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.