Minimum Distance Between Two Parallel Planes Given Points in Space

Question

I'm given a number of points (in this case 7) and need to calculate the minimum distance between two parallel planes which "sandwich" or encompass all of the given points. The goal is to calculate flatness of the specimen.

I'm not even sure how to approach this...

Surface flatness example

Answer 1

Hint: Compute the Hessian Normalform of the plane,given by $$\frac{ax+by+cz+d}{\pm\sqrt{a^2+b^2+c^2}}=0$$

Answer 2

You are seeking the width of the convex hull of your point set. Compute the convex hull $H$. Then the width is determined either by a vertex and a face of $H$ (i.e., one plane determined by a face, the other antipodal and parallel, touching at a vertex), or by two skew edges. Since you only have $n=7$ points, you can just check all {point,face} and {edge,edge} pairs: $O(n^2)$ complexity.

Added to illustrate the need for edge-edge width support.

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Answer 3

You can solve this problem using a linear programming approach

Given the points $p_k$ and a vector $\vec n = (a,b,c)$ we need to solve the linear problem

$$ \min_{a,b,c,d} d\ \ \text{s. t. }\ \ -d \le p_k\cdot \vec n \le d,\ \ \text{for}\ \ k=1,2,\cdots,n $$