In recent years, the field of active-contour based image segmentation have seen the emergence of two competing approaches. The first and oldest approach represents active contours in an explicit (or parametric) manner corresponding to the Lagrangian formulation. The second approach represent active contours in an implicit manner corresponding to the Eulerian framework. After comparing these two approaches, we describe several new topological and physical constraints applied on parametric active contours in order to combine the advantages of these two contour representations. We introduce three key algorithms for independently controlling active contour parameterization, shape and topology. We compare our result to the level-set method and show similar results with a significant speed-up.