# Geometric interpretations of midpoint algorithm, homogeneous linear least squares and nonlinear least squares method in 3D reconstruction?

In "Multiple View Geometry in Computer Vision" Chapter 12. Structure Computation, page 310-313, triangulation is used for point 3D reconstruction. There are three methods mentioned:

1. Midpoint method that "finds the midpoint of the common perpendicular to the two rays in space".

2. Linear triangulation methods that uses SVD to solve homogeneous equation Ax = 0. In this method algebraic error is minimized.

3. Nonlinear triangulation method that uses iterative optimization algorithms e.g. LM to solve a nonlinear least square equation Ax = b. The "gold standard" reprojection error is minimized.

My question is:

1. What are the geometric interpretations of the three methods?

2. What are the differences between the three triangulation methods above?

I did some research and found some useful information:

This post explains that algebraic error assumes noise comes from 3D points, while reprojection error assumes noise comes from 2D image

These slides introduce three methods but did not provide explicit geometric interpretations.