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In this diagram, F is the focus of the parabola, and T and U lie on its directrix. P is an arbitrary point on the parabola. PT is perpendicular to the directrix, and the line MP bisects angle ∠FPT. Q is another point on the parabola, with QU perpendicular to the directrix. We know that FP = PT and FQ = QU.
In geometry, focuses or foci (/ ˈ f oʊ k aɪ /; 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 .
Parabola: the set of points equidistant from a fixed point (the focus) and a line (the directrix). Hyperbola: the set of points for each of which the absolute value of the difference between the distances to two given foci is a constant. Ellipse: the set of points for each of which the sum of the distances to two given foci is a constant
From the point of view of projective geometry, an elliptic paraboloid is an ellipsoid that is tangent to the plane at infinity. Plane sections. The plane sections of an elliptic paraboloid can be: a parabola, if the plane is parallel to the axis, a point, if the plane is a tangent plane. an ellipse or empty, otherwise.
Parabolas have only one focus, so, by convention, confocal parabolas have the same focus and the same axis of symmetry. Consequently, any point not on the axis of symmetry lies on two confocal parabolas which intersect orthogonally (see below). A circle is an ellipse with both foci coinciding at the center.
The first degree polynomial equation = + is a line with slope a. A line will connect any two points, so a first degree polynomial equation is an exact fit through any two points with distinct x coordinates. If the order of the equation is increased to a second degree polynomial, the following results:
A family of conic sections of varying eccentricity share a focus point F and directrix line L, including an ellipse (red, e = 1/2), a parabola (green, e = 1), and a hyperbola (blue, e = 2). The conic of eccentricity 0 in this figure is an infinitesimal circle centered at the focus, and the conic of eccentricity ∞ is an infinitesimally ...
Successive parabolic interpolation is a technique for finding the extremum (minimum or maximum) of a continuous unimodal function by successively fitting parabolas (polynomials of degree two) to a function of one variable at three unique points or, in general, a function of n variables at 1+n(n+3)/2 points, and at each iteration replacing the "oldest" point with the extremum of the fitted ...