Explicit solution of the overdetermined three-dimensionalresection problem
Several procedures for solving in a closed form the three-dimensional resection problem have already been presented. In the present contribution, the over determined three-dimensional resection problem is solved in a closed form in two steps. In step one a combinatorial minimal subset of observation...
| Main Authors: | , |
|---|---|
| Format: | Journal Article |
| Published: |
Springer - Verlag
2003
|
| Subjects: | |
| Online Access: | http://hdl.handle.net/20.500.11937/12246 |
| _version_ | 1848748024462835712 |
|---|---|
| author | Awange, Joseph Grafarend, E. |
| author_facet | Awange, Joseph Grafarend, E. |
| author_sort | Awange, Joseph |
| building | Curtin Institutional Repository |
| collection | Online Access |
| description | Several procedures for solving in a closed form the three-dimensional resection problem have already been presented. In the present contribution, the over determined three-dimensional resection problem is solved in a closed form in two steps. In step one a combinatorial minimal subset of observations is constructed which is rigorously converted into station coordinates by means of the Groebner basis algorithm or the multipolynomial resultant algorithm. The combinatorial solution points in a polyhedron are then reduced to their barycentric in step two by means of their weighted mean. Such a weighted mean of the polyhedron points in R3 is generated via the Error Propagation law/variance-covariance propagation. The Fast Nonlinear Adjustment Algorithm was proposed by C.F. Gauss, whose work was published posthumously, and C.G.I. Jacobi. The algorithm, here referred to as theGauss-Jacobi Combinatorial algorithm, solves the over determined three-dimensional resection problem in a closed form without reverting to iterative or linearization procedures. Compared to the actual values, the obtained results are more accurate than those obtained from the closed-form solution of a minimano of three known stations. |
| first_indexed | 2025-11-14T06:58:28Z |
| format | Journal Article |
| id | curtin-20.500.11937-12246 |
| institution | Curtin University Malaysia |
| institution_category | Local University |
| last_indexed | 2025-11-14T06:58:28Z |
| publishDate | 2003 |
| publisher | Springer - Verlag |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | curtin-20.500.11937-122462017-09-13T16:04:16Z Explicit solution of the overdetermined three-dimensionalresection problem Awange, Joseph Grafarend, E. Gauss?Jacobi combinatorial algorithm Groebner basis Multipolynomial - resultants - Overdetermined three-dimensional resection Several procedures for solving in a closed form the three-dimensional resection problem have already been presented. In the present contribution, the over determined three-dimensional resection problem is solved in a closed form in two steps. In step one a combinatorial minimal subset of observations is constructed which is rigorously converted into station coordinates by means of the Groebner basis algorithm or the multipolynomial resultant algorithm. The combinatorial solution points in a polyhedron are then reduced to their barycentric in step two by means of their weighted mean. Such a weighted mean of the polyhedron points in R3 is generated via the Error Propagation law/variance-covariance propagation. The Fast Nonlinear Adjustment Algorithm was proposed by C.F. Gauss, whose work was published posthumously, and C.G.I. Jacobi. The algorithm, here referred to as theGauss-Jacobi Combinatorial algorithm, solves the over determined three-dimensional resection problem in a closed form without reverting to iterative or linearization procedures. Compared to the actual values, the obtained results are more accurate than those obtained from the closed-form solution of a minimano of three known stations. 2003 Journal Article http://hdl.handle.net/20.500.11937/12246 10.1007/s00190-002-0287-0 Springer - Verlag restricted |
| spellingShingle | Gauss?Jacobi combinatorial algorithm Groebner basis Multipolynomial - resultants - Overdetermined three-dimensional resection Awange, Joseph Grafarend, E. Explicit solution of the overdetermined three-dimensionalresection problem |
| title | Explicit solution of the overdetermined three-dimensionalresection problem |
| title_full | Explicit solution of the overdetermined three-dimensionalresection problem |
| title_fullStr | Explicit solution of the overdetermined three-dimensionalresection problem |
| title_full_unstemmed | Explicit solution of the overdetermined three-dimensionalresection problem |
| title_short | Explicit solution of the overdetermined three-dimensionalresection problem |
| title_sort | explicit solution of the overdetermined three-dimensionalresection problem |
| topic | Gauss?Jacobi combinatorial algorithm Groebner basis Multipolynomial - resultants - Overdetermined three-dimensional resection |
| url | http://hdl.handle.net/20.500.11937/12246 |