How to get an new cartesian coordinate system in 3D, starting from one vector.

if N.j > Epsilon ( N et j not orthogonal ) => N and k not parallel => k x N not null

Let's just define, arbitrarily,        u = k x N / || k x N ||

By construction, u is perpendicular to both k and N

then Let's define v = N x u /  || N x u ||

With u and v, we've got an orthogonal basis (hamel basis) within the plan normal to N

What if N and j are orthogonal ?

then N and j are not parallel => N x j not null => u = N x j

What about continuity issues (would it be better to use i : N x i ?)

An application : the envelope of a path

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