LineHomographyAxis2

Line homography axis II

Continuing the discussion from LineHomographyAxis.html , about a line connected with the homography between lines e, e' defined by x'=(a*x+b)/(c*x+d). There we saw that the geometric locus of intersection points P of lines XY', X'Y is a line f (red line). This line is the polar of the intersection point I of lines e, e', with respect to the conic (c) which envelopes all lines XX'. The generation of conics that way is the Chasles-Steiner method to define a conic. The easy construction of the line homography axis (through two points as P and Q in the figure below) leads to an easy construction of the conic (c). In fact observe first that (c) is a member of the bitangent family of conics passing through V, W (intersection points of e, e' with the homography axis) and tangent there to the lines IV, IW correspondingly. Next, from Pascal's theorem, follows that in the triangle IXX', the lines joining the vertices with the contact points on the opposite sides will pass all from a common point A. Thus, given I, V, W, X and X' one constructs B and consequently conic (c) being the unique member of the bitangent family through that point.

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 similarity relation (between the points of the two lines). The corresponding conic is then a parabola. This case is handled in the file LineSimilarityAxis.html .
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 x₀, x₀' with respect to e, e'. Then solving x₀'=F(x₁), we find the coordinate of W, and x₂=F(x₀) gives the coordinate of V.
Notice that the four quantities x₀, x₀', x₁ and x₂ play also a role in answering the question:
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*x₀'*x₁ + x₀'*d - a*x₁ - b = 0,
(ii) c*x₀*x₂ + x₂*d - a*x₀ - b = 0,
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.

Line homography axis II

See Also