Linear Regression in 3D
by Robert Szlizs, FixPoint

Describing the Problem

My friends visited me some time ago and described following problem: They have bought a precision mechanics measuring device they can measure coordinates of arbirarily selected points on a subject's surface with. The task is to fit a tube (with radius r and unlimited length) to this set of 3D points the way the sum of points' distances from the tube will be minimal. In other words: we have to find a line (we can call it regression line in the 3D space) thats' distance from these set of measured point is minimal.

A cloud of measured 3D-points that defines a tube.

The immediate idea is to use least square variational method to solve the problem. However, classical least square methods say: the set of measured points should be approximated by the closest hyperplane. So for 3d-points the least square method finds a 2D-plane that's distance from the set of points is minimal. But here we have to find an axis, a line located in the 3D-space that can not be described other way than a section of two 2D-planes or in a parametric form. We will use the second one.