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Research Papers

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

[+] Author and Article Information
Craig M. Shakarji

Mem. ASME
Physical Measurement Laboratory,
National Institute of Standards and Technology,
Gaithersburg, MD 20899
e-mail: craig.shakarji@nist.gov

Vijay Srinivasan

Fellow ASME
Engineering Laboratory,
National Institute of Standards and Technology,
Gaithersburg, MD 20899
e-mail: vijay.srinivasan@nist.gov

Contributed by the Computers and Information Division of ASME for publication in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscript received March 15, 2018; final manuscript received August 15, 2018; published online October 18, 2018. Assoc. Editor: Kristina Wärmefjord. This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Approved for public release; distribution is unlimited.

J. Comput. Inf. Sci. Eng 19(1), 011002 (Oct 18, 2018) (12 pages) Paper No: JCISE-18-1064; doi: 10.1115/1.4041226 History: Received March 15, 2018; Revised August 15, 2018

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.

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References

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Shakarji, C. M. , and Srinivasan, V. , 2017, “ Convexity and Optimality Conditions for Constrained Least-Squares Fitting of Planes and Parallel Planes to Establish Datums,” (accepted manuscript).

Figures

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Fig. 1

A simple example of (a) specification during design of a part and (b) establishment of a system of primary and secondary datum planes on a manufactured instance of the part. The secondary datum plane B is required to be perpendicular to the primary datum plane A.

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Fig. 2

Specification of a datum plane for (a) a slab (external width) and (b) a slot (internal width). In each case, the datum is the median plane (indicated by dashes and dots) of two parallel planes that are fitted to two sets of points on two planar surface features (indicated by extension lines) on a manufactured instance of the part.

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Fig. 3

Notations for (a) fitting a plane, and (b) fitting a line

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Fig. 4

Notations for fitting parallel lines

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Fig. 5

The outer convex hull (a) to a set of 2D points, and (b) showing supporting lines. The material is understood to be above the outer convex hull.

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Fig. 6

Gauss map obtained by mapping each unit normal of a supporting plane of an outer convex hull in space in (a) to a point on a unit sphere in (b). A similar Gauss map (shown as the thick arc of the circle) is obtained by mapping each unit normal of a supporting line of an outer convex hull in a plane in (c) to a point on a unit circle in (d).

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Fig. 7

Geometrical illustration of the linear mapping M of (a) a unit circle to (b) an ellipse using the SVD of M

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Fig. 8

Illustration of (a) an outer convex hull and (b) a composite ellipse using polar plots over a Gauss map

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Fig. 9

Illustration of datum for (a) a wedge, where the material is interior to the acute angle shown, and (b) an angular slot, where the material is external to the acute angle shown. The datum is a system consisting of the median line indicated with dashes and dots, and a point indicated with a filled circle on the median line.

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Fig. 10

(a) An example of upper and lower convex hulls and (b) the unit circle and the corresponding composite ellipse consisting of two connected components

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Fig. 11

Illustrations of (a) a polar plot, (b) a Cartesian plot of the objective function as being convex, and (c) a nonconvex objective function

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Fig. 12

Illustration for the interpretation of Eq. (11)

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Fig. 13

Illustration of three cases of transition and the related objective functions. The perpendiculars h from the centroid g to the supporting lines are shown dotted in the top row. The transition neighborhoods are shown within dotted circles in the h2-θ plots in the bottom row.

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Fig. 14

Illustration of local convexity of a function within an interval (x1, x2). The line segment joining a and b (shown dotted) lies entirely above f(x).

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Fig. 15

(a) An example of a composite ellipse in 2D and (b) plot of the objective function ru2+rl2 as a function of just one angle θ

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