Energy and B-spline interproximation

In this paper, we study B-spline curve interproximation with different energy forms and parametrization techniques, and present an interproximation scheme for B-spline surfaces. It shows that the energy form has a much bigger impact on the generated curve than the parametrization technique. With the same energy form, different parametrization techniques generate relatively small difference on the corresponding curves. With the same parametrization technique, however, different energy forms make significant difference on the shape and smoothness of the resulting curves. Furthermore, interproximating B-spline curves generated by minimizing approximated energy forms are far from being good approximations to the optimal curves. They tend to generate flatter regions and sharper turns than curves generated by minimizing the exact energy form. The interproximation scheme for surfaces is aimed at generating a smooth surface to interpolate a grid of data which could either be a point or a region. This is achieved by minimizing a strain energy based on squared principal curvatures for bicubic B-spline surfaces. The surface interproximation process is also studied with different energy forms and parametrization techniques. The test results of the surface interproximation process also show the same conclusion as the curve interproximation process.