ProjectiveRotations

Projective rotations

Here we continue the discussion initiated in TriangleProjectivitiesPlay.html - [8]. Start with a point A₁ on the outer conic e''(x) and draw tangents to the inner one e'(x). The second intersection points on these tangents with e''(x) form a triangle A₁B₁C₁ which is simultaneously inscribed in e''(x) and circumscribed in e'(x).

All the triangles constructed in this way have the same trilinear polar L(x) w.r. to x. They are simply the images of the other equilateral triangles inscribed in the same circle with the original one A₀B₀C₀. Thus they correspond to a sort of projective rotations of triangle ABC about the point x. Notice the locus of intersection points (like points C', B') of tangents at vertices A₁, B₁, C₁ which is a conic c₁ belonging to the family of conics generated by e'(x) and e''(x).
The term projective rotation is suggested from the fact that we build projectivities of the form H_t = H*R_t, which are compositions of H and a usual rotation R_t by an angle (t) about M₀. Thus H_t*H^-1 is conjugate to a usual rotation.

Projective rotations

See Also