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?


  • $\begingroup$ Does the determinant written in that form remind you of anything? $\endgroup$ – quicksort Jan 24 '17 at 12:14
  • $\begingroup$ Signed area of the triangle pqr? I dont know the theory. $\endgroup$ – Sami Jan 24 '17 at 12:17
  • 2
    $\begingroup$ It's exactly the crossproduct $(b-a) \times (c-a)$. $\endgroup$ – quicksort Jan 24 '17 at 12:25

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