IBM Journal of Research and Development
High accurate rational approximation of parametric curves
Selected papers of the international symposium on Free-form curves and free-form surfaces
Hermite interpolation by Pythagorean hodograph quintics
Mathematics of Computation
Geometric Hermite interpolation
Computer Aided Geometric Design
Geometric Hermite interpolation with maximal order and smoothness
Computer Aided Geometric Design
Surface design based on Hermite spline interpolation with tension control and optimal twist vectors
Neural, Parallel & Scientific Computations - computer aided geometric design
Direct highlight line modification on nurbs surfaces
Computer Aided Geometric Design
Geometric Hermite interpolation with Tschirnhausen cubics
Journal of Computational and Applied Mathematics
Hermite interpolation with Tschirnhausen cubic spirals
Computer Aided Geometric Design
Optimal geometric Hermite interpolation of curves
Proceedings of the international conference on Mathematical methods for curves and surfaces II Lillehammer, 1997
On the local existence of the quadratic geometric Hermite interpolant
Computer Aided Geometric Design
ACM Transactions on Mathematical Software (TOMS)
Planar cubic G1 interpolatory splines with small strain energy
Journal of Computational and Applied Mathematics
Unbalanced hermite interpolation with tschirnhausen cubics
CIS'04 Proceedings of the First international conference on Computational and Information Science
Geometric hermite curves based on different objective functions
VSMM'06 Proceedings of the 12th international conference on Interactive Technologies and Sociotechnical Systems
Technical Section: Euler arc splines for curve completion
Computers and Graphics
Constructive G1 connection of multiple freeform pipes in arbitrary poses
Computer Aided Geometric Design
Proceedings of the 5th International Conference on Automotive User Interfaces and Interactive Vehicular Applications
Geometric constraints on quadratic Bézier curves using minimal length and energy
Journal of Computational and Applied Mathematics
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The purpose of this paper is to provide yet another solution to a fundamental problem in computer aided geometric design, i.e., constructing a smooth curve satisfying given endpoint (position and tangent) conditions. A new class of curves, called optimized geometric Hermite (OGH) curves, is introduced. An OGH curve is defined by optimizing the magnitudes of the endpoint tangent vectors in the Hermite interpolation process so that the strain energy of the curve is a minimum. An OGH curve is not only mathematically smooth, i.e., with minimum strain energy, but also geometrically smooth, i.e., loop-, cusp- and fold-free if the geometric smoothness conditions and the tangent direction preserving conditions on the tangent angles are satisfied. If the given tangent vectors do not satisfy the tangent angle constraints, one can use a 2-segment or a 3-segment composite optimized geometric Hermite (COH) curve to meet the requirements. Two techniques for constructing 2-segment COH curves and five techniques for constructing 3-segment COH curves are presented. These techniques ensure automatic satisfaction of the tangent angle constraints for each OGH segment and, consequently, mathematical and geometric smoothness of each segment of the curve. The presented OGH and COH curves, combined with symmetry-based extension schemes, cover tangent angles of all possible cases. The new method has been compared with the high-accuracy Hermite interpolation method by de Boor et al. and the Pythagorean-hodograph (PH) curves by Farouki et al. While the other two methods both would generate unpleasant shapes in some cases, the new method generates satisfactory shapes in all the cases.