FIG. 13. Numerical Example 2
FIG. 34. Numerical Example 12
|
Abstract: Rapidly
convergent algorithms for fitting circles, parabolas, and clothoids to
measured points are developed and tested. A solution of the line fitting
problem is also presented for a complete treatment of the curves encountered
in civil engineering route design. The second order, reduced Hessian method,
broadly applicable to the class of scalable, C2
parametrizations, is orthogonal distance regression with four-parameter
similarity transformations. The local parameters, or state variables, are
implicitly eliminated, and second order solutions are rigorously computed
in the model parameter space (rank < 4). The algorithms are further
distinguished from earlier works by the inclusion of approximation procedures
that yield very good starting values. Additionally, a strong connection
between the Helmert transformation and the total least squares problem
is established, and a fixed point method is suggested.
The dissertation contains 5 tables, 34 figures, and 161 equations. There are 12 numerical examples and 69 distinct citations. FORTRAN programs comprise five appendices. Repeated here are the results of two examples that use the same data: example 2, a circle fit, and example 12, a clothoid fit. |
| x | 1.0 | 2.0 | 3.0 | 5.0 | 7.0 | 9.0 |
| y | 7.0 | 6.0 | 7.0 | 8.0 | 7.0 | 5.0 |
FIG. 13. Numerical Example 2
FIG. 34. Numerical Example 12
| Number of
iterations |
Scale
factor |
Origin | Angle | Objective
function |
|
| k | a | x0 | y0 | θ | F |
| (a) Numerical example 2 | |||||
| 0 | 5.103229 | 4.621360 | 2.491403 | -- | 0.630518 |
| 4 | 4.714226 | 4.739782 | 2.983533 | -- | 0.613800 |
| (b) Numerical example 12 | |||||
| 0 | 4.596706 | 4.629824 | 6.764230 | 0.029989 | 1.681458 |
| 10 | 3.727323 | 3.104647 | 7.012556 | 0.272679 | 0.553825 |