Active contours are very popular tools for video tracking and image segmentation. Parameterized contours are used due to their fast evolution and have become the method of choice in the Sobolev context. Unfortunately, these contours are not easily adaptable to topological changes, and they may sometimes develop undesirable loops, resulting in erroneous results. To solve such topological problems, one needs an algorithm for contour self-crossing detection. We propose a simple methodology via simple techniques from differential topology. The detection is accomplished by inspecting the total net change of a given contour's angle, without point sorting and plane sweeping. We discuss the efficient implementation of the algorithm. We also provide algorithms for locating crossings by angle considerations and by plotting the four-connected lines between the discrete contour points. The proposed algorithms can be added to any parametric active-contour model. We show examples of successful tracking in real-world video sequences by Sobolev active contours and the proposed algorithms and provide ideas for further research. © 2012 IEEE.