  In this paper we present three related results on the subject of fields of
parametrization. Let C be a rational curve over a field of characteristic zero.
Let K be a field finitely generated over Q, such that it is a field of
definition of C but not a field of parametrization. It is known that there are
quadratic extensions of K that parametrize C. First, we prove that there are
infinitely many quadratic extensions of K that are fields of parametrization of
C.
  As a consequence, we prove that the witness variety, that appear in the
context of the parametric Weil's descente method, is always a special curve
related to algebraic extensions, called hypercircle. It is possible that the
witness variety is not a hypercircle for the given extension, but for an
alternative one.
  We use these two facts to present an algorithm to solve the following optimal
reparametrization problem. Given a birational parametrization f(t) of a curve
C, compute the affine reparametrization at+b such f(at+b) has coefficients over
a field as small as possible. The main advantage of this algorithm is that it
does not need to compute any rational point on the curve.
