HyperbolaFromEllipse

Hyperbola from ellipse through projectivity

The generation of hyperbolas from the circle was discussed in the file HyperbolaGeneration2.html . Here we modify a little bit the recipe studied there and define the projectivity by starting with a screen-rectangle (sides vertical/horizontal) ABCD symmetric about the origin. Define the projectivity by prescribing its values at the vertices of the rectangle: f(A) = B, f(D) = C, f(B) = D and f(C) = A.
This completely defines projectivity f with the properties:
- It maps line [AD] to [BC] by acting as parallel translation (parallel to AB) at every point of line [AD].
- It maps line [BC] to [AD] acting by the symmetry at O at every point of line [BC].
- It maps the ellipse inscribed in the rectangle:

An easy calculation, using the recipe given in Projectivity.html , shows that the matrix describing the transformation (f) is:

Hyperbola from ellipse through projectivity

See Also