Focus (geometry)
In geometry, focuses or foci (/ˈfoʊkaɪ/; sg.: focus) are special points with reference to which any of a variety of curves is constructed. For example, one or two foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola. In addition, two foci are used to define the Cassini oval and the Cartesian oval, and more than two foci are used in defining an n-ellipse.
Cartesian and Cassini ovals[edit]
A Cartesian oval is the set of points for each of which the weighted sum of the distances to two given foci is constant. If the weights are equal, the special case of an ellipse results.
A Cassini oval is the set of points for each of which the product of the distances to two given foci is constant.
Generalizations[edit]
An n-ellipse is the set of points all having the same sum of distances to n foci (the n = 2 case being the conventional ellipse).
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.