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Polar object representations have proven to be a powerful shape model for many medical as well as other computer vision applications, such as interactive image segmentation or tracking. Inspired by recent work on Sobolev active contours we derive a Sobolev-type function space for polar curves. This so-called polar space is endowed with a metric that allows us to favor origin translations and scale changes over smooth deformations of the curve. Moreover, the resulting curve flow inherits the coarse-to-fine behavior of Sobolev active contours and is thus very robust to local minima. These properties make the resulting polar active contours a powerful segmentation tool for many medical applications, such as cross-sectional vessel segmentation, aneurysm analysis, or cell tracking.