Geometric characterization of planar regression

The geometric characterization of linear regression in terms of the ‘concentration ellipse’ by Galton Galton, F., 1886, Family likeness in stature (with Appendix by Dickson, J.D.H.). Proceedings of the Royal Society of London, 40, 42–73.] and Pearson Pearson, K., 1901, On lines and planes of closest fit to systems of points in space. Philosophical Magazine, 2, 559–572.] was extended to the case of unequal variances of the presumably uncorrelated errors in the experimental data McCartin, B.J., 2003, A geometric characterization of linear regression. Statistics, 37(2), 101–117.]. In this paper, this geometric characterization is further extended to planar (and also linear) regression in three dimensions where a beautiful interpretation in terms of the concentration ellipsoid is developed.