If I have an ordered 3-tuple of points in integer barycentric coordinates, how do I test orientation on them? (I want to know if the points are collinear, form a left turn or a right turn)
The "algorithm" has to be quite robust so I don't want to convert the coordinates to cartesians.
For cartesians, there is a very nice way to determine this using only multiplication and addition: https://www.cs.cmu.edu/~quake/robust.html
There is a similar way to find out if three points are collinear here, but I don't know if I can use it for this application: http://web.evanchen.cc/handouts/bary/bary-short.pdf
As the last paper says, for CCW base triangle ABC signed area for PQR is positive for CCW order (footnote at page 1), so triplet P, Q, R makes left turn if determinant
x1 y1 z1
x2 y2 z2
x3 y3 z3
has positive value (theorem 10), and points are collinear for zero determinant