A method to reconstruct symmetric B-spline curves and surfaces is presented. The symmetry property is realized by using symmetric knot vector and symmetric control points. Firstly, data points are divided into two parts based on the symmetry axis or symmetry plane extracted from data points. Then the divided data points are parameterized and a symmetric knot vector is selected in order to get symmetric B-spline basis functions. Constraint equations regarding the control points are deduced to keep the control points of the B-spline curve or surface to be symmetric with respect to the extracted symmetry axis or symmetry plane. Lastly, the constrained least squares fitting problem is solved with the Lagrange multiplier method. Two examples from industry are given to show that the proposed method is efficient, robust and able to meet the general engineering requirements.
For reconstructing a freeform feature from point cloud, a deformation-based method is proposed in this paper. The freeform feature consists of a secondary surface and a blending surface. The secondary surface plays a role in substituting a local region of a given primary surface. The blending surface acts as a bridge to smoothly connect the unchanged region of the primary surface with the secondary surface. The secondary surface is generated by surface deformation subjected to line constraints, i.e., character lines and limiting lines, not designed by conventional methods. The lines are used to represent the underlying informa-tion of the freeform feature in point cloud, where the character lines depict the feature’s shape, and the limiting lines determine its location and orientation. The configuration of the character lines and the extraction of the limiting lines are discussed in detail. The blending surface is designed by the traditional modeling method, whose intrinsic parameters are recovered from point cloud through a series of steps, namely, point cloud slicing, circle fitting and regression analysis. The proposed method is used not only to effectively and efficiently reconstruct the freeform feature, but also to modify it by manipulating the line constraints. Typical examples are given to verify our method.
