The inverse of a conic from any point not on the curve is a nodal
bicircular quartic. This is shown by inverting the general equation of
the conic
ax² + 2hxy + by² + 2gx + 2fy + c = 0;
x y
by substituting for x and y, ————————— and —————————— ,
(x² + y²) (x² + y²)
we get the equation
ax² + 2hxy + by² + 2(gx + fy)(x² + y²) + c(x² + y²)² = 0.
The origin is evidently a double point on the curve, and is a crunode,
acnode, or cusp according as the conic is a hyperbola, ellipse, or
parabola. The factors of the terms of the fourth degree, viz:
(x + iy) (x + iy) (x - iy) (x - iy), show that the two imaginary
circular points at infinity are double points on the quartic, which is
thus trinodal. Hence this nodal, bicircular quartic can be projected
into the most general form of the trinodal quartic. Trinodal quartics
are unicursal.
If the conic which we invert be a parabola, the quartic has two nodes
and one cusp. If the conic be inverted from a focus, the quartic has
the two circular points at infinity for cusps. This is best shown
analytically as follows: let the equation of the conic, origin being at
the focus, be written
x² y² 2aex b²
——— + ——— + ———— - ——— = 0.
a² b² a² a²