The arrows above indicate points and their images on lines e, e' under the homographic map. Conic (c) has been determined by the condition to be tangent to five lines. It could though be defined by the easier procedure described above. The segments OE, O'E' on e, e' respectively define the origin of coordinates on each line and the unit length. A particular case is the one for which c = 0. The homographic relation is then of the form x'=a*x+b and is called a

Notice that W and V could be determined from the relation x'=F(x)=(a*x+b)/(c*x+d) itself. In fact, assume that I has coordinates x

Notice that the four quantities x

How many homographic relations between the lines e, e' (fixing their coordinate systems) define the same points V and W on them?

The answer is given by solving the linear system:

(i) c*x

(ii) c*x

with respect to {a, b, c, d}. Since multiplying these coefficients with a constant defines the same F we obtain a one-parametric family of solutions. Each such solution [a, b, c, d] represents a member-conic of the bitangent family as envelope of lines XX' of the corresponding homographic relation.

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HomographicRelation.html

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