[alogo] Homographic Relation Example

The basics for homographic relations between two real variables are discussed in HomographicRelation.html . Here is the construction of such a relation based on a conic (the ellipse below) and two fixed tangents (a, b) of it. The relation is defined between the line coordinate x of a and the line coordinate y of b, using the following recipe:
For an arbitrary point C of the conic, consider the tangent (c) at C and its intersection points X(x) with (a) and Y(y) with (b). The correspondence y = f(x) between the line coordinates is a homographic relation. The figure below displays the graph of this function, which is a rectangular hyperbola, together with its asymptotic lines.
Note the location of the asymptotics corresponding to points B', C' symmetric to B, C w.r. to the center of the conic.

[0_0] [0_1] [0_2] [0_3]
[1_0] [1_1] [1_2] [1_3]
[2_0] [2_1] [2_2] [2_3]
[3_0] [3_1] [3_2] [3_3]

Note that this is, in some sense, the inverse of the construction of the conic as an envelope of lines, discussed in the file Chasles_Steiner_Envelope.html .

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