Geometric Hermite Curves With A Minimum Rate Of Change Of Curvature

Constructing a smooth curve with given endpoint conditions is a fundamental problem in computer aided geometric design (CAGD). The Hermite interpolation process is frequently used in the construction of such a curve and the resulting cubic polynomial curve is called a Hermite curve. The Hermite curves can be widely used in applications such as shape design and curve/surface fairing in geometric modeling. For example, in fairing the abnormal region of a NURBS surface uses such traditional Hermite curves to replace abnormal portions of the highlight lines in those regions and deforms the surface so that the new surface would have the modified highlight lines as the new highlight lines and consequently, achieves the fairing result.Hermite curve has the minimum strain energy among all C~1 cubic polynomial curves satisfying the same endpoint conditions. Hence, fairing a C~1 cubic spline curve with endpoint (position and tangent vector) constraints will eventually lead to a cubic Hermite curve. Unfortunately, the shape of the Hermite curve may be unpleasant. It may have loops, cusps or folds, namely, not geometrically smooth. Hence, additional degrees of freedom are needed to meet the geometric smoothness requirements. Obviously, adjusting the magnitudes of the given tangent vectors can achieve the goal, and such Hermite curve is known as geometric Hermite curve. The work of this paper is to discuss how to producing a G~1 geometric Hermite curve with a pleasing shape, i.e., it does not contain undesired features such as loops, cusps or folds.Minimum curvature variation is used as the new smoothness criterion of curve in this paper and the integrated squared third derivative of curve is chosen as the approximate form of the curvature variation, i.e., objective function. The extended definition of optimized geometric Hermite (OGH) curve is given. A curve in this class is defined by optimizing the magnitudes of the endpoint tangent vectors in the Hermite interpolation process so that the curvature variation of the curve is a minimum. An explicit formula for obtaining such a curve is presented. The tangent angle constraints (tangent direction preserving conditions and geometric smoothness...