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NESC9588 3DGEOELE.

3DGEOELE, 3-D Nonlinear Least Square Fit

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1. NAME OR DESIGNATION OF PROGRAM:  3DGEOELE.
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2. COMPUTERS
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Program name Package id Status Status date
3DGEOELE NESC9588/01 Tested 09-MAY-1989

Machines used:

Package ID Orig. computer Test computer
NESC9588/01 DEC PDP-11 DEC VAX 8810
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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.
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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.
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5. RESTRICTIONS ON THE COMPLEXITY OF THE PROBLEM:  Maximum of -, 30 iterations
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6. TYPICAL RUNNING TIME:
NESC9588/01
NEA-DB ran the test cases included in this package on a VAX 8810 computer in less than 8 second sof CPU time.
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7. UNUSUAL FEATURES OF THE PROGRAM:
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8. RELATED AND AUXILIARY PROGRAMS:
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9. STATUS
Package ID Status date Status
NESC9588/01 09-MAY-1989 Tested at NEADB
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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.
NESC9588/01, included references:
- S.R. Drake:
  Computer Programs for Nonlinear Least Squares Analysis of Various
  Three-Dimensional Geometrical Elements
  BDX-613-3741  (April 1987).
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11. MACHINE REQUIREMENTS:
NESC9588/01
The different modules run on a VAX 8810 computer in 22K bytes (43 pages) of main storage.
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12. PROGRAMMING LANGUAGE(S) USED
Package ID Computer language
NESC9588/01 FORTRAN-IV
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13. OPERATING SYSTEM UNDER WHICH PROGRAM IS EXECUTED:  RSX-11M.
NESC9588/01
V5.0-1 (VAX 8810).
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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.
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15. NAME AND ESTABLISHMENT OF AUTHORS

          S.R. Drake
          Bendix Kansas City Division
          Allied Corporation
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16. MATERIAL AVAILABLE
NESC9588/01
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
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17. CATEGORIES
  • P. General Mathematical and Computing System Routines

Keywords: computer-aided design, coordinates, least square fit, nonlinear problems, x-y-z.