We can think of this Pascal configuration as lying in the
plane *P: z=1*. Now we can use the origin to project this plane
onto any other plane *P'*. The projection will take points in
the *P* to points in the *P'*, and lines in *P* to
lines in *P'*. The same projection will take the circle from
*P* to some conic in *P'*. However, in *P'* the line
at infinity may intersect or be tangent to the conic; in other words, the
conic may become a hyperbola or a parabola.

Let *P'* have the line at infinity *l(inf)'*. This line
corresponds to some line *l* in *P*. Now consider the reverse
process: specifying the line *l* in *P* and allowing it to
determine the plane *P'*. This does not determine the plane
*P'* uniquely; it only restricts the possibilities for *P'* to
a family of parallel planes. However, this does determine
uniquely the projection of the Pascal configuration from *P*.
To be more precise: Say we take two planes from this family
and consider the projections
of the Pascal configuration in both of them. Then the two
projections can differ only in size; they are identical up to
a dilation. Thus, once we specify *l*, we have specified what
the Pascal configuration looks like in *P'*.

The nature of this specification becomes much more clear when
it is displayed graphically; this is the point of the program
called below. This program produces two graphs, one of which
is the already-described Pascal configuration in *P*. The second
graph shows the same Pascal configuration in *P'*; we specify *P'*
by specifying *l* in *P*, and we specify *l* using the
bottom half of the form.

If the projection in *P'* of our unit circle is a hyperbola, it
will contain two points on the line at infinity. These will
correspond to two points in *P* which lie both on the unit circle
and on *l*, and we may specify these two points by means of angles.
So, when the hyperbola option is selected on the form, the last
two inputs are taken as angles specifying these points, the
line *l*, and the plane *P'*.

If the projection in *P'* of our unit circle is an ellipse, it
will not intersect the line at infinity at all. So the line
*l* in *P* will not intersect the unit circle at all; we can not
specify *l* as we did in the hyperbolic case. Instead, we
imagine dropping a perpendicular from the origin to *l*; we
specify the line *l* by specifying this perpendicular. So,
when the elliptic option is selected on the form, the second-
to-last input is taken as the angle from the *x*-axis to
the perpendicular, and the last input is taken as the length
of that perpendicular.

Comments to:
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Created: Nov 30 1995 ---
Last modified: Thu Nov 30 15:58:53 1995