendental.Knowing , the area of the circle can be computed either geometrically or using calculus. As the..

Given a parabola with parametricequations(1)(2)the negative pedal curve for a pedal point has equation(3)(4)Taking the pedal point at the origin gives(5)(6)which is a semicubical parabola. Similarly, taking gives(7)(8)which is a Tschirnhausen cubic.

The pedal curve of an ellipse with parametric equations(1)(2)and pedal point is given by(3)(4)The pedal curve of an ellipse with pedal point at the focus is a circle (Hilbert and Cohn-Vossen 1999, pp. 25-26).For other pedal points, the pedal curves are more complicated, as illustrated above.

A curve of constant width constructed by drawing arcs from each polygon vertex of an equilateral triangle between the other two vertices. The Reuleaux triangle has the smallest area for a given width of any curve of constant width. Let the arc radius be . Since the area of each meniscus-shaped portion of the Reuleaux triangle is a circular segment with opening angle ,(1)(2)But the area of the central equilateral triangle with is(3)so the total area is then(4)Because it can be rotated inside a square, as illustratedabove, it is the basis for the Harry Watt square drill bit.When rotated inside a square of side length 2 having corners at ), the envelope of the Reuleaux triangle is a region of the square with rounded corners. At the corner , the envelope of the boundary is given by the segment of the ellipse with parametric equations(5)(6)for , extending a distance from the corner (Gleißner and Zeitler 2000). The ellipse has center , semimajor axis..

Curves which, when rotated in a square, make contact with all four sides. Such curves are sometimes also known as rollers.The "width" of a closed convex curve is defined as the distance between parallel lines bounding it ("supporting lines"). Every curve of constant width is convex. Curves of constant width have the same "width" regardless of their orientation between the parallel lines. In fact, they also share the same perimeter (Barbier's theorem). Examples include the circle (with largest area), and Reuleaux triangle (with smallest area) but there are an infinite number. A curve of constant width can be used in a special drill chuck to cut square "holes."A generalization gives solids of constant width. These do not have the same surface area for a given width, but their shadows are curves of constant width with the same width!..

Given a circle with center and radius , then two points and are inverse with respect to if . If describes a curve , then describes a curve called the inverse of with respect to the circle (with inversion center ). The Peaucellier inversor can be used to construct an inverse curve from a given curve.If the polar equation of is , then the inverse curve has polar equation(1)If and , then the inverse has equations(2)(3)curveinversion centerinverse curveArchimedean spiraloriginArchimedean spiralcardioidcuspparabolacircleany pointanother circlecissoid of Dioclescuspparabolacochleoidoriginquadratrix of HippiasepispiraloriginroseFermat's spiraloriginlituushyperbolacenterlemniscatehyperbolagraph vertexright strophoidhyperbola with graph vertexMaclaurin trisectrixlemniscatecenterhyperbolalituusoriginFermat's spirallogarithmic spiraloriginlogarithmic spiralMaclaurin trisectrixfocusTschirnhausen's..

There are at least two theorems known as Salmon's theorem. This first states that if and are two points, and are the perpendiculars from and to the polars of and , respectively, with respect to a circle with center , then (Durell 1928; Salmon 1954, §101, p. 93).The second Salmon's theorem states that, given a track bounded by two confocal ellipses, if a ball is rolled so that its trajectory is tangent to the inner ellipse, the ball's trajectory will be tangent to the inner ellipse following all subsequent caroms as well (Salmon 1954, §189, pp. 181-182).

A fast boat is overtaking a slower one when fog suddenly sets in. At this point, the boat being pursued changes course, but not speed, and proceeds straight in a new direction which is not known to the fast boat. How should the pursuing vessel proceed in order to be sure of catching the other boat?The amazing answer is that the pursuing boat should continue to the point where the slow boat would be if it had set its course directly for the pursuing boat when the fog set in. If the boat is not there, it should proceed in a spiral whose origin is the point where the slow boat was when the fog set in. The spiral must be constructed in such a way that, while circling the origin, the fast boat's distance from it increases at the same rate as the boat being pursued. The two courses must therefore intersect before the fast boat has completed one circuit. In order to make the problem reasonably practical, the fast boat should be capable of maintaining a speed four or five times..

The tractrix arises in the following problem posed to Leibniz: What is the path of an object starting off with a vertical offset when it is dragged along by a string of constant length being pulled along a straight horizontal line (Steinhaus 1999, pp. 250-251)? By associating the object with a dog, the string with a leash, and the pull along a horizontal line with the dog's master, the curve has the descriptive name "hundkurve" (dog curve) in German. Leibniz found the curve using the fact that the axis is an asymptote to the tractrix (MacTutor Archive).From its definition, the tractrix is precisely the catenary involute described by a point initially on the vertex (so the catenary is the tractrix evolute). The tractrix is sometimes called the tractory or equitangential curve. The tractrix was first studied by Huygens in 1692, who gave it the name "tractrix." Later, Leibniz, Johann Bernoulli, and others studied the curve.In..

If moves along a known curve, then describes a pursuit curve if is always directed toward and and move with uniform velocities. Pursuit curves were considered in general by the French scientist Pierre Bouguer in 1732, and subsequently by the English mathematician Boole.Under the name "path minimization," pursuit curves are alluded to by math genius Charlie Eppes in the Season 2 episode "Dark Matter" of the television crime drama NUMB3RS when considering the actions of the mysterious third shooter.The equations of pursuit are given by(1)which specifies that the tangent vector at point is always parallel to the line connecting and , combined with(2)which specifies that the point moves with constant speed (without loss of generality, taken as unity above). Plugging (2) into (1) therefore gives(3)The case restricting to a straight line was studied by Arthur Bernhart (MacTutor Archive). Taking the parametric equation..

In the mice problem, also called the beetle problem, mice start at the corners of a regular -gon of unit side length, each heading towards its closest neighboring mouse in a counterclockwise direction at constant speed. The mice each trace out a logarithmic spiral, meet in the center of the polygon, and travel a distanceThe first few values for , 3, ..., aregiving the numerical values 0.5, 0.666667, 1, 1.44721, 2, 2.65597, 3.41421, 4.27432, 5.23607, .... The curve formed by connecting the mice at regular intervals of time is an attractive figure called a whirl.The problem is also variously known as the (three, four, etc.) (bug, dog, etc.) problem. It can be generalized to irregular polygons and mice traveling at differing speeds (Bernhart 1959). Miller (1871) considered three mice in general positions with speeds adjusted to keep paths similar and the triangle similar to the original...

The first Brocard point is the interior point (also denoted or ) of a triangle with points labeled in counterclockwise order for which the angles , , and are equal, with the unique such angle denoted . It is not a triangle center, but has trilinear coordinates(1)(Kimberling 1998, p. 47).Note that extreme care is needed when consulting the literature, since reversing the order in which the points of the triangle are labeled results in exchanging the Brocard points.The second Brocard point is the interior point (also denoted or ) for which the angles , , and are equal, with the unique such angle denoted . It is not a triangle center, but has trilinear coordinates(2)(Kimberling 1998, p. 47).Moreover, the two angles are equal, and this angle is called the Brocard angle,(3)(4)The first two Brocard points are isogonal conjugates (Johnson 1929, p. 266). They were described by French army officer Henri Brocard in 1875, although they..

Given a ship with a known constant direction and speed , what course should be taken by a chase ship in pursuit (traveling at speed ) in order to intercept the other ship in as short a time as possible? The problem can be solved by finding all points which can be simultaneously reached by both ships, which is an Apollonius circle with . If the circle cuts the path of the pursued ship, the intersection is the point towards which the pursuit ship should steer. If the circle does not cut the path, then it cannot be caught.

The catacaustic of the natural logarithm specified parametrically as(1)(2)is a complicated expression for an arbitrary radiantpoint.However, for a point , the catacaustic becomes(3)(4)Making the substitution then gives the equivalent parametrization(5)(6)which is the equation of a catenary.

If is a simple closed curve in , then the Jordan curve theorem, also called the Jordan-Brouwer theorem (Spanier 1966) states that has two components (an "inside" and "outside"), with the boundary of each.The Jordan curve theorem is a standard result in algebraic topology with a rich history. A complete proof can be found in Hatcher (2002, p. 169), or in classic texts such as Spanier (1966). Recently, a proof checker was used by a Japanese-Polish team to create a "computer-checked" proof of the theorem (Grabowski 2005).

The normal to an ellipse at a point intersects the ellipse at another point . The angle corresponding to can be found by solving the equation(1)for , where and . This gives solutions(2)where(3)of which gives the valid solution. Plugging this in to obtain then gives(4)(5)(6)To find the maximum distance, take the derivative and set equal to zero,(7)which simplifies to(8)Substituting for and solving gives(9)(10)Plugging these into then gives(11)This problem was given as a Sangaku problem on a tablet from Miyagi Prefecture in 1912 (Rothman 1998). There is probably a clever solution to this problem which does not require calculus, but it is unknown if calculus was used in the solution by the original authors (Rothman 1998).

The polar coordinates (the radial coordinate) and (the angular coordinate, often called the polar angle) are defined in terms of Cartesian coordinates by(1)(2)where is the radial distance from the origin, and is the counterclockwise angle from the x-axis. In terms of and ,(3)(4)(Here, should be interpreted as the two-argument inverse tangent which takes the signs of and into account to determine in which quadrant lies.) It follows immediately that polar coordinates aren't inherently unique; in particular, will be precisely the same polar point as for any integer . What's more, one often allows negative values of under the assumption that is plotted identically to .The expression of a point as an ordered pair is known as polar notation, the equation of a curve expressed in polar coordinates is known as a polar equation, and a plot of a curve in polar coordinates is known as a polar plot.In much the same way that Cartesian curves can be plotted on..

A plot of a function expressed in polar coordinates, with radius as a function of angle . Polar plots can be drawn in the Wolfram Language using PolarPlot[r, t, tmin, tmax]. The plot above is a polar plot of the polar equation , giving a cardioid.Polar plots of give curves known as roses, while polar plots of produce what's known as Archimedes' spiral, a special case of the Archimedean spiral corresponding to . Other specially-named Archimedean spirals include the lituus when , the hyperbolic spiral when , and Fermat's spiral when . Note that lines and circles are easily-expressed in polar coordinates as(1)and(2)for the circle with center and radius , respectively. Note that equation () is merely a particular instance of the equation(3)defining a conic section of eccentricity and semilatus rectum . In particular, the circle is the conic of eccentricity , while yields a general ellipse, a parabola, and a hyperbola.The plotting of a complex number..

The dual of Pascal's theorem (Casey 1888, p. 146). It states that, given a hexagon circumscribed on a conic section, the lines joining opposite polygon vertices (polygon diagonals) meet in a single point.In 1847, Möbius (1885) gave a statement which generalizes Brianchon's theorem: if all lines (except possibly one) connecting two opposite vertices of a ()-gon circumscribed on a conic section meet in one point, then the same is true for the remaining line.

The Tschirnhausen cubic is a plane curve given by the polarequation(1)Letting gives the parametric equations(2)(3)or(4)(5)(Lawrence 1972, p. 88).Eliminating from the above equations gives the Cartesian equations(6)(7)(Lawrence 1972, p. 88).The curve is also known as Catalan's trisectrix and l'Hospital's cubic. The name Tschirnhaus's cubic is given in R. C. Archibald's 1900 paper attempting to classify curves (MacTutor Archive).The curve has a loop, illustrated above, corresponding to in the above parametrization. The area of the loop is given by(8)(9)(10)(11)(Lawrence 1972, p. 89).In the first parametrization, the arc length, curvature, and tangential angle as a function of are(12)(13)(14)The curve has a single ordinary double point located at in the parametrization of equations (◇) and (◇).The Tschirnhausen cubic is the negative pedal curve of a parabola with respect to..

The polar curve(1)that can be used for angle trisection. It was devised by Ceva in 1699, who termed it the cycloidum anomalarum (Loomis 1968, p. 29). It has Cartesian equation(2)It has area(3)and arc length(4)(5)(OEIS A138497), with , where , , and are complete elliptic integrals of the first, second, and third, respectively.The arc length function is a slightly complicated expression that can be expressed in closed form in terms of elliptic functions, and the curvature is given by(6)

An equation of the form(1)where the three roots of the equation coincide (and aretherefore real), i.e.,(2)Loomis (1968, p. 28) considers a cubical parabola of the form(3)which can be used for angle trisection.

The Maclaurin trisectrix is a curve first studied by Colin Maclaurin in 1742. It was studied to provide a solution to one of the geometric problems of antiquity, in particular angle trisection, whence the name trisectrix. The Maclaurin trisectrix is an anallagmatic curve, and the origin is a crunode.The Maclaurin trisectrix has Cartesian equation(1)or the parametric equations(2)(3)The asymptote has equation , and the center of the loop is at . If is a point on the loop so that the line makes an angle of with the negative y-axis, then the line will make an angle of with the negative y-axis.The Maclaurin trisectrix is given in polar coordinatesas(4)(5)Another form of the polar equation is the polarequation(6)which is a version shifted by two units along the -axis so that the origin is inside the loop.The tangents to the curve at the origin make angles of with the x-axis. The area and arc length of the loop are(7)(8)(9)(OEIS A138499), where is an elliptic..

The limaçon is a polar curve ofthe form(1)also called the limaçon of Pascal. It was first investigated by Dürer, who gave a method for drawing it in Underweysung der Messung (1525). It was rediscovered by Étienne Pascal, father of Blaise Pascal, and named by Gilles-Personne Roberval in 1650 (MacTutor Archive). The word "limaçon" comes from the Latin limax, meaning "snail."If , the limaçon is convex. If , the limaçon is dimpled. If , the limaçon degenerates to a cardioid. If , the limaçon has an inner loop. If , it is a trisectrix (but not the Maclaurin trisectrix).For , the inner loop has area(2)(3)(4)where . Similarly the area enclosed by the outer envelope is(5)(6)(7)Thus, the area between the loops is(8)In the special case of , these simplify to(9)(10)(11)Taking the parametrization(12)(13)gives the arc length as a function of as(14)where is an elliptic..

A curve with polar coordinates,(1)studied by the Greek mathematician Nicomedes in about 200 BC, also known as the cochloid. It is the locus of points a fixed distance away from a line as measured along a line from the focus point (MacTutor Archive). Nicomedes recognized the three distinct forms seen in this family for , , and . (For , it obviously degenerates to a circle.)The conchoid of Nicomedes was a favorite with 17th century mathematicians and could be used to solve the problems of cube duplication, angle trisection, heptagon construction, and other Neusis constructions (Johnson 1975).In Cartesian coordinates, the conchoid ofNicomedes may be written(2)or(3)The conchoid has as an asymptote, and the area between either branch and the asymptote is infinite.A conchoid with has a loop for , where , giving area(4)(5)(6)The curvature and tangentialangle are given by(7)(8)..

The quadratrix was discovered by Hippias of Elias in 430 BC, and later studied by Dinostratus in 350 BC (MacTutor Archive). It can be used for angle trisection or, more generally, division of an angle into any integral number of equal parts, and circle squaring.It has polar equation(1)with corresponding parametric equation(2)(3)and Cartesian equation(4)Using the parametric representation, the curvatureand tangential angle are given by(5)(6)for .

The 60 Pascal lines of a hexagon inscribed in a conic intersect three at a time through 20 Steiner points, and also three at a time in 60 points known as Kirkman points. Each Steiner point lies together with three Kirkman points on a total of 20 lines known as Cayley lines. There is a reciprocity relationship between the 60 Kirkman points and the 60 Pascal lines (Hesse, quoted in Salmon 1960), although the relationship is not one of duality in the commonly accepted meaning of that word.

A semicubical parabola is a curve of the form(1)(i.e., it is half a cubic, and hence has power ). It has parametric equations(2)(3)and the polar equation(4)The evolute of the parabola is a particular case of the semicubical parabola also called Neile's parabola or the cuspidal cubic. In Cartesian coordinates, it has equation(5)which can also be written(6)The Tschirnhausen cubic catacausticis also a semicubical parabola.The semicubical parabola is the curve along which a particle descending under gravity describes equal vertical spacings within equal times, making it an isochronous curve. It was discovered by William Neile in 1657 and was the first nontrivial algebraic curve to have its arc length computed. Wallis published the method in 1659, giving Neile the credit (MacTutor Archive). The problem of finding the curve having this property had been posed by Leibniz in 1687 and was also solved by Huygens (MacTutor Archive).The semicubical..

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..

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