## A Loe4l /Upruentation B4IJed on Shape Parameter

- Brian A. Barsky, TM Bdo-.pline
- and Fund4mental Geometric Me4lJure., Ph.D. Thesi…
- 1981

- Published 2015

Parametric spline curves and surfaces are typically construded so that 80me number of derivatives matcb wbere tbe curve segments or surface patcbes abut. If derivativel up to order n are continuous, tbe segment. or patcbes are said to meet with C", or nUl order pGnundric continuity. It ha. been shown previously that parametric continuity;' suJficient, but not necessary, for geometric smootbnea. Tbe geometric measures of .nit tangent aDd curvature vectors for curves, and tangent plane and Dupin indictltri: for surfacell, have been u.ed to define fim and second order geometric continuit,. III tb;' paper, we extend tbe notion of geometric continuity to arbitrary order n (a") for curves and surfaces, and present an intuitive development of constraint equatiou tbat are nece!IBary and sufficient for it, The constraints (bown as tbe Beta corutruint,) result from a direct application of tbe univariate cbain rule for CUTVell and tbe bivariate cbain rule for surface.. For fiBt and second order continuity, tbe Beta cOllBtraints are equivalent to requiring continuity of tbe geometric measures detJCribed above. Tbe Beta COllBtraillt. provide for tbe introduction of quantities bown a. ,luapc parumeter,. If two curve segmeDts are to meet witb G" continuity, n shape pa--

@inproceedings{DeRoae2015AnIA,
title={An Intuitive Approach to Geometric Continuity for Parametric Curves and Surfaces},
author={Tony D. DeRoae and Briaa A. Bank},
year={2015}
}