Constructing the axes, asymptotes, and foci of a circumconic
For a circumconic with perspector P.
To have asymptotes, P must be outside the inner Steiner ellipse.
Notation: t stands for the istotomic conjugate operation.
m and d for the medial and antimedial operations.
~ stands for the dual which is the tripolar of the isotomic conjugate.
A—B is the line determined by points A and B.
a•b is the point determined by lines a and b.
The relationships described by these operations are affinely
invariant.
The equations of the asymptotes of a general hyperbola are derived here.
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 is to construct the points where nE1— nE2 meets the conic. The lines from G to these two points are parallel to the asymptotes.
To construct the Axes
1. Construct line l = D—dP through dP (or D = dK) parallel to PK, where K is the symmedian point.
2. Find the intersections E1, E2 of line l with the Steiner ellipse.
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 axes through G.
5. The axes are parallel to the tripolars though the center mtdP.
This is a maximally affine construction. The only non-affine information needed is the existence of K, the symmedian point.
The axes of the P-circumconic are parallel to the asymptotes of the ~(dP—D) rectangular hyperbola.
The point ~(dP—D) = ~dP•~D (property of duality), hence the perspector is the intersection of the line ~dP with ~D, the polar axis.
The construction of the axes is very much like the construction of the asymptotes. In fact if the asymptotes are constructed for a circumconic of perspector ~(dP—D) = ~dP•~D, these asymptotes will be parallel to the axes of a conic of perspector P.
To construct the Foci.
Draw the bounding box of the ellipse. (Not all these lines shown in figure).
Find an intersection of the ellipse with the minor axis.
Construct circle from this point tangent to the perpendiculars to major axis from major vertices.
This circle will intersect the major axes at the foci.
Figure: The P-circumconic is red; its auxiliary conic in blue. The circumcircle is in blue and the Steiner ellipse in black. The foci are constructed using the bounding box of the conic. E1 and E2 are the intersection of dP—D with the Steiner ellipse. F1 and F2 the foci.