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

PROJECTION OF THE CIRCULAR POINTS.

237

The alternative formulae are
g cos e - t sin e + x sin e.
x...........5(2).
y _ xq sin e + y (g cos e - 6 sin e)
cos e - f sin e + sine
355. Any two points can be projected into the circular points
at infinity and vice versa.
Let P and Q be two points in the plane '. Take a point 0
as origin in the line of intersection of the planes z and z', such
that OPQ is an isosceles triangle whose vertex is 0, and let the
equations of OP, OQ be y'= + mx'; and let the equation of PQ
be x'=1.
Through PQ draw a plane parallel to the plane z, and let the
vertex V be a point in this plane whose coordinates referred to 0
are:, 0, 1 sin e. Then the equation of the projection of OP on
the plane z is
y (I cos e - I) = mlx,
and the projection of P is the intersection of this line with the
line at infinity on z. If this point is the circular point x= ty, we
must have
I cos e - = mnl.
Let e = -r - l/3; then cos e = t sinh 3, whence
t= Lt (sinh/3 - mn),
7= 0,
= I cosh f3.
Accordingly (2) become
lmi +.
- only.(3),.........................(3
tml + x
which give
iLmlx \
___0.. (4)........................... (4 ).
Y=I -x'
7 = -- --
tY r- J ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~(>