A parabola (plural "parabolas"; Gray 1997, p. 45) is the set of all points in the plane equidistant from a given line (the conic section directrix) and a given point not on the line (the focus). The focal parameter (i.e., the distance between the directrix and focus) is therefore given by , where is the distance from the vertex to the directrix or focus. The surface of revolution obtained by rotating a parabola about its axis of symmetry is called a paraboloid.
The parabola was studied by Menaechmus in an attempt to achieve cube duplication. Menaechmus solved the problem by finding the intersection of the two parabolas and . Euclid wrote about the parabola, and it was given its present name by Apollonius. Pascal considered the parabola as a projection of a circle, and Galileo showed that projectiles falling under uniform gravity follow parabolic paths. Gregory and Newton considered the catacaustic properties of a parabola that bring parallel rays of light to a focus (MacTutor Archive), as illustrated above.
For a parabola opening to the right with vertex at (0, 0), the equation in Cartesian coordinates is
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(2)
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The quantity is known as the latus rectum.
If the vertex is at instead of (0, 0), the equation of the parabola with latus rectum is
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A parabola opening upward with vertex is at and latus rectum has equation
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Three points uniquely determine one parabola with directrix parallel to the -axis and one with directrix parallel to the -axis. If these parabolas pass through the three points , , and , they are given by equations
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and
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In polar coordinates, the equation of a parabola with parameter and center (0, 0) is given by
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(left figure). The equivalence with the Cartesian form can be seen by setting up a coordinate system and plugging in and to obtain
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Expanding and collecting terms,
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so solving for gives (◇). A set of confocal parabolas is shown in the figure on the right.
In pedal coordinates with the pedal point at the focus, the equation is
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The parabola can be written parametrically as
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or
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A segment of a parabola is a Lissajous curve.
A parabola may be generated as the envelope of two concurrent line segments by connecting opposite points on the two lines (Wells 1991).
In the above figure, the lines , , and are tangent to the parabola at points , , and , respectively. Then (Wells 1991). Moreover, the circumcircle of passes through the focus (Honsberger 1995, p. 47). In addition, the foot of the perpendicular to a tangent to a parabola from the focus always lies on the tangent at the vertex (Honsberger 1995, p. 48).
Given an arbitrary point located "outside" a parabola, the tangent or tangents to the parabola through can be constructed by drawing the circle having as a diameter, where is the focus. Then locate the points and at which the circle cuts the vertical tangent through . The points and (which can collapse to a single point in the degenerate case) are then the points of tangency of the lines and and the parabola (Wells 1991).
The curvature, arc length, and tangential angle are
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The tangent vector of the parabola is
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The plots below show the normal and tangent vectors to a parabola.