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Let    $$X\subset \mathbb {P}^3$$  X ⊂ P 3 be an integral and non-degenerate curve. We say that    $$q\in \mathbb {P}^3\setminus X$$  q ∈ P 3 \ X has  X -rank 3 if there is no line    $$L\subset \mathbb {P}^3$$  L ⊂ P 3 such that    $$q\in L$$  q ∈ L and    $$\#(L\cap X)\ge 2$$  # ( L ∩ X ) ≥ 2 . We prove that for all hyperelliptic curves of genus    $$g\ge 5$$  g ≥ 5 there is a degree    $$g+3$$  g + 3 embedding    $$X\subset \mathbb {P}^3$$  X ⊂ P 3 with exactly    $$2g+2$$  2 g + 2 points with  X -rank 3 and another embedding without points with  X -rank 3 but with exactly    $$2g+2$$  2 g + 2 points    $$q\in \mathbb {P}^3$$  q ∈ P 3 such that there is a unique pair of points of  X  spanning a line containing  q . We also prove the non-existence of points of  X -rank 3 for general curves of bidegree ( a ,  b ) in a smooth quadric (except in known exceptional cases) and we give lower bounds for the number of pairs of points of  X  spanning a line containing a fixed    $$q\in \mathbb {P}^3\setminus X$$  q ∈ P 3 \ X . For all integers    $$g\ge 5$$  g ≥ 5 ,    $$x\ge 0$$  x ≥ 0 we prove the existence of a nodal hyperelliptic curve  X  with geometric genus  g , exactly  x  nodes,    $$\deg (X) = x+g+3$$  deg ( X ) = x + g + 3 and having at least    $$x+2g+2$$  x + 2 g + 2 points of  X -rank 3.

Space curves, X-ranks and cuspidal projections