Editing Parabola (section)

== Focal length and radius of curvature at the vertex ==
The focal length of a parabola is half of its [[Radius of curvature (mathematics)|radius of curvature]] at its vertex.

;Proof:
<gallery widths="300px" heights="200px">
File:Huygens + Snell + van Ceulen - regular polygon doubling.svg|Image is inverted. AB is {{mvar|x}} axis. C is origin. O is center. A is {{math|(''x'', ''y'')}}. OA = OC = {{mvar|R}}. PA = {{mvar|x}}. CP = {{mvar|y}}. OP = {{math|(''R'' − ''y'')}}. Other points and lines are irrelevant for this purpose.
File:Parabola circle.svg|The radius of curvature at the vertex is twice the focal length. The measurements shown on the above diagram are in units of the latus rectum, which is four times the focal length.
File:Concave mirror.svg
</gallery>
Consider a point {{math|(''x'', ''y'')}} on a circle of radius {{mvar|R}} and with center at the point {{math|(0, ''R'')}}. The circle passes through the origin. If the point is near the origin, the [[Pythagorean theorem]] shows that
<math display="block">\begin{align}
 x^2 + (R - y)^2 &= R^2, \\[1ex]
 x^2 + R^2 - 2Ry + y^2 &= R^2, \\[1ex]
 x^2 + y^2 &= 2Ry.
\end{align}</math>

But if {{math|(''x'', ''y'')}} is extremely close to the origin, since the {{mvar|x}} axis is a tangent to the circle, {{mvar|y}} is very small compared with {{mvar|x}}, so {{math|''y''<sup>2</sup>}} is negligible compared with the other terms. Therefore, extremely close to the origin
{{NumBlk||<math display="block">x^2 = 2Ry.</math>|{{EquationRef|1}}}}

Compare this with the parabola
{{NumBlk||<math display="block">x^2 = 4fy,</math>|{{EquationRef|2}}}}
which has its vertex at the origin, opens upward, and has focal length {{mvar|f}} (see preceding sections of this article).

Equations {{EquationNote|(1)}} and {{EquationNote|2|(2)}} are equivalent if {{math|1=''R'' = 2''f''}}. Therefore, this is the condition for the circle and parabola to coincide at and extremely close to the origin. The radius of curvature at the origin, which is the vertex of the parabola, is twice the focal length.

; Corollary:
A concave mirror that is a small segment of a sphere behaves approximately like a parabolic mirror, focusing parallel light to a point midway between the centre and the surface of the sphere.