Parabola

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 $ M $ in the plane for each of which the distance to a given point $ F $( the focus of the parabola) is equal to the distance to a certain given line $ d $( the directrix). Thus, a parabola is a conic with eccentricity one. The distance $ p $ 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

$$ y ^ {2} = 2px . $$

The equation of the tangent to a parabola at the point $ ( x _ {0} , y _ {0} ) $ is

$$ yy _ {0} = p( x + x _ {0} ) . $$

The equation of a parabola in polar coordinates $ ( \rho , \phi ) $ is

$$ \rho = \frac{p}{1 - \cos \phi } ,\ \textrm{ where } 0 < \phi < 2 \pi . $$

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

References

[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)