HyperbolaGeneration2

Generation of a hyperbola from a circle - II

In the file HyperbolaGeneration.html we saw the construction of a (rectangular) hyperbola out from the circle (x/r)² + (y/r)² = 1, by applying to it a projectivity. The projectivity was described with the matrix:

The minor change in the transformation representation changes radically its behaviour. Lines XY (Y = f(X)) do not pass through a fixed point any more. Instead, they are tangent to some curve which is rational of degree seven (quotient of two polynomials of degree at most seven).
- Besides, one can easily see that (f) restricted on line t_B coincides with the symmetry at O and maps the line to line t_C.
- On line t_C in turn, (f) coincides with the parallel translation along the (directed) CB.
- These two remarks suffice to construct projectivity (f) through a recipe on the vertices of square ADEG. It is the unique projectivity mapping A-->G, E--> and G-->E, D-->A.
-The connection between the two maps can be easily seen by multiplying the matrices A'*A^-1 = A'*A (since A² = 1), which is the matrix

representing the reflexion on the x-axis. Thus A' = W*A, where W is an isometry of the circle. One could generalize by taking W to be an arbitrary isometry of the circle.

A slightly modified configuration, producing a hyperbola from an ellipse is studied in HyperbolaFromEllipse.html .

Generation of a hyperbola from a circle - II

See Also