# List of Famous Curves

Here you can find, as a curiosity, a list of curves that made history in mathematics. In each one of them, you will be able to consult the name of the mathematician(s) to whom the discovery was attributed, as well as its equation and the graphical representation of the curve.

#### Astroid

DescriptionEquationGraphic
The astroid was first discussed by Johann Bernoulli in 1691-92. It also appears in Leibniz's correspondence of 1715. It is sometimes called the tetracuspid for the obvious reason that it has four cusps. The astroid only acquired its present name in 1836 in a book published in Vienna. It has been known by various names in the literature, even after 1836, including cubocycloid and paracycle. x^(2/3) + y^(2/3) = a^(2/3)

#### Bicorn

DescriptionEquationGraphic
The bicorn (also called the cocked-hat) is the name of a collection of quartic curves studied by Sylvester in 1864. The same curves were the studied by Cayley in 1867. The particular bicorn given by Sylvester and Cayley is a different quartic from the one given here but this one, with a simpler formula, has essentially the same shape. y^2(a^2 - x^2) = (x^2 + 2ay - a^2)^2

#### Cardioid

DescriptionEquationGraphic
The cardioid, a name first used by de Castillon in a paper in the Philosophical Transactions of the Royal Societyin 1741, is a curve that is the locus of a point on the circumference of circle rolling round the circumference of a circle of equal radius. Of course the name means 'heart-shaped'. Its length had been found by La Hire in 1708, and he therefore has some claim to be the discoverer of the curve. In the notation given above the length is 16a. It is a special case of the Limacon of Pascal (Etienne Pascal) and so, in a sense, its study goes back long before Castillon or La Hire. (x^2 + y^2 - 2ax)^2 = 4a^2(x^2 + y^2)

#### Cartesian Oval

DescriptionEquationGraphic
This curve C consists of two ovals so it should really be called Cartesian Ovals. It is the locus of a point P whose distances s and t from two fixed points S and T satisfy s + mt = a. When c is the distance between S and T then the curve can be expressed in the form given above. The curves were first studied by Descartes in 1637 and are sometimes called the 'Ovals of Descartes'. The curve was also studied by Newton in his classification of cubic curves. ((1 - m^2)(x^2 + y^2) + 2m^2cx + a^2 - m^2c^2)^2
= 4a^2(x^2 + y^2)

#### Cassinian Ovals

DescriptionEquationGraphic
The curve was first investigated by Giovanni Cassini in 1680 when he was studying the relative motions of the Earth and the Sun. Cassini believed that the Sun travelled round the Earth on one of these ovals, with the Earth at one focus of the oval. Cassini actually introduced his curves 14 years before Jacob Bernoulli described his lemniscate. (x^2 + y^2)^2 - 2a^2(x^2 - y^2) + a^4 - c^4 = 0

#### Catenary

DescriptionEquationGraphic
The catenary is the shape of a perfectly flexible chain suspended by its ends and acted on by gravity. Its equation was obtained by Leibniz, Huygens and Johann Bernoulli in 1691. They were responding to a challenge put out by Jacob Bernoulli to find the equation of the 'chain-curve'. Huygens was the first to use the term catenary in a letter to Leibniz in 1690 and David Gregory wrote a treatise on the catenary in 1690. Jungius (1669) disproved Galileo's claim that the curve of a chain hanging under gravity would be a parabola. The catenary is the locus of the focus of a parabola rolling along a straight line. y = a cosh(x/a)

#### Cayley's Sextic

DescriptionEquationGraphic
This was first discovered by Maclaurin but 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. The evolute of Cayley's Sextic is a nephroid curve. 4(x^2 + y^2 - ax)^3 = 27a^2(x^2 + y^2)^2

#### Circle

DescriptionEquationGraphic
The study of the circle goes back beyond recorded history. The invention of the wheel is a fundamental discovery of properties of a circle. The first theorems relating to circles are attributed to Thales around 650 BC. Book III of Euclid's Elements deals with properties of circles and problems of inscribing and escribing polygons. One of the problems of Greek mathematics was the problem of finding a square with the same area as a given circle. Several of the 'famous curves' in this stack were first studied in an attempt to solve this problem. Anaxagoras in 450 BC is the first recored mathematician to study this problem. x^2 + y^2 = a^2

#### Cissoid of Diocles

DescriptionEquationGraphic
This curve (meaning 'ivy-shaped') was invented by Diocles in about 180 BC in connection with his attempt to duplicate the cube by geometrical methods. The name first appears in the work of Geminus about 100 years later. Fermat and Roberval constructed the tangent in 1634. Huygens and Wallis found, in 1658, that the area between the curve and its asymptote was 3πa2. From a given point there are either one or three tangents to the cissoid. The Cissoid of Diocles is the roulette of the vertex of a parabola rolling on an equal parabola. y^2 = x^3/(2a - x)

#### Conchoid

DescriptionEquationGraphic
The name means shell form and was studied by the Greek mathematician Nicomedes in about 200 BC in relation to the problem of duplication of the cube. Nicomedes recognised the three distinct forms seen in this family. Nicomedes was a minor geometer who worked around 180 BC. His main invention was the conchoid ascribed to him by Pappus. It was a favourite with 17 Century mathematicians and could be used, as Nicomedes had intended, to solve the problems of duplicating the cube and trisecting an angle. (x - b)^2(x^2 + y^2) - a^2x^2 = 0

#### Cycloid

DescriptionEquationGraphic
The cycloid was first studied by Cusa when he was attempting to find the area of a circle by integration. Mersenne gave the first proper definition of the cycloid and stated the obvious properties such as the length of the base equals the circumference of the rolling circle. Mersenne attempted to find the area under the curve by integration but failed. He posed the question to other mathematicians. The curve was named by Galileo in 1599. In 1639 he wrote to Torricelli about the cycloid, saying that he had been studying its properties for 40 years. x = at - h sin(t)
y = a - h cos(t)

#### Devil's Curve

DescriptionEquationGraphic
The Devil's Curve was studied by Gabriel Cramer in 1750 and Lacroix in 1810. It appears in Nouvelles Annalesin 1858. Cramer (1704-1752) was a Swiss mathematician. He became professor of mathematics at Geneva and wrote on work related to physics; also on geometry and the history of mathematics. He is best known for his work on determinants (1750) but also made contributions to the study of algebraic curves (1750). y^4 - x^4 + a y^2 + b x^2 = 0

#### Ellipse

DescriptionEquationGraphic
The ellipse was first studied by Menaechmus. Euclid wrote about the ellipse and it was given its present name by Apollonius. The focus and directrix of an ellipse were considered by Pappus. Kepler, in 1602, said he believed that the orbit of Mars was oval, then he later discovered that it was an ellipse with the sun at one focus. In fact Kepler introduced the word "focus" and published his discovery in 1609. The eccentricity of the planetary orbits is small (i.e. they are close to circles). In 1705 Halley showed that the comet, which is now called after him, moved in an elliptical orbit round the sun. x^2/a^2 + y^2/b^2 = 1

#### Epicycloid

DescriptionEquationGraphic
There are four curves which are closely related. These are the epicycloid, the epitrochoid, the hypocycloid and the hypotrochoid and they are traced by a point P on a circle of radius b which rolls round a fixed circle of radius a. For the epicycloid, an example of which is shown above, the circle of radius b rolls on the outside of the circle of radius a. The point P is on the circumference of the circle of radius b. For the example drawn here a = 8 and b = 5. x = (a + b) cos(t) - b cos((a/b + 1)t)
y = (a + b) sin(t) - b sin((a/b + 1)t)

#### Epitrochoid

DescriptionEquationGraphic
An example of an epitrochoid appears in Dürer's work Instruction in measurement with compasses and straight edge(1525). He called them spider lines because the lines he used to construct the curves looked like a spider. These curves were studied by la Hire, Desargues, Leibniz, Newton and many others. x = (a + b) cos(t) - c cos((a/b + 1)t)
y = (a + b) sin(t) - c sin((a/b + 1)t)

#### Equiangular Spiral

DescriptionEquationGraphic
The equiangular spiral was invented by Descartes in 1638. Torricelli worked on it independently and found the length of the curve. If P is any point on the spiral then the length of the spiral from P to the origin is finite. In fact, from the point P which is at distance d from the origin measured along a radius vector, the distance from P to the pole is d sec b. Jacob Bernoulli in 1692 called it spira mirabilis and it is carved on his tomb in Basel. r = a exp(theta cot b)

#### Fermat's Spiral

DescriptionEquationGraphic
This spiral was discussed by Fermat in 1636. For any given positive value of θ there are two corresponding values of r, one being the negative of the other. The resulting spiral will therefore be symmetrical about the line y = -x as can be seen from the curve displayed here. r^2 = a^2 theta

#### Folium of Descartes

DescriptionEquationGraphic
This folium was first discussed by Descartes in 1638 but, although he found the correct shape of the curve in the positive quadrant, he believed that this leaf shape was repeated in each quadrant like the four petals of a flower. The problem to determine the tangent to the curve was proposed to Roberval who also wrongly believed the curve had the form of a jasmine flower. His name of fleur de jasminwas later changed. x^3 + y^3 = 3axy

#### Frequency Curve

DescriptionEquationGraphic
This curve, the normal curve of error, originated with de Moivre in 1733. It was also studied with Laplace and Gauss. The name frequency curve also applies to a great variety of other curves. y = sqrt(2 pi) exp(-x^2/2)

#### Hiperbola

DescriptionEquationGraphic
A special case of the hyperbola was first studied by Menaechmus. This special case was xy = ab where the asymptotes are at right angles and this particular form of the hyperbola is called a rectangular hyperbola. Euclid and Aristaeus wrote about the general hyperbola but only studied one branch of it while the hyperbola was given its present name by Apollonius who was the first to study the two branches of the hyperbola. x^2/a^2 - y^2/b^2 = 1

#### Hyperbolic Spiral

DescriptionEquationGraphic
The hyperbolic spiral originated with Pierre Varignon in 1704. It was studied by Johann Bernoulli between 1710 and 1713 and it was also studied by Cotes in 1722. Pierre Varignon (1654-1722) was professor of mathematics at Collège Mazarin and later at Collège Royal. Led into mathematics by reading Euclid he also read Descartes' Géométrieand thereafter devoted himself to the mathematical sciences. He was one of the first French scholars to recognise the value of the calculus. His chief contributions were to mechanics. r = a/theta

#### Hypocycloid

DescriptionEquationGraphic
There are four curves which are closely related. These are the epicycloid, the epitrochoid, the hypocycloid and the hypotrochoid and they are traced by a point P on a circle of radius b which rolls round a fixed circle of radius a. For the hypocycloid, an example of which is shown here, the circle of radius b rolls on the inside of the circle of radius a. The point P is on the circumference of the circle of radius b. For the example a = 5 and b = 3. x = (a - b) cos(t) + b cos((a/b - 1)t)
y = (a - b) sin(t) - b sin((a/b - 1)t)

#### Hypotrochoid

DescriptionEquationGraphic
For the hypotrochoid, an example of which is shown here, the circle of radius b rolls on the inside of the circle of radius a. The point P is at distance c from the centre of the circle of radius b. For this example a = 5, b = 7 and c = 2.2. These curves were studied by la Hire, Desargues, Leibniz, Newton and many others. x = (a - b) cos(t) + c cos((a/b -1)t)
y = (a - b) sin(t) - c sin((a/b -1)t)

#### Involute of a Circle

DescriptionEquationGraphic
The involute of a circle is the path traced out by a point on a straight line that rolls around a circle. It was studied by Huygens when he was considering clocks without pendulums that might be used on ships at sea. He used the involute of a circle in his first pendulum clock in an attempt to force the pendulum to swing in the path of a cycloid. Finding a clock which would keep accurate time at sea was a major problem and many years were spent looking for a solution. The problem was of vital importance since if GMT was known from a clock then, since local time could be easily computed from the Sun, longitude could be easily computed. x = a(cos(t) + t sin(t))
y = a(sin(t) - t cos(t))

#### Kampyle of Eudoxus

DescriptionEquationGraphic
A curve studied by Eudoxus also in relation to the classical problem of duplication of the cube. Eudoxus was a pupil of Plato. His main work was in astronomy. He was the first to describe the constellations and invented the astrolabe. He introduced the study of mathematical astronomy into Greece. Eudoxus found formulas for measuring pyramids cones and cylinders. His work contains elements of the calculus with a rigorous study of the method of exhaustion. a^2x^4 = b^4(x^2 + y^2)

#### Kappa Curve

DescriptionEquationGraphic
The kappa curve is also called Gutschoven's curve. It was first studied by G. van Gutschoven around 1662. The curve was studied by Newton and, some years later, by Johann Bernoulli. y^2(x^2 + y^2) = a^2x^2

#### Lame Curves

DescriptionEquationGraphic
In 1818 Lamé discussed the curves with equation given above. He considered more general curves than just those where n is an integer. If n is a rational then the curve is algebraic but, for irrational n, the curve is transcendental. The curve drawn above is the case n = 4. For even integers n the curve becomes closer to a rectangle as n increases. For odd integer values of n the curve looks like the even case in the positive quadrant but goes to infinity in both the second and fourth quadrants. (x/a)^n + (y/b)^n = 1

#### Lemniscate of Bernoulli

DescriptionEquationGraphic
In 1694 Jacob Bernoulli published an article in Acta Eruditorumon a curve "shaped like a figure 8, or a knot, or the bow of a ribbon" which he called by the Latin word lemniscus ('a pendant ribbon'). Jacob Bernoulli was not aware that the curve he was describing was a special case of a Cassinian Oval which had been described by Cassini in 1680. The general properties of the lemniscate were discovered by Giovanni Fagnano in 1750. Euler's investigations of the length of arc of the curve (1751) led to later work on elliptic functions. (x^2 + y^2)^2 = a^2(x^2 - y^2)

#### Limacon of Pascal

DescriptionEquationGraphic
Limacon of Pascal was discovered by Étienne Pascal (father of Blaise Pascal) and named by another Frenchman Gilles-Personne Roberval in 1650 when he used it as an example of his methods of drawing tangents i.e. differentiation. The name 'limacon' comes from the Latin limax meaning 'a snail'. Étienne Pascal corresponded with Mersenne whose house was a meeting place for famous geometers including Roberval. (x^2 + y^2 - 2ax)^2 = b^2(x^2 + y^2)

#### Lissajous Curves

DescriptionEquationGraphic
Lissajous curves or Lissajous figures are sometimes called Bowditch curves after Nathaniel Bowditch who considered them in 1815. They were studied in more detail (independently) by Jules-Antoine Lissajous in 1857. Lissajous curves have applications in physics, astronomy and other sciences. x = a sin(nt + c)
y = b sin(t)

#### Lituus

DescriptionEquationGraphic
The lituus curve originated with Cotes in 1722. Lituus means a crook, for example a bishop's crosier. Maclaurin used the term in his book Harmonia Mensurarumin 1722. The lituus is the locus of the point P moving in such a manner that the area of a circular sector remains constant. Roger Cotes (1682-1716) died at the age of 34 having only published two memoirs during his lifetime. Appointed professor at Cambridge at the age of 24 his work was published only after his death. r^2 = a^2/theta

#### Neile's Semi-cubical Parabola

DescriptionEquationGraphic
This curve, sometimes called the semi-cubical parabola, was discovered by William Neile in 1657. It was the first algebraic curve to have its arc length computed. Wallis published the method in 1659 giving Neile the credit. The Dutch writer Van Heuraet used the curve for a more general construction. William Neile was born at Bishopsthrope in 1637. He was a pupil of Wallis and showed great promise. Neile's parabola was the first algebraic curve to have its arc length calculated; only the arc lengths of transcendental curves such as the cycloid and the logarithmic spiral had been calculated before this. Unfortunately Neile died at a young age in 1670 before he had achieved many further results. y^3 = a x^2

#### Parabola

DescriptionEquationGraphic
The parabola was studied by Menaechmus who was a pupil of Plato and Eudoxus. He attempted to duplicate the cube, namely to find side of a cube that has a volume double that of a given cube. Hence he attempted to solve x3 = 2 by geometrical methods. In fact the geometrical methods of ruler and compass constructions cannot solve this (but Menaechmus did not know this). y = ax^2 + bx + c

#### Pearls of Sluze

DescriptionEquationGraphic
The curves with the equation given above, where n, p and m are integers, were studied by de Sluze between 1657 and 1698. The name Pearls of Sluze was given to these curves by Blaise Pascal. y^n = k(a - x)^px^m

#### Pursuit Curve

DescriptionEquationGraphic
If A moves along a known curve then P describes a pursuit curve if P is always directed towards A and A and P move with uniform velocities. These were considered in general by the French scientist Pierre Bouguer in 1732. The case here is where A is on a straight line and was studied by Arthur Bernhart. Pierre Bouguer was a French scientist who was the first to attempt to measure the density of the Earth using the deflection of a plumb line due to the attraction of a mountain. He made measurements in Peru in 1740. y = cx^2 - log(x)

DescriptionEquationGraphic
The quadratrix was discovered by Hippias of Elis in 430 BC. It may have been used by him for trisecting an angle and squaring the circle. The curve may be used for dividing an angle into any number of equal parts. Later it was studied by Dinostratus in 350 BC who used the curve to square the circle. Hippias of Elis was a statesman and philosopher who travelled from place to place taking money for his services. Plato describes him as a vain man being both arrogant and boastful. He had a wide but superficial knowledge. His only contribution to mathematics seems to be the quadratrix. y = x cot((pi x)/(2a))

#### Straight Line

DescriptionEquationGraphic
The straight line must be one of the earliest curves studied, but Euclid in his Elementsalthough he devotes much study to the straight line, does not consider it a curve. In fact nobody attempted a general definition of a curve until Jordan in his Cours d'Analysein 1893. y=mx + b

#### Right Strophoid

DescriptionEquationGraphic
The strophoid first appears in work by Isaac Barrow in 1670. However Torricelli describes the curve in his letters around 1645 and Roberval found it as the locus of the focus of the conic obtained when the plane cutting the cone rotates about the tangent at its vertex. y^2 = x^2(a - x)/(a + x)

#### Serpentine

DescriptionEquationGraphic
This curve was named and studied by Newton in 1701. It is contained in his classification of cubic curves which appears in Curves by Sir Isaac Newton in Lexicon Technicumby John Harris published in London in 1710. Harris's introduction to the article states: "The incomparable Sir Isaac Newton gives this following Ennumeration of Geometrical Lines of the Third or Cubick Order; in which you have an admirable account of many Species of Curves which exceed the Conick-Sections, for they go no higher than the Quadratick or Second Order." x^2y + aby - a^2x = 0

#### Spiral of Archimedes

DescriptionEquationGraphic
This spiral was studied by Archimedes in about 225 BC in a work On Spirals. It had already been considered by his friend Conon. Archimedes was able to work out the lengths of various tangents to the spiral. It can be used to trisect an angle and square the circle. The curve can be used as a cam to convert uniform angular motion into uniform linear motion. 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 centre will result in uniform linear motion of the point where it crosses the y-axis. r = a + b theta

#### Tractrix

DescriptionEquationGraphic
The tractrix is sometimes called a tractory or equitangential curve. It was first studied by Huygens in 1692 who gave it its name. Later Leibniz, Johann Bernoulli and others studied the curve. The study of the tractrix started with the following problem being posed to Leibniz: "What is the path of an object dragged along a horizontal plane by a string of constant length when the end of the string not joined to the object moves along a straight line in the plane?" x = 1/cosh(t)
y = t - tanh(t)

#### Tricuspoid

DescriptionEquationGraphic
The tricuspoid or deltoid was first considered by Euler in 1745 in connection with an optical problem. It was also investigated by Steiner in 1856 and is sometimes called Steiner's hypocycloid. The length of the tangent to the tricuspoid, measured between the two points P, Q in which it cuts the curve again is constant and equal to 4a. If you draw tangents at P and Q they are at right angles. (x^2+y^2)^2+18a^2(x^2+y^2)-27a^4 = 8a(x^3-3xy^2)

#### Trident of Newton

DescriptionEquationGraphic
This curve was investigated by Newton and also by Descartes. It is sometimes called the 'Parabola of Descartes' even although it is not a parabola. The name trident is due to Newton. The curve occurs in Newton's study of cubics. It is contained in his classification of cubic curves which appears in Curves by Sir Isaac Newtonin Lexicon Technicumby John Harris published in London in 1710. Newton was the first to undertake such a systematic study of cubic equations and he classified them into 72 different cases. In fact he missed six cases in his classification. The trident is the 66th species in his classification and Newton gives the graph essentially looking identical to the graph given above. xy = cx^3 + dx^2 + ex + f

#### Trisectrix of Maclaurin

DescriptionEquationGraphic
This was first studied by Colin Maclaurin in 1742. Like so many curves it was studied to provide a solution to one of the ancient Greek problems, this one is in relation to the problem of trisecting an angle. The name trisectrix arises since it can be used to trisect angles. The trisectrix of Maclaurin is an anallagmatic curve. y^2(a + x) = x^2(3a - x)

#### Tschirnhaus Cubic

DescriptionEquationGraphic
This curve was investigated by Tschirnhaus, de L'Hôpital and Catalan. As well as Tschirnhaus's cubic it is sometimes called de L'Hôpital's cubic or the trisectrix of Catalan. The name Tschirnhaus's cubic is given in R C Archibald's paper written in 1900 where he attempted to classify curves. Tschirnhaus's cubic is the negative pedal of a parabola with respect to the focus of the parabola.  3a y^2 = x(x-a)^2

#### Witch of Agnesi

DescriptionEquationGraphic
This was studied and named versiera by Maria Agnesi in 1748 in her book Istituzioni Analitiche. It is also known as Cubique d'Agnesi or Agnésienne. There is a discussion on how it came to be called witch in Agnesi's biography. The curve had been studied earlier by Fermat and Guido Grandi in 1703. y(x^2 + a^2) = a^3