What Are Bifocals?

In geometry, focus (focus or foci) (UK: / foka /, USA: / fosa /), focus refers to a particular point in a curve. For example, one or two focal points can be used to define a conical section, and its four types are circular, elliptical, parabolic, and hyperbolic. In addition, two focal points are used to define the Cassini ellipse and Cartesian ellipse, and more than two focal points are used to define the n-ellipse.

In geometry, focus (focus or foci) (UK: / foka /, USA: / fosa /), focus refers to a particular point in a curve. For example, one or two focal points can be used to define a conical section. The four types are

An n-ellipse is a set of points with the same sum of distances from n focal points. (The case of n = 2 is the traditional ellipse)

The concept of focus can be generalized to arbitrary algebraic curves. Let C be a m-like curve, and let I and J represent infinitely far-flung dots. Draw a m tangent to C through each of I and J. There are two sets of m rows that will have m ² intersections, which in some cases differ due to singularities. These intersections are defined as the focus, in other words, if both PI and PJ are tangent to C, then the point P is the focus. When C is a real curve, only the intersection of the conjugate pair is true, so at the actual focus and the m ² -m imaginary focus. When C is a quadratic curve, the true focus defined in this way is exactly the focus that can be used for the geometric construction of C. ^[2]

Let P1, P2, ..., Pm be the focus of the curve C of class m. Let P be the product of tangent equations of these points, and Q be the product of tangent equations of infinite circular points. Then all lines of the common tangent of P = 0 and Q = 0 are tangent to C. Therefore, by AF + BG theorem, the tangent equation of C has the form of HP + KQ = 0. Since C has a rank m, H must be a constant K but less than or equal to m-2. The case where H = 0 can be eliminated as degradation, so the tangent equation of C can be written as P + fQ = 0, where f is an arbitrary polynomial m-2

For example, let P1 = (1,0) and P2 = (-1,0). The tangent equation is X + 1 = 0 and X-1 = 0, so P = X ² -1 = 0. The tangent equation of the infinite loop point is X + iY = 0 and X-iY = 0, so Q = X ² + Y ² . Therefore, the tangent equation of the quadratic curve for a given focus is X ² -1 + c (X ² + Y ² ) = 0 or (1 + c) X ² + cY ² = 1, where c is an arbitrary constant. At point coordinates this becomes