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THE MATHEMATICAL REPRESENTION OF GEOMETRIC FORM

Objective



The project consists in developing a mathematical theory which would enable the advantages and disadvantages of alternative mathematical representations for a specified geometry to be compared for any given orientation of the geometric form in the coordinate system of the measuring machine. This must inevitably relate uncertainty in the constants in the mathematical representation of any geometric form to the current position and orientation of the form in the coordinate system. In addition to leading to direct improvements in the uncertainties associated with data fits, it should be possible to give practical advice for the orientation of workpieces on 3 cmm when setting out to measure specific geometries. The following geometries have been considered in all orientations in an x-y-z coordinate system:
straight line (2d), straight line (3d), flat plane, circle (2d)
circle (3d), sphere, cylinder, cone

RESULTS

The theory developed describes how the above geometric elements can reliably be represented in terms of sets of algebraic parameters. For any element, it enables to determine whether a possible parameterisation is good or not and gives a quantitative measure for the degree of goodness.

Two sets of parametrisations are recommended. The first set applies to full elements, ie elements that are to be derived from measurements distributed throughout the feature. The second applies to partial elements derived from measurements over a relatively small portion of a feature. The parametrisation recommended is optimal or near-optimal in terms of the above measure.
Considerations have been given to the relationship between recommended parametrisation and algorithms used to determine numerical values of geometric element parameters. In addition, ways are recommended on how to algebraically represent the distance of a point from each of the main geometric elements, the formulae being given in terms of the recommended parameters.
A detailed description can be found in EUR Report 13517.

Funding Scheme

CSC - Cost-sharing contracts

Coordinator

National Physical Laboratory (NPL)
Address
Queen's Road
TW11 0LW Teddington
United Kingdom

Participants (1)

PHYSIKALISCH-TECHNISCHE BUNDESANSTALT
Germany
Address
Bundesallee 100
Braunschweig