Inmathematics, ageneralized conic is ageometrical object defined by a property which is ageneralization of some defining property of the classicalconic. For example, inelementary geometry, anellipse can be defined as thelocus of a point which moves in a plane such that the sum of its distances from two fixed points – thefoci – in the plane is a constant. The curve obtained when the set of two fixed points is replaced by an arbitrary, but fixed,finite set of points in the plane is called ann–ellipse and can be thought of as a generalized ellipse. Since an ellipse is theequidistant set of two circles, where one circle is inside the other, the equidistant set of two arbitrary sets of points in a plane can be viewed as a generalized conic. In rectangularCartesian coordinates, the equationy =x² represents aparabola. The generalized equationy =x^r, forr ≠ 0 andr ≠ 1, can be treated as defining a generalized parabola. The idea of generalized conic has found applications inapproximation theory andoptimization theory.^[1]

Among the several possible ways in which the concept of a conic can be generalized, the most widely used approach is to define it as a generalization of theellipse. The starting point for this approach is to look upon an ellipse as a curve satisfying the 'two-focus property': an ellipse is a curve that is the locus of points the sum of whose distances from two given points is constant. The two points are the foci of the ellipse. The curve obtained by replacing the set of two fixed points by an arbitrary, but fixed, finite set of points in the plane can be thought of as a generalized ellipse. Generalized conics with three foci are called trifocal ellipses. This can be further generalized to curves which are obtained as the loci of points such that someweighted sum of the distances from a finite set of points is a constant. A still further generalization is possible by assuming that the weights attached to the distances can be of arbitrary sign, namely, plus or minus. Finally, the restriction that the set of fixed points, called the set of foci of the generalized conic, be finite may also be removed. The set may be assumed to be finite or infinite. In the infinite case, the weighted arithmetic mean has to be replaced by an appropriate integral. Generalized conics in this sense are also calledpolyellipses,egglipses, or,generalized ellipses. Since such curves were considered by the German mathematicianEhrenfried Walther von Tschirnhaus (1651 – 1708) they are also known asTschirnhaus'sche Eikurve.^[2] Also such generalizations have been discussed byRené Descartes^[3] and by James Clerk Maxwell.^[4]

René Descartes in hisLa Géométrie (1637) set apart a section of about 15 pages to discuss what he called bifocal ellipses. A bifocal oval was defined there as the locus of a pointP which moves in a plane such that${\displaystyle AP+\lambda BP=c}$ whereA andB are fixed points in the plane andλ andc are constants which may be positive or negative. Descartes introduced these ovals, which are now known asCartesian ovals, to determine the surfaces of glass such that afterrefraction the rays meet at the same point. Descartes also recognized these ovals as generalizations of central conics, because for certain values ofλ these ovals reduce to the familiar conics, namely, the circle, the ellipse or the hyperbola.^[3]

Multifocal ovals were rediscovered byJames Clerk Maxwell (1831–1879) while he was still a school student. At the age of 15, Maxwell wrote a scientific paper on these ovals with the title "Observations on circumscribed figures having a plurality of foci, and radii of various proportions". It was presented byJames David Forbes in a meeting of theRoyal Society of Edinburgh in 1846. Forbes also published an account of the paper in theProceedings of the Royal Society of Edinburgh.^[4][5] In his paper, though Maxwell did not use the term "generalized conic", he was considering curves which were generalizations of the defining condition of an ellipse.

A multifocal oval is a curve which is defined as the locus of a point moving such that

${\displaystyle {\lambda }_1{A}_{1}P+{\lambda }_2{A}_{2}P+\cdots +{\lambda }_n{A}_{n}P=c}$

whereA₁,A₂, . . . ,A_n are fixed points in a plane andλ₁,λ₂, . . . ,λ_n are fixed rational numbers andc is a constant. He gave simple pin-string-pencil methods for drawing such ovals.

The method for drawing the oval defined by the equation${\displaystyle AP+2BP=c}$ illustrates the general approach adopted by Maxwell for drawing such curves. Fix two pins at the fociA andB. Take a string whose length isc +AB and tie one end of the string to the pin atA. A pencil is attached to the other end of the string and the string is passed round the pin at the focusB. The pencil is then moved guided by the bight of the string. The curve traced by the pencil is the locus ofP. His ingenuity is more visible in his description of the method for drawing a trifocal oval defined by an equation of the form${\displaystyle AP+BP+CP=c}$. Let three pins be fixed at the three fociA,B,C. Let one end of the string be fixed at the pin atC and let the string be passed around the other pins. Let the pencil be attached to the other end of the string. Let the pencil catch a bight in the string betweenA andC and then stretch toP. The pencil is moved such that the string is taut. The resulting figure would be a part of a trifocal ellipse. The positions of the string may have to adjusted to get the full oval.

In the two years after his paper was presented to the Royal Society of Edinburgh, Maxwell systematically developed the geometrical and optical properties of these ovals.^[5]

As a special case of Maxwell's approach, consider then-ellipse—the locus of a point which moves such that the following condition is satisfied:

${\displaystyle A_{1}P+A_{2}P+\cdots +A_{n}P=c.}$

Dividing byn and replacingc/n byc, this defining condition can be stated as

${\displaystyle \frac{1}{n}(A_{1}P+A_{2}P+\cdots +A_{n}P)=c}$

This suggests a simple interpretation: the generalised conic is a curve such that the average distance of every pointP on the curve from the set {A₁,A₂, . . . ,A_n} has the same constant value. This formulation of the concept of a generalized conic has been further generalised in several different ways.

• Change the definition of the average. In the formulation, the average was interpreted as the arithmetic mean. This may be replaced by other notions of averages like geometric mean of the distances. If the geometric mean is used to specify the average, the resulting curves turn out to belemniscates. "Lemniscates are sets all of whose points have the same geometric mean of the distances (i.e. their product is constant). Lemniscates play a central role in the theory of approximation. The polynomial approximation of aholomorphic function can be interpreted as the approximation of the level curves with lemniscates. The product of distances corresponds to the absolute value of the root-decomposition of polynomials in thecomplex plane."^[6]
• Change thecardinality of the focal set. Modify the definition so that the definition can be applied even in the case where the focal set infinite. This possibility was first introduced by C. Gross and T.-K. Strempel [2] and they posed the problem whether which results (of the classical case) can be extended to the case of infinitely many focal points or to continuous set of foci.^[7]
• Change the dimension of the underlying space. The points may be assumed to lie in somed-dimensional space.
• Change the definition of the distance. Traditionally euclidean definitions are employed. in its place, other notions of distance liketaxicab distance, may be used.^[6][8] Generalized conics with this notion of distance have found applications in geometrictomography.^[6][9]

The formulation of the definition of the generalized conic in the most general case when the cardinality of the focal set is infinite involves the notions of measurable sets andLebesgue integration. All these have been employed by different authors and the resulting curves have been studied with special emphasis on applications.

Let${\displaystyle d:R^n\to R}$ be a metric and${\displaystyle \mu }$ a measure on a compact set${\displaystyle K\subset R^n}$ with${\displaystyle \mu (K)>0}$. The unweighted generalized conic function${\displaystyle f_K}$ associated with${\displaystyle K}$ is

${\displaystyle f_K(x)={\int }_{K}g(x,y)d(x,y)d_{\mu }y}$

where${\displaystyle g:R^n\times R^n\to R}$ is a kernel function associated with${\displaystyle f_K}$.${\displaystyle K}$ is the set of foci. The level sets are called generalized conics.^[6]

Given a conic, by choosing afocus of the conic as thepole and the line through the pole drawn parallel to thedirectrix of the conic as the polar axis, thepolar equation of theconic can be written in the following form:

${\displaystyle r=\frac{ed}{1+e\sin \theta }}$

Heree is theeccentricity of the conic andd is the distance of the directrix from the pole.Tom M. Apostol andMamikon A. Mnatsakanian in their study of curves drawn on the surfaces of right circular cones introduced a new class of curves which they called generalized conics.^[10][11] These are curves whose polar equations are similar to the polar equations of ordinary conics and the ordinary conics appear as special cases of these generalized conics.

For constantsr₀ ≥ 0,λ ≥ 0 and realk, aplane curve described by the polar equation

${\displaystyle r=\frac{r_0}{1+\lambda \sin (k\theta )}}$

is called ageneralized conic.^[11] The conic is called a generalized ellipse, parabola or hyperbola according asλ < 1,λ = 1, orλ > 1.

• In the special case whenk = 1, the generalized conic reduces to an ordinary conic.
• In the special case whenk > 1, there is a simple geometrical method for the generation of the corresponding generalized conic.^[11]

Letα be an angle such that sinα = 1/k. Consider a right circular cone with semi-vertical angle equal toα. Consider the intersection of this cone by a plane such that the intersection is a conic with eccentricityλ. Unwrap the cone to a plane. Then the curve in the plane to which the conic section of eccentricityλ is unwrapped is a generalized conic with polar equation as specified in the definition.

• In the special case whenk < 1, the generalized conic cannot be obtained by unwrapping a conic section. In this case there is another interpretation.

Consider an ordinary conic drawn on a plane. Wrap the plane to form a right circular cone so that the conic becomes a curve in three-dimensional space. The projection of the curve onto a plane perpendicular to the axis of the cone will be a generalized conic in the sense of Apostol and Mnatsakanian withk < 1.

In 1996, Ruibin Qu introduced a new notion of generalized conic as a tool for generating approximations to curves.^[12] The starting point for this generalization is the result that the sequence of points${\displaystyle \{P_k:k=0,1,2,\dots \}}$ defined by

${\displaystyle P_{k+1}=2\alpha P_k-P_{k-1}}$

lie on a conic. In this approach, the generalized conic is now defined as below.

A generalized conic is such a curve that if the two points${\displaystyle P_0}$ and${\displaystyle P_1}$ are on it, then the points${\displaystyle \{P_k:k>1\}}$ generated by the recursive relation

${\displaystyle P_{k+1}=2\alpha P_k+\beta P_{k-1}}$

for some${\displaystyle \alpha }$ and${\displaystyle \beta }$ satisfying the relations

${\displaystyle \alpha \beta \ne 0,(\alpha ,\beta )\ne (\pm 1,-1),{\alpha }^2+\beta \ne 0}$

are also on it.

Let (X,d) be ametric space and letA be anonempty subset ofX. Ifx is a point inX, the distance ofx fromA is defined asd(x,A) = inf{d(x,a):a inA}. IfA andB are both nonempty subsets ofX then the equidistant set determined byA andB is defined to be the set {x inX:d(x,A) =d(x,B)}. This equidistant set is denoted by {A =B }. The term generalized conic is used to denote a general equidistant set.^[13]

Classical conics can be realized as equidistant sets. For example, ifA is a singleton set andB is a straight line, then the equidistant set {A =B } is a parabola. IfA andB are circles such thatA is completely withinB then the equidistant set {A =B } is an ellipse. On the other hand, ifA lies completely outsideB the equidistant set {A =B } is a hyperbola.

A similar approach considers a generalization of the focus/directrix/eccentricity interpretation of conics, by retaining a single pointF for the focus, any differentiable curved serving as the directrix, ande > 0, the eccentricity. LetX be a variable point on d. The resultant generalized conic is the set of pointsP (each lying on a normal tod throughX) for which the distancesPF andPX satisfy the ratioPF/PX = e. Norman^[14] and Poplin^[15] referred to these curves as pseudoconics and the constraint that the distance fromP to the directrix be minimal has been discarded.

If one retains the minimality requirement, then the set of pointsP satisfying this requirement are considered to be the primary pseudoconic, and the remainder of the curve is the secondary branch of the pseudoconic. Similar examples of generalized parabolas can be found in Josephet al..^[16]

• For a detailed discussion of generalized conics from the viewpoint of differential geometry, see the chapter on generalized conics in the book Convex Geometry by Csaba Vincze available online.^[1]

• Conic sections
• Algebraic curves