with at least one of A, B, C, D, E
not equal to zero.
This equation has 15 constants. However,
it can be multiplied by
any non-zero constant
without changing the curve; thus by the choice
of an appropriate constant of multiplication,
any one of the coefficients can be set
to 1,
leaving only 14 constants. Therefore, the space
of quartic curves can be
identified with the real
projective space
RP14 It also follows, from Cramer's
theorem
on algebraic curves, that there is exactly one
quartic curve that passes through a set of
14
distinct points in general
position, since a quartic
has 14 degrees of freedom.
One may also consider
quartic curves over
other fields (or even rings), for instance
the complex numbers. In this way, one
gets
m Riemann surfaces,
which are
one-dimensional
objects over C, but are two-dimensional over R.
An example is the Klein quartic.
Additionally,
one can loo k at curves in the projective
plane,
given by homogeneous polynomials.
