Here (u=0) represents line VW, (v=0) represents line UW and (w=0) represents line UV.

The family, some times called

Among the family members there is a parabola c

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 c

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*u

More is true: Consider the line e

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.

ParabolaSkew.html

HomographyAxis.html

HomographicRelation.html

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