Stationary values of mean tangent rotation angles are used in the intrinsic~urve design method which defines curves for CA D by specify/no curvature and/or torsion profiles ~ 2 This paper discusses these mean angles (~end ~) and makes use of ~ to derive approximate formulas for fast Integration of curvature profiles s .
This paper develops two ideas
Approximate formulas ~=Β’~ and Β’=/3, which are the key controlling parameters in the intrinsic-curve design method t,2, are more formally discussed and shown to be stationary points Curvature integration (integration of curvature and torsion profilest, 2 ) uses Fresnel integrals which are usually evaluated by numerical integration 3, series expansions 4,s , or by using tables 6 . All of these methods need long computation times and new formulas are proposed. These formulas, within the limitations mentioned later, are faster than the best available series method by a factor of almost 4. The approximation ~=~ is used to derive these formulas.
NOTATION
Variables with parentheses are used to denote the values at a specific arc length. Subscripts 1 and 2 refer to initial and final points and subscripts / and h denote lower and upper bounds, respectively. d do E e F,G K L La r S s t, n, b U Chord vector Approximate chord vector Error (=ld -dal/S) Infinitesimal quantity Functionals (defined in text) Curvature Chord length Approximate chord length Position vector of a point Total arc length Variable arc length Tangent, normal, binormal at a general point Normalized arc length (varies from 0 to 1)