Confusion related to this algorithm about finding the orientation given 3 points

Question

I have a confusion related to an algorithm for finding the orientation given three points, i.e., determining whether one line is clockwise or counter-clockwise of another.

Let's say the points are $P_1=(x_1,y_1)$, $P_2=(x_2,y_2)$, $P_3(x_3,y_3)$. The algorithm says

• Let the slope of line $P_1P_2$ be $m_1$ and slope of $P_2P_3$ be $m_2$.
• If $m_2 > m_1$, the orientation is counterclockwise and if $m_1 > m_2$ it is clockwise.

However lets say the points are $(0,0)$, $(4,4)$ and $(2,4)$. This give $m_1=1$, $m_2=0$. The orientation is counter clockwise but $m_2 < m_1$. So I am wondering if there is any issue with the algorithm.

Any suggestions?

Answer 1

The orientation of three points in the plane is classically done with a determinant. In the linked article the approach is done on a polygon thus they use a 3x3 determinant. For only three points, i.e., a triangle, a 2x2 determinat is sufficient.

Let $p_1=(x_1,y_1), p_2=(x_2,y_2)$, and $p_3=(x_3,y_3)$ be three points in the plane. Then we compute the orientation vectors at $p_1$, that is $p_{1,2}=p_2-p_1$ and $p_{1,3}=p_3-p_1$. Now we use the determinant to find the orientation: $$\mathrm{det}\begin{bmatrix}x_2-x_1&y_2-y_1\\x_3-x_1&y_3-y_1\end{bmatrix}=(x_2-x_1)(y_3-y_1)-(y_2-y_1)(x_3-x_1).$$

If the result is smaller than zero then $(p_1,p_2,p_3)$ are in counter-clockwise order, if larger than zero in clockwise order, and if zero then they are collinear.