keywords: Math, Plane Side, Point

Method 1

Let \(A*x+B*y+C*z+D=0\) be the equation determining your plane.

Substitute the \((X, Y, Z)\) coordinates of a point into the left hand side of the equation (I mean the \(A*x+B*y+C*z+D\)) and look at the sign of the result.

The points having the same sign are on the same side of the plane.

Origin:
Check which side of a plane points are on
https://stackoverflow.com/questions/15688232/check-which-side-of-a-plane-points-are-on

Plane Equation:
If point \(P_0 = (x_0, y_0, z_0)\), and vector \(n = (a, b, c)\), Plane Equation is:

\[a*x + b*y + c*z = a*x_0 + b*y_0 + c*z_0\]

Method 2

The plane passing through \(P_1(X_1, Y_1, Z_1)\), \(P_2(X_2, Y_2, Z_2)\), and \(P_3(X_3, Y_3, Z_3)\), the point to check is \(A(X_a, Y_a, Z_a)\).
So we can get the normal vector of plane using cross product:

\[\overrightarrow{n}=\overrightarrow{P_1P_2}\times\overrightarrow{P_1P_3}\]

then calculate the dot product of \(\overrightarrow{n}\) and \(\overrightarrow{P_1A}\):

\[ a=\overrightarrow{n}\cdot\overrightarrow{P_1A}\]

Point \(A\) is on one side if \(a\) > 0, otherwise it's on the another side.


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