  We present a new method for visualizing implicit real algebraic curves inside
a bounding box in the $2$-D or $3$-D ambient space based on numerical
continuation and critical point methods. The underlying techniques work also
for tracing space curve in higher-dimensional space. Since the topology of a
curve near a singular point of it is not numerically stable, we trace only the
curve outside neighborhoods of singular points and replace each neighborhood
simply by a point, which produces a polygonal approximation that is
$\epsilon$-close to the curve. Such an approximation is more stable for
defining the numerical connectedness of the complement of the projection of the
curve in $\mathbb{R}^2$, which is important for applications such as solving
bi-parametric polynomial systems. The algorithm starts by computing three types
of key points of the curve, namely the intersection of the curve with small
spheres centered at singular points, regular critical points of every connected
components of the curve, as well as intersection points of the curve with the
given bounding box. It then traces the curve starting with and in the order of
the above three types of points. This basic scheme is further enhanced by
several optimizations, such as grouping singular points in natural clusters,
tracing the curve by a try-and-resume strategy and handling "pseudo singular
points". The effectiveness of the algorithm is illustrated by numerous
examples. This manuscript extends our preliminary results that appeared in CASC
2018.
