NESC9588 3DGEOELE.

1. NAME OR DESIGNATION OF PROGRAM:  3DGEOELE.

3. DESCRIPTION OF PROGRAM OR FUNCTION

3DGEOELE is a collection of six programs - LINE, PLANE, CYL, CONE, RING, and TOROID - for nonlinear  least squares fitting to a line, plane, circular cylinder, circular  cone, ring, and toroid, respectively, in three-dimensional space of  a set of measured x, y, z coordinates. The measurements are made by  multi-axis coordinate measuring machines. The analysis is based on the residuals being calculated in a direction perpendicular to the surface or line. All six types involve nonlinear relationships, but  in the case of the straight line or plane, nonlinear equations are avoided by using linear equations iteratively in repeated two- dimensional fits. In the case of the cylinder, cone, ring, and toroid, the equation of the geometrical shape (element) in the general position is found by taking the standard equation of the element relative to a coordinate system where the principal axis of  the element is the z axis of that coordinate system and then by means of transformation equations, converting the equation to be relative to the original coordinate system. These transformation equations are based on three shift constants (in the x, y, z direction) and on two angles of rotation (first counter-clockwise about x, and then counter-clockwise about y). There are two modes of execution: experimental and production. The experimental mode is for checking each program to ensure proper operation and sufficient accuracy in the final results. The production mode is used with measured data where the answers are unknown in advance.

4. METHOD OF SOLUTION

The least squares solution used minimizes the sum of the squares of the perpendicular deviations or residuals from the fitted line to the data points. The nonlinear normal equations from the least squares analysis are solved using Brown's iteration method, which is Newton-like and based on Gaussian elimination but converges faster than Newton's method. If the residuals are in the realm of hundreds of microinches or less, convergence will occur in  less than 20 iterations to 12 significant figures.

5. RESTRICTIONS ON THE COMPLEXITY OF THE PROBLEM:  Maximum of -, 30 iterations

6. TYPICAL RUNNING TIME:

NEA-DB ran the test cases included in this package on a VAX 8810 computer in less than 8 second sof CPU time.

7. UNUSUAL FEATURES OF THE PROGRAM:

8. RELATED AND AUXILIARY PROGRAMS:

10. REFERENCES

- 3DGEOELE, NESC No. 9588 Tape Description,
  National Energy Software Center Note 87-69, May 21, 1987.
- K.M. Brown
  A Quadratically Convergent Newton-like Method Based Upon Gaussian
  Elimination,
  SIAM Journal of Numerical Analysis, Vol. 6, No. 4, pp. 560-569,
  December 1969.

- S.R. Drake:
  Computer Programs for Nonlinear Least Squares Analysis of Various
  Three-Dimensional Geometrical Elements
  BDX-613-3741  (April 1987).

11. MACHINE REQUIREMENTS:

The different modules run on a VAX 8810 computer in 22K bytes (43 pages) of main storage.

13. OPERATING SYSTEM UNDER WHICH PROGRAM IS EXECUTED:  RSX-11M.

V5.0-1 (VAX 8810).

14. OTHER PROGRAMMING OR OPERATING INFORMATION OR RESTRICTIONS

The measured coordinates must be available in a stored data file prior to execution; all additional information required is supplied manually by the user and is menu driven.

15. NAME AND ESTABLISHMENT OF AUTHORS

          S.R. Drake
          Bendix Kansas City Division
          Allied Corporation

File name	File description	Records
NESC9588_01.001	Information file	148
NESC9588_01.002	FORTRAN-77 source program of LINE	388
NESC9588_01.003	FORTRAN-77 source program of PLANE	396
NESC9588_01.004	FORTRAN-77 source program of CYL	693
NESC9588_01.005	FORTRAN-77 source program of CONE	722
NESC9588_01.006	FORTRAN-77 source program of RING	695
NESC9588_01.007	FORTRAN-77 source program of TOROID	746
NESC9588_01.008	Sample data of experimental mode of LINE	5
NESC9588_01.009	Sample data of experimental mode of PLANE	5
NESC9588_01.010	Sample data of experimental mode of CYL	9
NESC9588_01.011	Sample data of experimental mode of CONE	9
NESC9588_01.012	Sample data of experimental mode of RING	5
NESC9588_01.013	Sample data of experimental mode of TOROID	17
NESC9588_01.014	Sample data of production mode of LINE	7
NESC9588_01.015	Sample data of production mode of PLANE	8
NESC9588_01.016	Sample data of production mode of CYL	9
NESC9588_01.017	Sample data of production mode of CONE	17
NESC9588_01.018	Sample data of production mode of RING	17
NESC9588_01.019	Sample data of production mode of TOROID	33
NESC9588_01.020	Output of experimental mode of LINE	75
NESC9588_01.021	Output of experimental mode of PLANE	59
NESC9588_01.022	Output of experimental mode of CYL	99
NESC9588_01.023	Output of experimental mode of CONE	97
NESC9588_01.024	Output of experimental mode of RING	90
NESC9588_01.025	Output of experimental mode of TOROID	141
NESC9588_01.026	Output of production mode of LINE	32
NESC9588_01.027	Output of production mode of PLANE	34
NESC9588_01.028	Output of production mode of CYL	54
NESC9588_01.029	Output of production mode of CONE	67
NESC9588_01.030	Output of production mode of RING	65
NESC9588_01.031	Output of production mode of TOROID	100
NESC9588_01.032	Original PDP-11 source, not tested, LINE	366
NESC9588_01.033	Original PDP-11 source, not tested, PLANE	374
NESC9588_01.034	Original PDP-11 source, not tested, CYL	673
NESC9588_01.035	Original PDP-11 source, not tested, CONE	702
NESC9588_01.036	Original PDP-11 source, not tested, RING	675
NESC9588_01.037	Original PDP-11 source, not tested, TOROID	720