Search

The constrained orthogonal Procrustes problem is the least-squares problem that calls for a rotation matrix that optimally aligns two matrices of the same order. Over past decades, the algorithm of choice for solving this problem has been the Kabsch-Umeyama algorithm, which is effectively no more than the computation of the singular value decomposition of a particular matrix. Its justification, as presented separately by Kabsch and Umeyama, is not totally algebraic since it is based on solving the minimization problem via Lagrange multipliers. In order to provide a more transparent alternative, it is the main purpose of this paper to present a purely algebraic justification of the algorithm through the exclusive use of simple concepts from linear algebra. For the sake of completeness, a proof is also included of the well known and widely used fact that the orientation-preserving rigid motion problem, i.e., the least-squares problem that calls for an orientation-preserving rigid motion that optimally aligns two corresponding sets of points in d -dimensional Euclidean space, reduces to the constrained orthogonal Procrustes problem.

A scheme is presented and software is documented for representing as integers input decimal numbers that have been stored in a computer as double precision floating point numbers and for carrying out multiplications, additions and subtractions based on these numbers in an exact manner. The input decimal numbers must not have more than nine digits to the left of the decimal point. The decimal fractions of their floating point representations are all first rounded off at a prespecified location, a location no more than nine digits away from the decimal point. The number of digits to the left of the decimal point for each input number besides not being allowed to exceed nine must then be such that the total number of digits from the leftmost digit of the number to the location where round-off is to occur does not exceed fourteen.

A numerical procedure is described in which the aerodynamic drag and torque in low Earth orbit are calculated for a prototype Space Shuttle external tank and its components, the "LO2" and "LH2" tanks, carrying liquid oxygen and hydrogen, respectively, for any given angle of attack. Calculations assume the hypersonic limit of free molecular flow theory. Each shell of revolution is assumed to be described by a series of parametric equations for their respective contours. It is discretized into circular cross sections perpendicular to the axis of revolution, which yield a series of ellipses when projected according to the given angle of attack. The drag profile, that is, the projection of the entire shell is approximated by the convex envelope of those ellipses. The area of the drag profile, that is, the drag area, and its center of area moment, that is, the drag center, are then calculated and permit determination of the drag vector and the eccentricity vector from the center of gravity of the shell to the drag center. The aerodynamic torque is obtained as the cross product of those vectors. The tanks are assumed to be either evacuated or pressurized with a uniform internal gas distribution: dynamic shifting of the tank center of mass due to residual propellant sloshing is not considered.

Lavery splines are examined in the univariate and bivariate cases. In both instances relaxation based algorithms for approximate calculation of Lavery splines are proposed. Following previous work Gilsinn, et al. [7] addressing the bivariate case, a rotationally invariant functional is assumed. The version of bivariate splines proposed in this paper also aims at irregularly spaced data and uses Hseih-Clough-Tocher elements based on the triangulated irregular network (TIN) concept. In this paper, the univariate case, however, is investigated in greater detail so as to further the understanding of the bivariate case.