  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.
