Optimality Conditions for Constrained Least-Squares Fitting of Circles, Cylinders, and Spheres to Establish Datums

2018 ◽  
Vol 18 (3) ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Optimality Conditions ◽  
Necessary Conditions ◽  
Three Dimensional ◽  
Necessary Condition ◽  
Constrained Least Squares ◽  
Least Squares Fitting ◽  
Candidate Solution ◽  
Cylinders And Spheres ◽  
New Algorithms

This paper addresses the combinatorial characterizations of the optimality conditions for constrained least-squares fitting of circles, cylinders, and spheres to a set of input points. It is shown that the necessary condition for optimization requires contacting at least two input points. It is also shown that there exist cases where the optimal condition is achieved while contacting only two input points. These problems arise in digital manufacturing, where one is confronted with the task of processing a (potentially large) number of points with three-dimensional coordinates to establish datums on manufactured parts. The optimality conditions reported in this paper provide the necessary conditions to verify if a candidate solution is feasible, and to design new algorithms to compute globally optimal solutions.

Optimality Conditions for Constrained Least-Squares Fitting of Circles, Cylinders, and Spheres to Establish Datums

2017 ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Optimality Conditions ◽  
Necessary Conditions ◽  
Three Dimensional ◽  
Necessary Condition ◽  
Constrained Least Squares ◽  
Least Squares Fitting ◽  
Candidate Solution ◽  
Cylinders And Spheres ◽  
New Algorithms

This paper addresses the combinatorial characterizations of the optimality conditions for constrained least-squares fitting of circles, cylinders, and spheres to a set of input points. It is shown that the necessary condition for optimization requires contacting at least two input points. It is also shown that there exist cases where the optimal condition is achieved while contacting only two input points. These problems arise in digital manufacturing, where one is confronted with the task of processing a (potentially large) number of points with three-dimensional coordinates to establish datums on manufactured parts. The optimality conditions reported in this paper provide the necessary conditions to verify if a candidate solution is feasible, and to design new algorithms to compute globally optimal solutions.

Convexity and Optimality Conditions for Constrained Least-Squares Fitting of Planes and Parallel Planes to Establish Datums

2018 ◽  
Vol 19 (1) ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Optimality Conditions ◽  
Global Optimum ◽  
Objective Functions ◽  
Constrained Least Squares ◽  
Least Squares Fitting ◽  
Optimum Solution ◽  
Set Of Points ◽  
Theoretical Issues ◽  
Global Optimum Solution

This paper addresses some important theoretical issues for constrained least-squares fitting of planes and parallel planes to a set of points. In particular, it addresses the convexity of the objective function and the combinatorial characterizations of the optimality conditions. These problems arise in establishing planar datums and systems of planar datums in digital manufacturing. It is shown that even when the set of points (i.e., the input points) are in general position, (1) a primary planar datum can contact 1, 2, or 3 input points, (2) a secondary planar datum can contact 1 or 2 input points, and (3) two parallel planes can each contact 1, 2, or 3 input points, but there are some constraints to these combinatorial counts. In addition, it is shown that the objective functions are convex over the domains of interest. The optimality conditions and convexity of objective functions proved in this paper will enable one to verify whether a given solution is a feasible solution, and to design efficient algorithms to find the global optimum solution.

Convexity and Optimality Conditions for Constrained Least-Squares Fitting of Planes and Parallel Planes to Establish Datums

2017 ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Optimality Conditions ◽  
Feasible Solution ◽  
Global Optimum ◽  
Objective Functions ◽  
Constrained Least Squares ◽  
Least Squares Fitting ◽  
Optimum Solution ◽  
Theoretical Issues ◽  
Global Optimum Solution

This paper addresses some important theoretical issues for constrained least-squares fitting of planes and parallel planes to a set of input points. In particular, it addresses the convexity of the objective function and the combinatorial characterizations of the optimality conditions. These problems arise in establishing planar datums and systems of planar datums in digital manufacturing. It is shown that even when the input points are in general position: (1) a primary planar datum can contact 1, 2, or 3 input points, (2) a secondary planar datum can contact 1 or 2 input points, and (3) two parallel planes can each contact 1, 2, or 3 input points, but there are some constraints to these combinatorial counts. In addition, it is shown that the objective functions are convex over the domains of interest. The optimality conditions and convexity of objective functions proved in this paper will enable one to verify whether a given solution is a feasible solution, and to design efficient algorithms to find the global optimum solution.

scholarly journals Three-dimensional GPS ionospheric tomography over Japan using constrained least squares

2014 ◽  
Vol 119 (4) ◽  
pp. 3044-3052 ◽  
Author(s):  
Gopi K. Seemala ◽  
Mamoru Yamamoto ◽  
Akinori Saito ◽  
Chia-Hung Chen
Keyword(s):  
Least Squares ◽  
Three Dimensional ◽  
Constrained Least Squares ◽  
Ionospheric Tomography

scholarly journals Optimal Control and Necessary Optimality Conditions for Nonlinear and Perturbed Dynamic Problems

2018 ◽  
Vol 10 (6) ◽  
pp. 63
Author(s):  
Gossan D. Pascal Gershom ◽  
Bailly Balè ◽  
Yoro Gozo
Keyword(s):  
Optimal Control ◽  
Optimality Conditions ◽  
Variational Formulation ◽  
Necessary Conditions ◽  
Three Dimensional ◽  
Necessary Optimality Conditions ◽  
Cancer Cell Proliferation ◽  
Dynamic Problems ◽  
First Order ◽  
Tumor Growth Model

The main goal of this paper is to establish the first order necessary optimality conditions for a tumor growth model that evolves due to cancer cell proliferation. The phenomenon is modeled by a system of three-dimensional partial differential equations. We prove the existence and uniqueness of optimal control and necessary conditions of optimality are established by using the variational formulation.

Computational Investigations for a New, Constrained Least-Squares Datum Definition for Circles, Cylinders, and Spheres

2016 ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Stable Solution ◽  
Computational Investigation ◽  
Constrained Least Squares ◽  
Least Squares Solution ◽  
Full Contact ◽  
Least Squares Fit ◽  
Engineering Drawings ◽  
Cylinders And Spheres

For engineering drawings and CAD definitions, the problem of a suitable datum definition for datum features of circles, spheres, and cylinders has been sought by standards writers over decades. The maximum-inscribed and minimum-circumscribed definitions that have often been used have known problems relating to stability in many common, industrial cases. Examples of these problem cases include cylindrical datum features having an hourglass shape, barrel shape, or the shape of a tapered shaft and circular or spherical datum features that are dimpled. For this cause, many resort to using a least-squares fit whose diameter is scaled to be just inside (or just outside) the datum feature. However, we show this shifted least-squares solution has its own drawbacks. This paper investigates a new datum definition based on a constrained least-squares criterion. The use of this definition for datum planes has already elegantly solved the problem of providing a full contact solution when that solution is stable, while providing a balanced, stable solution in the case of rocker conditions. With that success as motivation, we now investigate using this definition for circles, spheres, and cylinders. We demonstrate that the constrained least-squares is an excellent choice for several known problematic cases. This datum definition maintains stability in cases where the maximum-inscribed fits are not unique and thus not stable. Yet they also maintain close adherence to the maximum-inscribed solution when such solutions are stable. We also show that the constrained least-squares solution has clear benefits over the shifted least squares solution. This is the first computational investigation into the behavior of the constrained least-squares as a possible datum definition for these features. While not being fully comprehensive, these initial findings indicate that the constrained least-squares appears to be a safe and advantageous datum definition choice and provide substantial optimism that results in future investigated cases will be pleasing as well.

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