TITLE: Point on the left or right side of a plane in 3D space
QUESTION [9 upvotes]: I have an alpha plane determined by 3 points in space. How can I check if another point in space is on the left side of the plane or on the right side of it?
For example if the plane is determined by points $A(0,0,0)$, $B(0,1,0)$ and $C(0,0,1)$ then point $X(-1, 0, 0)$ is on the left side of the plane and point $Y(1,0,0)$ is on the right side of the plane. 
I need a fast solution for plug-in development for a 3D application and I'm not very good at math.

REPLY [4 votes]: I'm guessing from your comment that your planes intersect the $x$-axis in exactly one point, which determines the left and right sides.
Let $A,B,C$ be the points that determine the plane. Then the cross product $(B-A) \times (C-A)$ gives us a normal ${\bf n}$ to the plane. Now consider a test point $(x,0,0)$ where $x$ is a huge positive number. This should be on the right side of the plane. Now $((x,0,0) - A) \cdot {\bf n}$ is just the first coordinate of ${\bf n}$ times $x$ minus some constant. For large enough $x$ the sign of the dot product is then just the sign of the first coordinate of ${\bf n}$. Thus: If the sign of first coordinate of $n$ is positive, then the right side consists of the points $P$ with $(P - A) \cdot {\bf n} > 0$ and the left side consists of the points with $(P - A) \cdot {\bf n} < 0$. The inequalities are reversed if the sign of the first coordinate of $n$ is negative.