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：采用残量方法，我们完整描述了给定切线方向的有理空间曲线空间。当方向场的范数为完全平方时，有理Pythagorean速端曲线作为特例出现。曲线空间基的显式表达式被给出。由此，该空间中的一系列插值问题（$G^1$、$C^1$、$C^2$、$C^1/G^2$）转化为线性问题，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成为二次型，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验证。