Three-dimensional rational cubic B-Spline approximation of space curve data given in three-dimensional coordinates or two-dimensional projection coordinates

Many new mechanical designs involve modification of existing product information, much of which exists only on hardcopy drawings. With industries shifting from manual design methods to computer aided design, drawing conversion systems that convert drawings to CAD databases would be very useful. Commercially available scanners that convert drawings to raster and/or rough vector data are available. A technique to smooth and convert these data to a more complex and compact 2-D object format is proposed and developed. This technique approximates the rough vector data of free-form planar or space curves with uniform non-periodic rational parametric cubic B-Splines represented by the homogeneous coordinates of their polygon vertices. A 2-D to 3-D conversion capability is also developed so that multiple 2-D vector data of projections of space curves can be converted to three dimensional B-Spline models. By properly integrating these procedures into CAD systems, it would also be possible to use 'CAD drafting' methods to create planar or space curves and then convert them to B-Spline formats. Another application of these procedures would be to obtain B-Spline representations of planar or spatial locations constrained by time as in motion generation and control.;The approach for the 2-D to 3-D conversion is similar except that the distances involved in the index of performance are from the vector data points in the 2-D views to the projections onto the views of the 3-D B-spline model curve.;Computer implementations of the procedures indicate that engineering approximations are practically realizable for a variety of free-form curves. Many examples are given. Recommendations for further development of the procedures are given.;The approach to achieving the approximations that involve no conversion minimizes an index of performance which is the sum of the squares of the shortest distances between the vector data points and the B-Spline model curve. Homogeneous coordinates of polygon vertices and the parameters corresponding to the vector data points are the variables in the minimization. Fletcher-Reeves conjugate gradient method has been employed for the optimization.