Editing Parabola (section)

=== 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>