Constructing the Asymptotes and Foci of a Circumhyperbola

If the circumconic has perspector P,

To construct the asymptotes

1. Construct the dual of P.
2. Find the intersections E₁, E₂ of this line with the Steiner ellipse. If the dual does not intersect the SE, there are no real asymptotes. If tangent, the conic is a parabola.
3. Construct nE₁, nE₂, the reflections of E₁ and E₂ in the centroid G.
4. The tripolars ~tnE₁ and ~tnE₂ of nE₁, nE₂ are parallel to the asymptotes through G.
5. The asymptotes are parallels to the tripolars though the center mtdP.

A second construction it to construct the points where nE₁ and nE₂ meet the conic. The lines from G to these two points are parallel to the asymptotes.

To construct the Foci

Construct a line perpendicular to the major axis from a vertex.
Construct the circle from the hyperbola center to the meet of the perpendicular with an asymptote.
This circle intersects the major axis in the two foci.