# Orientation of 3 points in 2d plane

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?

Thanks

• Does the determinant written in that form remind you of anything? – quicksort Jan 24 '17 at 12:14
• Signed area of the triangle pqr? I dont know the theory. – Sami Jan 24 '17 at 12:17
• It's exactly the crossproduct $(b-a) \times (c-a)$. – quicksort Jan 24 '17 at 12:25