Fig. 1. An “active” curve moving towards some data points (three time steps).

Fig. 2. Precomputed distance fields.

Fig. 3. Evolution of a parametric curve towards a point cloud.

Fig. 4. Top: A T-spline grid. We use four-fold knots at boundaries. Bottom: The knot vectors for three selected control points.

Fig. 5. Adapting the topology by dual evolution of an implicitly defined (gray) and a parametric curve (black).

Fig. 6. The assumptions of Lemma 1.

Fig. 7. Conditions for the minimum of g on S. Case 1 (left) and Case 2 (right).

Fig. 8. Detection of topological changes.

Fig. 9. Synchronization without topological changes.

Fig. 10. Synchronization with topological changes. The target is defined by a smooth curve (top) or by a point cloud (bottom).

Fig. 11. Approximation with (right) and without (left) range constraints.

Fig. 12. Dual evolution with range constraints for a target defined by an algebraic curve.

Fig. 13. Dual evolution with range constraints for a target defined by three point clouds.

Fig. 14. Dual evolution with volume constraints.

Fig. 15. Approximation with (right) and without (left) convexity constraints.