Generalized conic

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

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

where A₁, A₂, . . . , A_n are fixed points in a plane and λ₁, λ₂, . . . , λ_n are fixed rational numbers and c 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 foci A and B. Take a string whose length is c + AB and tie one end of the string to the pin at A. A pencil is attached to the other end of the string and the string is passed round the pin at the focus B. The pencil is then moved guided by the bight of the string. The curve traced by the pencil is the locus of P. 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 foci A, B, C. Let one end of the string be fixed at the pin at C 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 between A and C and then stretch to P. 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]

Let ${\displaystyle d:R^{n}\rightarrow R}$ 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 ${\displaystyle g:R^{n}\times R^{n}\rightarrow R}$ is a kernel function associated with ${\displaystyle f_{K}}$. ${\displaystyle K}$ is the set of foci. The level sets ${\displaystyle \{x\in R^{n}:f_{K}(x)<c\}}$ are called generalized conics.[6]

For constants r₀ ≥ 0, λ ≥ 0 and real k, a plane curve described by the polar equation

${\displaystyle r={\frac {r_{0}}{1+\lambda \sin(k\theta )}}}$

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

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

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

• In the special case when k < 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 with k < 1.

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

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

are also on it.

Let (X, d) be a metric space and let A be a nonempty subset of X. If x is a point in X, the distance of x from A is defined as d(x, A) = inf{ d(x, a): a in A}. If A and B are both nonempty subsets of X then the equidistant set determined by A and B is defined to be the set {x in X: 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, if A is a singleton set and B is a straight line, then the equidistant set { A = B } is a parabola. If A and B are circles such that A is completely within B then the equidistant set { A = B } is an ellipse. On the other hand, if A lies completely outside B the equidistant set { A = B } is a hyperbola.