HyperbolaGeneration

Generation of a hyperbola from a circle

One can generate a hyperbola from a circle through a (projective) transformation of the circle-points as follows.
Consider a point B on a circle c(A,|OB|=r), and the tangent t_C at the antipodal point C of B. For every point X take Y on the line BX, such that the cross ratio (B,D,X,Y) = -1, D being the intersection point of BX with t_C.
The resulting transform (f) is a harmonic perspectivity, fixing the points of line t_C and the point B. Further (f) preserves the tangent t_B by reflecting its points on B and sends line OA to infinity.
With the figure centered at the origin, as shown, (f) maps the circle (x/r)²+(y/r)² = 1 onto the rectangular hyperbola (y/r)² -(x/r)² = 1.

The brakets in [A] denoting the class of all non-zero multiples of the matrix. The transformation is given, as usual, in the projectification of the real plane:

Map (f) reminds to the stereographic projection (from B onto the line t_B), but that is another story, concerning inversions. An analogous transformation, producing a parabola, is discussed in ParabolaProjectFromCircle.html . The more general transformations [V], defined above, although not involutive as f (f²=1), produce arbitrary hyperbolas (centered at the origin) from the same circle. Taking all the isometries of the circle onto itself and conjugating with V we can see that the group of all affinities leaving invariant an hyperbola (or a parabola, using the above reference) is isomorphic to the group of all symmetries of the circle.
An other projectivity, mapping the same circle to the same hyperbola with a different recipe is examined in HyperbolaGeneration2.html .

Generation of a hyperbola from a circle

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