BitangentConics

Families of bitangent conics and their homographies

We discuss the family consisting of conics defined by the equation: v*w - k*u² = 0.
Here (u=0) represents line VW, (v=0) represents line UW and (w=0) represents line UV.
The family, some times called family of bitangent conics, consists of all conics tangent to lines (v=0, w=0) at W, V correspondingly. In some sense the present discussion is a continuation of that started in HomographyAxis.html .
Among the family members there is a parabola c₀ which can be constructed by following the recipe explained in ParabolaSkew.html .
The members of the family of conics are invariant under the homographies (projectivities) F which can be defined by the following properties:
(a) F fixes points U, V, W and
(b) F maps an arbitrary point A of a member (c) of the family to an arbitrary point A' of the same member.
For each F defined in this way there is a line (g) mapping to infinity by F. The line is tangent to the parabola c₀, since F maps this line to a tangent (at infinity) to the parabola (the line at infinity).

Notice that line (g) completely determines the homography F by determining its action on lines u=0, v=0, w=0. In fact, the intersection points of (g) with these lines map via F to the corresponding points at infinity of the line, hence on each of these three lines we know the restriction of F. This because we know the images under F of three points (the two fixed points on each and the point mapping to infinity). This suffices to the determination of the whole F, since finding F(X) for a given X can be reduced to line intersections and their transforms with the three basic lines (u=0, v=0, w=0). Thus, the points B of the parabola, through the corresponding tangent (g) naturally parametrize the set of homographies preserving each member of the family of conics v*w - k*u² = 0.
More is true: Consider the line e₁=BD parallel to the parabola-axis. This is mapped by F to a parallel line e₂, symmetric to e₁ with respect to the axis through U (parallel to parabola axis passing through U). To see this notice that point B maps via F to the point at infinity of e₁. Take also the cross ratio (UVCG) which by applying F preserves its value while becoming equal to (U-F(C))/(V-F(C)) since G maps to infinity on line UV. This, setting D=F(C), ((U-C)/(V-C)):((U-G)/(V-G)) = (U-D)/(V-D). But from the elementary properties of tangents to parabolas, we have that (V-G)/(V-C) = 1/2, hence (U-C)/(U-G) = 2(U-D)/(V-D), from which follows that D is symmetric to C with respect to U. This proves that e₂, the image of e₁ under F is parallel to e₁. Thus, homographies preserving the above family of conics are parameterized by ordered pairs of isotomic points ( (X,X') different from (U,V), (V,U) ) along the basis VW of triangle UVW.
After this discussion it is clear that we could define homography F by its property to fix U, V, W and map X to X'. Obviously also the map mapping X' to X is the inverse of F.

Families of bitangent conics and their homographies

See Also