How to find plane equation from 8 3D points with least square method

Question

I have been working on school projects to find a plane equation from 8 3D points. Normally from 3 points, we can create a plane equation but when we have a lot of points, we want to find a good fitting plane for it by using Least Square Method but I’m getting stuck with the procedure to find it. Do we have any ways to find it? Thank you

Answer 1

Assume that the equation of the best fitting plane for $n$ points $(x_1,y_1,z_1),(x_2,y_2,z_2),\ldots,(x_n,y_n,z_n)\in\mathbb{R}^3$ can be expressed as $$z=\alpha x + \beta y + \gamma$$ for some coefficients $\alpha,\beta,\gamma\in\mathbb{R}$.

To formulate this as a least squares problem let $A\in M_{n \times 3}(\mathbb{R})$ and $b\in\mathbb{R}^n$ be defined as: $$ A = \begin{pmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ \vdots & \vdots & \vdots \\ x_n & y_n & 1 \\ \end{pmatrix} \quad b = \begin{pmatrix} z_1 \\ z_2 \\ \vdots \\ z_n \\ \end{pmatrix} $$ Now your optimization goal is to find $\min_{x\in\mathbb{R}^3} \left\Vert Ax-b \right\Vert^2$. The components of the result vector $x$ will be the coefficents of the plane equation stated above i.e. $x=(\alpha,\beta,\gamma)$.