Reducing control points in surface interpolation

Surface interpolation to rectangularly arranged points is an integral part of surface design and modeling in CAD/CAM and graphics. Using B-spline surfaces, the process involves curve interpolations through rows of data points and through columns of control points. This method, as well tuned as it is, proves inadequate for recent problems such as those of reverse engineering. Data acquisition devices, such as scanners, may be used to return rows of data points, but it's not guaranteed that each row contains the same number of points. The problem then arises of passing a smooth surface through these points (assuming that interpolation is justified, meaning the number of points isn't large). Since each row contains different numbers of points, regular data interpolation can't be used. One method to solve this problem is to interpolate each row with B-spline curves and to pass a smooth surface through these curves via surface skinning. While this is a legitimate solution, the number of control points tends to become prohibitively large, especially if the number of rows is large. This article addresses the problem of how to reduce the number of control points while maintaining precise interpolation. The idea is to give the knots some flexibility so that each row can be interpolated with as few new knots added as possible.