Fig. 1. Representation of procedures that insert a point and an edge in a CDT. Left, point p is added to the original triangulation; right, edge ab is enforced, obtaining the final CDT.

Fig. 2. Inserting a constraining edge in a CDT. Left, triangles cut by the new constraining edge ab are removed from the CDT, giving the upper and lower pseudo-polygons. Right, retriangulation of the upper pseudo-polygon. In each recursive call the empty circumellipse criterion (dotted line) with respect to the bounding points of the pseudo-polygon is tested. The chosen ellipse is the one asigned to point c to fulfill Proposition 2.

Fig. 3. Example of application of the algorithm for the case that the ellipse E is unique. Top, original graph and usual CDT; bottom, output triangulation of the modified algorithm and deforming ellipse used.

Fig. 4. Example of application of the algorithm in the case the ellipse depends on the point of the plane. The figure on the left shows the ellipses evaluated at several points of the plane; the resulting triangulation is drawn on the right.

Fig. 5. Simple example that shows that the output triangulation produced by the modified algorithm in case II depends on the order in which the points are inserted. Points in the left triangulation have been inserted in the reverse order to those in the right triangulation.

Fig. 6. Second example. Top, original graph and usual CDT; bottom, deforming ellipses evaluated at several points of the plane and output triangulation.

Fig. 7. Example of an application of the directional CDT algorithm on a surface meshing. Notice the adaptability of the resulting triangulation to the curvature: while large triangles are produced in flat regions, small triangles recover very curved regions. Furthermore, triangles tend to be aligned according to the curvature directions.