In geometry, the foci (//; singular focus) are a pair of special points with reference to which any of a variety of curves is constructed. For example, foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola. In addition, foci are used to define the Cassini oval and the Cartesian oval.
Conics in geometry
Defining conics in terms of two foci
An ellipse can be defined as the locus of points for each of which the sum of the distances to two given foci is a constant.
A circle is the special case of an ellipse in which the two foci coincide with each other. Thus, a circle can be more simply defined as the locus of points each of which is a fixed distance from a single given focus. A circle can also be defined as the circle of Apollonius, in terms of two different foci, as the set of points having a fixed ratio of distances to the two foci.
A parabola is a limiting case of an ellipse in which one of the foci is a point at infinity.
A hyperbola can be defined as the locus of points for each of which the absolute value of the difference between the distances to two given foci is a constant.
Defining conics in terms of a focus and a directrix
It is also possible to describe all the conic sections in terms of a single focus and a single directrix, which is a given line not containing the focus. A conic is defined as the locus of points for each of which the distance to the focus divided by the distance to the directrix is a fixed positive constant, called the eccentricity e. If e is between zero and one the conic is an ellipse; if e=1 the conic is a parabola; and if e>1 the conic is a hyperbola. If the distance to the focus is fixed and the directrix is a line at infinity, so the eccentricity is zero, then the conic is a circle.
Defining conics in terms of a focus and a directrix circle
It is also possible to describe all the conic sections as loci of points that are equidistant from a single focus and a single, circular directrix. For the ellipse, both the focus and the center of the directrix circle have finite coordinates and the radius of the directrix circle is greater than the distance between the center of this circle and the focus; thus, the focus is inside the directrix circle. The ellipse thus generated has its second focus at the center of the directrix circle, and the ellipse lies entirely within the circle.
For the parabola, the center of the directrix moves to the point at infinity (see projective geometry). The directrix 'circle' becomes a curve with zero curvature, indistinguishable from a straight line. The two arms of the parabola become increasingly parallel as they extend, and 'at infinity' become parallel; using the principles of projective geometry, the two parallels intersect at the point at infinity and the parabola becomes a closed curve (elliptical projection).
To generate a hyperbola, the radius of the directrix circle is chosen to be less than the distance between the center of this circle and the focus; thus, the focus is outside the directrix circle. The arms of the hyperbola approach asymptotic lines and the 'right-hand' arm of one branch of a hyperbola meets the 'left-hand' arm of the other branch of a hyperbola at the point at infinity; this is based on the principle that, in projective geometry, a single line meets itself at a point at infinity. The two branches of a hyperbola are thus the two (twisted) halves of a curve closed over infinity.
In projective geometry, all conics are equivalent in the sense that every theorem that can be stated for one can be stated for the others.
Cartesian and Cassini ovals
A Cassini oval is the set of points for each of which the product of the distances to two given foci is a constant.
The concept of a focus can be generalized to arbitrary algebraic curves. Let C be a curve of class m and let I and J denote the circular points at infinity. Draw the m tangents to C through each of I and J. There are two sets of m lines which will have m2 points of intersection, with exceptions in some cases due to singularities, etc. These points of intersection are the defined to be the foci of C. In other words, a point P is a focus if both PI and PJ are tangent to C. When C is a real curve, only the intersections of conjugate pairs are real, so there are m in a real foci and m2−m imaginary foci. When C is a conic, the real foci defined this way are exactly the foci which can be used in the geometric construction of C.
Let P1, P2, …, Pm be given as foci of a curve C of class m. Let P be the product of the tangential equations of these points and Q the product of the tangential equations of the circular points at infinity. Then all the lines which are common tangents to both P=0 and Q=0 are tangent to C. So, by the AF+BG theorem, the tangential equation of C has the form HP+KQ=0. Since C has class m, H must be a constant and K but have degree less than or equal to m−2. The case H=0 can be eliminated as degenerate, so the tangential equation of C can be written as P+fQ=0 where f is an arbitrary polynomial of degree m−2.
For example, let P1=(1,0), P2=(−1,0). The tangential equations are X+1=0 and X−1=0 so P= X2-1=0. The tangential equations for the circular points at infinity are X+iY=0 and X−iY=0 so Q=X2+Y2. Therefore the tangential equation for a conic with the given foci is X2-1+c(X2+Y2)=0, or (1+c)X2+cY2=1 where c is an arbitrary constant. In point coordinates this becomes
- Follows Hilton p. 69 with an appeal to AF+BG for simplification.
- Hilton, Harold (1920). Plane Algebraic Curves. Oxford. p. 69.