Parameterizing and extending trimmed regions for tensor-product surface fitting

Abstract Our goal is to approximate the data points of an irregular, trimmed triangular mesh by standard, tensor-product surfaces, such as NURBS. This is a difficult and ambiguous task driven by several parameters, including tolerances, knot vectors, smoothing terms, etc. One of the most crucial issues is the parameterization of the data points that will have a strong influence on the quality of the surface to be fitted. We propose new techniques that attempt to establish some correspondence between the given trimmed region and the unknown four-sided surface. In the first phase a four-sided virtual guiding frame is created in 3D using labeled boundary segments. Then a 2D parameterization is computed that minimizes distortion while satisfying constraints implicitly defined by the labels and the frame. In the second phase new data points are inserted to fill in the domain rectangle, and an inverse mapping to 3D is performed. This leads to a four-sided mesh that extends the original trimmed region in a smooth and natural manner, and is well-suited for fitting an untrimmed tensor-product surface. We will discuss several examples to illustrate the benefits of our techniques using a “black-box” surface fitter. Benefits include even curvature distribution, natural surface extensions in the vicinity of the trimming boundaries, avoiding wiggles for the full surface and a tight bounding box.