Editing Parabola (section)

== As the affine image of the unit parabola ==
[[File:Parabel-aff-s.svg|thumb|Parabola as an affine image of the unit parabola]]
Another definition of a parabola uses [[affine transformation]]s:
{{block indent | em = 1.5 | text = Any ''parabola'' is the affine image of the unit parabola with equation <math>y = x^2</math>.}}

=== Parametric representation ===
An affine transformation of the Euclidean plane has the form <math>\vec x \to \vec f_0 + A \vec x</math>, where <math>A</math> is a regular matrix ([[determinant]] is not 0), and <math>\vec f_0</math> is an arbitrary vector. If <math>\vec f_1, \vec f_2</math> are the column vectors of the matrix <math>A</math>, the unit parabola <math>(t, t^2),\ t \in \R</math> is mapped onto the parabola
<math display="block">\vec x = \vec p(t) = \vec f_0 +\vec f_1 t +\vec f_2 t^2,</math>
where
* <math>\vec f_0</math> is a ''point'' of the parabola,
* <math>\vec f_1</math> is a ''tangent vector'' at point <math>\vec f_0</math>,
* <math>\vec f_2</math> is ''parallel to the axis'' of the parabola (axis of symmetry through the vertex).

=== Vertex ===
In general, the two vectors <math>\vec f_1, \vec f_2</math> are not perpendicular, and <math>\vec f_0</math> is ''not'' the vertex, unless the affine transformation is a [[Similarity (geometry)|similarity]].

The tangent vector at the point <math>\vec p(t)</math> is <math>\vec p'(t) = \vec f_1 + 2t \vec f_2</math>. At the vertex the tangent vector is orthogonal to <math>\vec f_2</math>. Hence the parameter <math>t_0</math> of the vertex is the solution of the equation
<math display="block">\vec p'(t) \cdot \vec f_2 = \vec f_1 \cdot \vec f_2 + 2t f_2^2 = 0,</math>
which is
<math display="block">t_0 = -\frac{\vec f_1 \cdot \vec f_2}{2 f_2^2},</math>
and the ''vertex'' is
<math display="block">\vec p(t_0) = \vec f_0 - \frac{\vec f_1 \cdot \vec f_2}{2 f_2^2} \vec f_1 + \frac{(\vec f_1 \cdot \vec f_2)^2}{4(f_2^2)^2} \vec f_2.</math>

=== Focal length and focus ===
The ''focal length'' can be determined by a suitable parameter transformation (which does not change the geometric shape of the parabola). The focal length is
<math display="block">f = \frac{f_1^2 \, f_2^2 - (\vec f_1 \cdot \vec f_2)^2}{4|f_2|^3}.</math>
Hence the ''focus'' of the parabola is
<math display="block">F:\ \vec f_0 - \frac{\vec f_1 \cdot \vec f_2}{2 f_2^2} \vec f_1 + \frac{f_1^2 \, f_2^2}{4(f_2^2)^2} \vec f_2.</math>

=== Implicit representation ===
Solving the parametric representation for <math>\; t, t^2\;</math> by [[Cramer's rule]] and using <math>\;t\cdot t-t^2 =0\; </math>, one gets the implicit representation
<math display="block">\det(\vec x\!-\!\vec f\!_0,\vec f\!_2)^2-\det(\vec f\!_1,\vec x\!-\!\vec f\!_0)\det(\vec f\!_1,\vec f\!_2) = 0.</math>

=== Parabola in space ===
The definition of a parabola in this section gives a parametric representation of an arbitrary parabola, even in space, if one allows <math>\vec f\!_0, \vec f\!_1, \vec f\!_2</math> to be vectors in space.