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Geometric Hermite interpolation by spatial Pythagorean-hodograph cubics. (English) Zbl 1065.68103
Summary: It is shown that, depending upon the orientation of the end tangents $\bold t_0$, $\bold t_1$ relative to the end point displacement vector $\Delta \bold p = \bold p_1 - \bold p_0$, the problem of $G^1$ Hermite interpolation by PH cubic segments may admit zero, one, or two distinct solutions. For cases where two interpolants exist, the bending energy may be used to select among them. In cases where no solution exists, we determine the minimal adjustment of one end tangent that permits a spatial PH cubic Hermite interpolant. The problem of assigning tangents to a sequence of points $\bold p_0,\ldots,\bold p_n$ in $R^3$, compatible with a $G^1$ piecewise-PH-cubic spline interpolating those points, is also briefly addressed. The performance of these methods, in terms of overall smoothness and shape-preservation properties of the resulting curves, is illustrated by a selection of computed examples.

68U05Computer graphics; computational geometry
Full Text: DOI
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