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Tolerances in geometric constraint problems. (English) Zbl 1075.65025
Geometric constraint solving is a term of computer-aided design and means the problems which arise when the location of geometric objets is described via geometric relations between them. The issues crucial for engineering applications are solvability of constraint problems and the sensitivity to errors.
The authors deal with the propagation of errors through implicit constraints. Based on the concept of tolerance zone, they show how to handle error propagation in the form of tolerance zones through implicit constraints in a way independent of solvability. The usage of tolerance zones generalizes interval arithmetic in the sense that intervals are tolerance zones of real numbers.
More precisely, authors assume that a certain number of geometric objects is given imprecisely, each of them is only known to be contained in a certain tolerance zone. Other geometric objects are located via constraints, and one wants to give tolerance zones for them. This is done by linearizing the system of constraints and giving upper bounds for the linearization error. Depending on the problem, there is a maximum size of tolerance zone for which this method is applicable. Computing this radius of validity and the linearization error requires upper bounds on the constraints’ second derivatives in the form of bilinear mappings. It turns out that such bounds are found easily of the constraints are quadratic, which is very often the case for geometric constraints.

65D17 Computer-aided design (modeling of curves and surfaces)
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