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Suppose a system of Cartesian coordinates is used such that the vertex of the parabola is at the origin, and the axis of symmetry is the y axis. The parabola opens upward. It is shown elsewhere in this article that the equation of the parabola is 4fy = x 2, where f is the focal length.
In the geometry of plane curves, a vertex is a point of where the first derivative of curvature is zero. [1] This is typically a local maximum or minimum of curvature, [ 2 ] and some authors define a vertex to be more specifically a local extremum of curvature. [ 3 ]
A parabola has only one focus, and can be considered as a limit curve of a set of ellipses (or a set of hyperbolas), where one focus and one vertex are kept fixed, while the second focus is moved to infinity. If this transformation is performed on each conic in an orthogonal net of confocal ellipses and hyperbolas, the limit is an orthogonal ...
A parabola, showing arbitrary line (L), focus (F), and vertex (V). Generic points marked for explanation. Adapted from this image, which shows L as the directrix of ...
Then the top parabola is rolled without slipping along the bottom one, and its successive positions are shown in the animation. Then the path traced by the vertex of the top parabola as it rolls is a roulette shown in red, which is the cissoid of Diocles. The pedal curve of a parabola with respect to its vertex is a cissoid of Diocles. [3]
A parabolic segment is the region bounded by a parabola and line. To find the area of a parabolic segment, Archimedes considers a certain inscribed triangle. The base of this triangle is the given chord of the parabola, and the third vertex is the point on the parabola such that the tangent to the parabola at that point is parallel to the chord.
The vertex of a parabola is the place where it turns; hence, it is also called the turning point. If the quadratic function is in vertex form, the vertex is ( h , k ) . Using the method of completing the square, one can turn the standard form
The equation for a conic section with apex at the origin and tangent to the y axis is + (+) = alternately = + (+) where R is the radius of curvature at x = 0. This formulation is used in geometric optics to specify oblate elliptical ( K > 0 ), spherical ( K = 0 ), prolate elliptical ( 0 > K > −1 ), parabolic ( K = −1 ), and hyperbolic ( K ...