To fit a plane or a line to a set of points by least squares

Abstract

EPITAXIAL RELATIONSHIPS OF CUPROIYS OXIDEatomic scale (111) facets which make up the structure of the (012) plane. The electron microscope pictures, in addition to confirming the orientations determined by electron diffraction, gave specific data on the shapes of the oxide growth and the particular faces present. 0nly the three most densely packed faces occurred on the oxide.The authors would like to thank Dr A. T. Gwathmey and Dr N. Cabrera for many helpful discussions. This work was sponsored by the Office of Naval Research.References BRADLEY, D. E. (1954). J. Inst. Met. 83, 35. DONY-HENAVLT, O. (1910). Bull. Soc. Chem. Belg. The fitting of a plane or a line to a set of points by least squares is discussed, and a convenient numerical method is given.In the description of a crystal structure, it is sometimes desired to fit a least-squares plane to the positions found for some approximately coplanar set of atoms.Because it seems that an incorrect method is often used for doing this, we would like to discuss the problem and recommend an alternative method that is both correct in principle and convenient in computation. It becomes evident that the problem of the plane is essentially equivalent to the problem of finding the principal plane of least inertia for a set of point masses and that the problem of the best line is similarly equivalent to the very closely related problem of the least axis of inertia. The discussion therefore naturally covers line as well as plane and essentially recapitulates parts of a classical mechanical theory in deriving what is special to the present application. We first formulate the problem of the plane and present the recommended alternative method of solution, including a detailed numerical example, then discuss the prevalent incorrect method as well as various special cases, and finally consider the problem of the line and give a convenient method for handling it. * Contribution No. 2287 from the Gates and Crellin Laboratories.We find it convenient to use both ordinary vector notation, as in equations (1), (2), and (3), and matrix notation, as in equation (11), sometimes side by side. We also use two summation conventions: the Gaussian bracket [ ], to express summation over a set of points (cf., e.g., Whittaker & Robinson, 1940), and the convention of dropping the operator Z whenever it applies to repeated alphabetic indices. Definitions we often express as identities. The least-squares planeWhat is desired is to find the plane that minimizes , the weighted sum of squares of distances D~ of points k from the plane sought. These points are defined by the vectors r -xlal +x2a2+xSas -= x'a,.(The plane is defined by its unit normal and by the origin-to-plane distance d, whereupon the distance from the plane to a point is

Comparison of the dimensions and conformation of the sulfaguanidine moiety in sulfaguanidine monohydrate and trans-dichlorobis(sulfaguanidine)palladium(II)

“…The Pd(SG)2CI2 molecule ( Fig. 3) can be described in terms of a number of planar or nearly planar groups of atoms (Table 9; planes and interplanar angles calculated by the methods of Schomaker, Waser, Marsh & Bergman (1959) and Waser, Marsh & Cordes (1973)]. …”

The crystal structure of ninhydrin

“…1 togethel with deviations from a least-squares plane passed through the six carbon atoms of the benzene ring. The method of determining the plane is that of Schomaker, Waser, Marsh & Bergman (1959) and the equation of the plane with respect to the a, b, and c axes is: 5.616x+ 3.787y + 3.006z = 4.179 ~.…”

Separation of close isomers by enclathration

“…A large peak at 28 = 11.6° is observed in both the host trace and the guest trace. However, the [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] intensity of this peak is very low in the XRD trace of the complex. The peak at 29 = 11.6° is due to the scattering from the (002) plane.…”