# 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. – bbgodfrey Jan 13 '17 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[
Show[ContourPlot3D[#, {x, -20, 20}, {y, -20, 20}, {z, -20, 20},
Mesh -> None],
Graphics3D[{Red, PointSize[0.04], Point[#2]}], PlotLabel -> #1]
&, {eqs, pts}], 5], Frame -> All];


• 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. – minhthien_2016 Jan 13 '17 at 6:58
• @toandhsp I suggest you try for yourself. I have found this the best way to learn. If you get in trouble post a question with what you tried, what you want, where it failed. If you look at function 'plane' you can isolate LHS you desire. The most reduced form is another matter. Good luck:) – ubpdqn Jan 13 '17 at 7:03

With list is sequence of three points. I tried:

Table[{pA, pB, pC} = points;
u = pB - pA;
v = pC - pA;
w = Cross[u, v];
k = GCD[w[[1]], w[[2]], w[[3]]];
m = {x, y, z};
If[w[[1]] != 0, n = Sign[w[[1]]] w/k,
If[w[[2]] != 0, n = Sign[w[[2]]] w/k,
If[w[[3]] != 0, n = Sign[w[[3]]] w/k]]];
{pA, pB, pC, TraditionalForm[Expand[n.(m - pA)]] == 0}, {points,
list}]