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In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several other superficially different mathematical descriptions, which can all be proved to define exactly the same curves.

One description of a parabola involves a point (the focus) and a line (the directrix). The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane parallel to another plane that is tangential to the conical surface.[a]

The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola through the middle) is called the "axis of symmetry". The point where the parabola intersects its axis of symmetry is called the "vertex" and is the point where the parabola is most sharply curved. The distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length". The "latus rectum" is the chord of the parabola that is parallel to the directrix and passes through the focus. Parabolas can open up, down, left, right, or in some other arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola—that is, all parabolas are geometrically similar.

Parabolas have the property that, if they are made of material that reflects light, then light that travels parallel to the axis of symmetry of a parabola and strikes its concave side is reflected to its focus, regardless of where on the parabola the reflection occurs. Conversely, light that originates from a point source at the focus is reflected into a parallel ("collimated") beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other waves. This reflective property is the basis of many practical uses of parabolas.

The parabola has many important applications, from a parabolic antenna or parabolic microphone to automobile headlight reflectors and the design of ballistic missiles. They are frequently used in physics, engineering, and many other areas.

Dual parabola and Bezier curve of degree 2 (right: curve point and division points ${\displaystyle Q_{0},Q_{1}}$ for parameter ${\displaystyle t=0.4}$)

A dual parabola consists of the set of tangents of an ordinary parabola.

The Steiner generation of a conic can be applied to the generation of a dual conic by changing the meanings of points and lines:

• Let be given two point sets on two lines ${\displaystyle u,v}$, and a projective but not perspective mapping ${\displaystyle \pi }$ between these point sets, then the connecting lines of corresponding points form a non degenerate dual conic.

In order to generate elements of a dual parabola, one starts with

1. three points ${\displaystyle P_{0},P_{1},P_{2}}$ not on a line,
2. divides the line sections ${\displaystyle {\overline {P_{0}P_{1}}}}$ and ${\displaystyle {\overline {P_{1}P_{2}}}}$ each into ${\displaystyle n}$ equally spaced line segments and adds numbers as shown in the picture.
3. Then the lines ${\displaystyle P_{0}P_{1},P_{1}P_{2},(1,1),(2,2),\dotsc }$ellipses and hyperbolas.

A dual parabola consists of the set of tangents of an ordinary parabola.

The Steiner generation of a conic can be applied to the generation of a dual conic by changing the meanings of points and lines:

• Let be given two point sets on two lines ${\displaystyle u,v}$, and a projective but not perspective mapping ${\displaystyle \pi }$ between these point sets, then the connecting lines of corresponding points form a non degenerate dual conic.

In order to generate elements of a dual parabola, one starts with

1. three points ${\displaystyle P_{0},P_{1},P_{2}}$${\displaystyle P_{0},P_{1},P_{2}}$ not on a line,
2. divides the line sections