Currently I am taking a computational geometry course and I have learnt that:
Orientation is formally defined as the sign of the determinant of the points given in homoge- neous coordinates. For example, in the plane, we have Orient(p,q,r) = determinant of the matrix:
1 ax ay 1 bx by 1 cx cy
What is the mathematical explanation for this formula?
I have expand the determinant and come up with this:
det(P) = (bx - ax)(cy - ay) - (cx - ax)(by - ay)
I also know an additional information which may lead to something useful, if the crossproduct (b-a)x(c-a) results in the +z direction then the rotation is positive. However, I am not able to derive anything from there. What should be my next step?