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Coplaner points calculator

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Point 1 X1  Y1  Z1 
 
 
Point 2 X2  Y2  Z2 
 
 
Point 3 X3  Y3  Z3 
 
 
 
Point 4 X4  Y4  Z4 
 
 
 
Point 5 X5  Y5  Z5 
 
 
 
Point 6 X6  Y6  Z6 
 
 
 
Point 7 X7  Y7  Z7 
 
 
 
Point 8 X8  Y8  Z8 
 
 
 
Point 9 X9  Y9  Z9 
 
 
 
Point 10 X10  Y10  Z10 
 
 
 
Coplanar test:
Plane equation:
nomenclature: Plane definition points
 
 Coplanar points
 
 Distance from plane
 
              

Coplanar points summary

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Coplanar points
Coplanar points
Points that are located on a plane are coplanar.
3 points always creates a plane so they are always coplanar.
4 points are coplanar if the volume created by the points is 0.
Volume = Volume
In vector format the points are coplanar if the vectors connecting one point to the other three points satisfy the scalar triple product:
Triple product
The vectors are:
v1 = (x2 − x1)i + (y2 − y1)j + (z2 − z1)k
v2 = (x3 − x1)i + (y3 − y1)j + (z3 − z1)k
v3 = (x4 − x1)i + (y4 − y1)j + (z4 − z1)k
If any three points determine a plane, then additional points can be checked for coplanarity by measuring the distance of the points from the plane, if the distance is 0 then the point is coplanar.
Plane equation: Ax + By + Cz + D = 0
The distance of a point (xn , yn , zn) from the plane is:
Distance of a point to plane

Coplanar points - Example

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Example: Find if the following points are coplanar: (3, -3, 2), (1, 0, 1), (1, 1, 0), (0, 1, 1).
Solution: the three vectors are:
v1 = − 2i + 3j − k
v2 = − 2i + 4j − 2k
v3 = − 3j + 4j − k
In order to find if all 4 points are coplanar we have to check the triple product:     v1 ∙(v2 × v3)
Distance of a point to plane
And the triple product is equal to: v1 ∙(v2 × v3) = (-2i + 3j - k)∙(i + j + k) = − 2 + 3 - 1 = 0
The resulting 0 indicates that the points are coplanar.
Another way to check if the points are coplanar is by calculating the volume produced by the 4 points, this process is tedious because we have to calculate the determinant of a 4th degree matrix:
Volume between 4 points the volume is zero therefore these 4 points are coplanar.
Now any additional points can be checked by calculating the distance of the points from the plane which is defined by the previous points, for example if we have another point let say the point (-2, 3, 1).
The coefficients of the equation of the plane are:       A = 1       B = 1       C = 1       D = − 2
x + y + z − 2 = 0
And the distance from this plane to the point (-2, 3, 1) is:
Distance of a point from plane
Because the distance is zero this additional point is also coplanar related to the other points.