  Converting a parametric curve into the implicit form, which is called
implicitization, has always been a popular but challenging problem in geometric
modeling and related applications. However, the existing methods mostly suffer
from the problems of maintaining geometric features and choosing a reasonable
implicit degree. The present paper has two contributions. We first introduce a
new regularization constraint(called the weak gradient constraint) for both
polynomial and non-polynomial curves, which efficiently possesses shape
preserving. We then propose two adaptive algorithms of approximate
implicitization for polynomial and non-polynomial curves respectively, which
find the ``optimal'' implicit degree based on the behavior of the weak gradient
constraint. More precisely, the idea is gradually increasing the implicit
degree, until there is no obvious improvement in the weak gradient loss of the
outputs. Experimental results have shown the effectiveness and high quality of
our proposed methods.
