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Steiner generation of a hyperbola

The following method to construct single points of a hyperbola relies on the Steiner generation of a non degenerate conic section:

For the generation of points of the hyperbola x 2 a 2 − y 2 b 2 = 1 {\displaystyle {\tfrac {x^{2}}{a^{2}}}-{\tfrac {y^{2}}{b^{2}}}=1} one uses the pencils at the vertices V 1, V 2 {\displaystyle V_{1},V_{2}}. Let P = (x 0, y 0) {\displaystyle P=(x_{0},y_{0})} be a point of the hyperbola and A = (a, y 0), B = (x 0, 0) {\displaystyle A=(a,y_{0}),B=(x_{0},0)}. The line segment B P ¯ {\displaystyle {\overline {BP}}} is divided into n equally-spaced segments and this division is projected parallel with the diagonal A B {\displaystyle AB} as direction onto the line segment A P ¯ {\displaystyle {\overline {AP}}} (see diagram). The parallel projection is part of the projective mapping between the pencils at V 1 {\displaystyle V_{1}} and V 2 {\displaystyle V_{2}} needed. The intersection points of any two related lines S 1 A i {\displaystyle S_{1}A_{i}} and S 2 B i {\displaystyle S_{2}B_{i}} are points of the uniquely defined hyperbola.

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