# How can I write the equation of the plane passing through the list of three points?

At here Equation of the plane passing through the three points I can write the equation of the plane passing through three points. Now I have a list of three points.

{{{-12, 2, -1}, {-11, 1, -5}, {-10, -2, 3}}, {{-12, 2, -1}, {-11,
1, -5}, {-10, 6, 3}}, {{-12, 2, -1}, {-11, 1, -5}, {-9,
5, -7}}, {{-12, 2, -1}, {-11, 1, -5}, {-9, 8, -4}}, {{-12,
2, -1}, {-11, 1, -5}, {-7, -6, -2}}, {{-12, 2, -1}, {-11,
1, -5}, {-7, -2, -8}}, {{-12, 2, -1}, {-11, 1, -5}, {-7, -2,
6}}, {{-12, 2, -1}, {-11, 1, -5}, {-7, 3, -9}}, {{-12,
2, -1}, {-11, 1, -5}, {-7, 3, 7}}, {{-12, 2, -1}, {-11,
1, -5}, {-7, 6, -8}}, {{-12, 2, -1}, {-11, 1, -5}, {-7, 9,
3}}, {{-12, 2, -1}, {-11, 1, -5}, {-7, 10, -2}}, {{-12,
2, -1}, {-11, 1, -5}, {-6, -4, -7}}, {{-12, 2, -1}, {-11,
1, -5}, {-6, -4, 5}}, {{-12, 2, -1}, {-11, 1, -5}, {-6,
8, -7}}, {{-12, 2, -1}, {-11, 1, -5}, {-6, 8, 5}}, {{-12,
2, -1}, {-11, 1, -5}, {-4, -6, 3}}, {{-12, 2, -1}, {-11,
1, -5}, {-4, -2, -9}}, {{-12, 2, -1}, {-11, 1, -5}, {-4, -2,
7}}, {{-12, 2, -1}, {-11, 1, -5}, {-4, 6, -9}}, {{-12,
2, -1}, {-11, 1, -5}, {-4, 6, 7}}, {{-12, 2, -1}, {-11,
1, -5}, {-4, 10, 3}}, {{-12, 2, -1}, {-11, 1, -5}, {-2, -6,
3}}, {{-12, 2, -1}, {-11, 1, -5}, {-2, 6, -9}}, {{-12,
2, -1}, {-11, 1, -5}, {-2, 6, 7}}, {{-12, 2, -1}, {-11,
1, -5}, {-2, 10, 3}}, {{-12, 2, -1}, {-11, 1, -5}, {3,
5, -7}}, {{-12, 2, -1}, {-11, 1, -5}, {3, 8, -4}}, {{-12,
2, -1}, {-11, 1, -5}, {4, -2, 3}}, {{-12, 2, -1}, {-11, 1, -5}, {4,
6, 3}}, {{-12, 2, -1}, {-11, 1, -5}, {5, 6, -2}}, {{-12,
2, -1}, {-11, 1, 3}, {-10, -2, -5}}, {{-12, 2, -1}, {-11, 1,
3}, {-10, 6, -5}}, {{-12, 2, -1}, {-11, 1, 3}, {-9, 5, -7}}, {{-12,
2, -1}, {-11, 1, 3}, {-9, 8, -4}}, {{-12, 2, -1}, {-11, 1,
3}, {-7, -6, -2}}, {{-12, 2, -1}, {-11, 1,
3}, {-7, -2, -8}}, {{-12, 2, -1}, {-11, 1, 3}, {-7, -2, 6}}, {{-12,
2, -1}, {-11, 1, 3}, {-7, 6, -8}}, {{-12, 2, -1}, {-11, 1,
3}, {-7, 9, -5}}}


How can I write all the equations of the plane passing through three points of this list. And, If the equation has the form $a x + b y + c z + d =1$, I want the form of equation has always $GCD[a,b,c,d]=1$.

• Please include the code you have tried in order to accomplish your goals. Jan 13, 2017 at 5:50

You can visualize planes using InfinitePlane.

You can define your own function, e.g. using pts as the points provided

plane[p_, q_, r_] :=
With[{u = q - p, v = r - p},
FullSimplify[Expand[Cross[u, v].({x, y, z} - p)] == 0]]


You can, for example, then visualize the planes and label with equation:

eqs = plane @@@ pts;
grd = Grid[
Partition[

• How can I get the equation of all planes in the form $a x + b y + c z + d=0$? Please see the last equation, have not my form. Jan 13, 2017 at 6:58