"H
ere's What Happened"
at this stage in the course -
Assumptions
about the curve
:
A
non-degenerate conic curve
(
NDCC
),
C
, is
the intersection of
a plane
, $\pi$,
with
a cone
that
does not pass through
the vertex of the cone,
$O$
.
We can assign
3 dimensional Cartesian coordinates
so that $O = (0,0,0)$ and $\pi =\{(x,y,z): z=1\}$.
C
is a NDCC
if and only if
there is a
real irreducible polynomial
$P(x,y)=Ax^2 + Bxy + Cy^2 + Dx + Ey + F$
with
$A, B$, and $C$
not all 0
AND
C
$ = \{(x,y,1): P(x,y) = 0\}.$