An elementary treatise on cubic and quartic curves, by A. B. Basset.

242

THEORY OF PROJECTION.

Quartic Curves.
366. We shall first consider the projection of a quartic with
three double points.
We have already shown that any two imaginary points may be
projected into a pair of real points; if therefore a quartic has a
pair of imaginary nodes or cusps, the curve may be projected into
a quartic having a pair of real nodes or cusps; also the triangle
whose vertices are the double points may be projected into an
equilateral triangle, such that any given point in its plane
coincides with any given point in the plane of the original
triangle. Whence:Any tricuspidal quartic may be projected into a three-cusped
hypocycloid or into a cardioid.
The projection may be accomplished in the first case by
projecting the cuspidal triangle into a real equilateral triangle,
whose centre of gravity coincides with the point of intersection of
the three cuspidal tangents; whilst in the second case, two of the
cusps must be projected into the circular points at infinity.
367. We shall now give some examples.
We have shown in ~ 325 that:-If any tangent to a threecusped hypocycloid cuts the curve in P and Q, the tangents at these
points intersect at right angles on a circle which touches the hypocycloid at three points; also the line at infinity is the only double
tangent, and the points of contact are the circular points. Whence,by projection:(i) If any tangent to a tricuspidal quartic cuts the curve in
P and Q, the tangents at these points intersect on a conic, which
(a) touches the quartic at three points; (/3) intersects it at the points
of contact of the double tangent; (7) also the point of intersection
of the three cuspidal tangents is the pole of the double tangent with
respect to the conic.
From ~ 357, it follows that:(ii) The two tangents, together with the lines joining their
point of intersection with the points of contact of the double tangent,
form a harmonic pencil.