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Similarity to the unit parabola

Two objects in the Euclidean plane are similar if one can be transformed to the other by a similarity, that is, an arbitrary composition of rigid motions (translations and rotations) and uniform scalings.

A parabola P {\displaystyle {\mathcal {P}}} with vertex V = (v 1, v 2) {\displaystyle V=(v_{1},v_{2})} can be transformed by the translation (x, y) → (x − v 1, y − v 2) {\displaystyle (x,y)\to (x-v_{1},y-v_{2})} to one with the origin as vertex. A suitable rotation around the origin can then transform the parabola to one that has the y axis as axis of symmetry. Hence the parabola P {\displaystyle {\mathcal {P}}} can be transformed by a rigid motion to a parabola with an equation y = a x 2, a ≠ 0 {\displaystyle y=ax^{2},\ a\neq 0}. Such a parabola can then be transformed by the uniform scaling (x, y) → (a x, a y) {\displaystyle (x,y)\to (ax,ay)} into the unit parabola with equation y = x 2 {\displaystyle y=x^{2}}. Thus, any parabola can be mapped to the unit parabola by a similarity.

A synthetic approach, using similar triangles, can also be used to establish this result.

The general result is that two conic sections (necessarily of the same type) are similar if and only if they have the same eccentricity. Therefore, only circles (all having eccentricity 0) share this property with parabolas (all having eccentricity 1), while general ellipses and hyperbolas do not.

There are other simple affine transformations that map the parabola y = a x 2 {\displaystyle y=ax^{2}} onto the unit parabola, such as (x, y) → (x, y a) {\displaystyle (x,y)\to \left(x,{\tfrac {y}{a}}\right)}. But this mapping is not a similarity, and only shows that all parabolas are affinely equivalent (see § As the affine image of the unit parabola).