- مبلغ: ۸۶,۰۰۰ تومان
- مبلغ: ۹۱,۰۰۰ تومان
Minkowski Pythagorean hodograph (MPH) curves provide a means for representing domains with rational boundaries via the medial axis transform. Based on the observation that MPH curves are not the only curves that yield rational envelopes, we define and study rational envelope (RE) curves that generalise MPH curves while maintaining the rationality of their associated envelopes. To demonstrate the utility of RE curves, we design a simple interpolation algorithm using RE curves, which is in turn used to produce rational surface blends between canal surfaces. Additionally, we initiate the study of rational envelope surfaces as a surface analogy to RE curves.
We have presented rational envelope curves as a generalisation of MPH curves. RE curves, although containing square roots, yield rational envelopes and can be constructed by simpler methods than those for MPH curves. To demonstrate the utility of RE curves, we proposed a simple interpolation algorithm for RE curves, which in turn can be used for canal surface blending using rational blends. While the curve case with (M)PH and RE curves is well understood now, their surface analogies still pose many challenges. For instance, there are no direct algorithms for interpolation with PN surfaces, especially in the polynomial setting, which could then be utilised in the MOS case. On the one hand, rational envelope surfaces seem to avoid this limitation, but on the other hand, further research needs to be conducted in this area to place the bivariate case on the same firm footing as the univariate setting.