Using a residuum approach, we provide a complete description of the space of the rational spatial curves of given tangent directions. The rational Pythagorean hodograph curves are obtained as a special case when the norm of the direction field is a perfect square. The basis for the curve space is given explicitly. Consequently a number of interpolation problems ($G^1$, $C^1$, $C^2$, $C^1/G^2$) in this space become linear, cusp avoidance can be encoded by linear inequalities, and optimization problems like minimal energy or optimal length are quadratic and can be solved efficiently via quadratic programming. We outline the interpolation/optimization strategy and demonstrate it on several examples.

翻译：采用残差方法，我们给出了给定切线方向的有理空间曲线的完整空间描述。当方向场的范数为完全平方时，有理Pythagorean速端曲线作为特例被导出。我们显式给出了曲线空间的基。因此，该空间中的一系列插值问题（$G^1$、$C^1$、$C^2$、$C^1/G^2$）变为线性问题，尖点规避可通过线性不等式编码，而最小能量或最优长度等优化问题为二次型，可通过二次规划高效求解。我们概述了插值/优化策略，并通过多个实例加以说明。