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A plane curve obtained as the intersection of a circular cone with a plane not passing through the vertex of the cone and parallel to one of its tangent planes. A parabola is a set of points in the plane for each of which the distance to a given point (the focus of the parabola) is equal to the distance to a certain given line (the directrix). Thus, a parabola is a conic with eccentricity one. The distance from the focus of the parabola to the directrix is called the parameter. A parabola is a symmetric curve; the point of intersection of a parabola with its axis of symmetry is called the vertex of the parabola, the axis of symmetry is called the axis of the parabola. A diameter of a parabola is any straight line parallel to its axis, and can be defined as the locus of the midpoints of a set of parallel chords.

Figure: p071150a

A parabola is a non-central second-order curve. Its canonical equation has the form

The equation of the tangent to a parabola at the point is

The equation of a parabola in polar coordinates is

A parabola has an optical property: Light rays emanating from the focus travel, after reflection in the parabola, parallel to the axis.



[a1] M. Berger, "Geometry" , II , Springer (1987) pp. Chapt. 17
[a2] J. Coolidge, "A history of the conic sections and quadric surfaces" , Dover, reprint (1968)
[a3] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963)
How to Cite This Entry:
Parabola. A.B. Ivanov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098