The concept provides a means to characterise the solution spaces of fully defined geometrical constraint solving problems defining the relative position of points, lines and planes implicitly by nonlinear geometrical constraints such as angle and distance.

In general, multiple solutions exist for each of those problems. The geometrical interpretation of each solution is referred to as variational geometry. While all variational geometries are equivalent from the perspective of the problem formulation not all variational geometries are equivalent in other interpretation contexts. The characterisation of variational geometries aims at their differentiation in order to allow for their selection, also known as solution selection. In order to disambiguate two sets of the solution space the chirality relation of two geometric elements is used as minimal information.

For computer-aided design the concept is applicable to variational geometries in two-dimensional sketches and three-dimensional assembly models. Other possible application areas include the interpretation of sensor-based measurements. Constraints within variational geometries do not include direction information that is necessary to identify which geometric element defines which other geometric element.

A deficit of existing approaches for solution selection consists in taking the direction information of constraints for granted. Other deficits are their limitation to points and the lacking proof of their completeness and correctness.

To overcome this deficit, a solution selection is required that enables the selection of geometrically unique solutions. In order to guarantee geometrical uniqueness such a solution selection needs to be founded on a concept which enables the characterisation of all variational geometries, given a specific geometrical problem. The proposed concepts for solution selection and characterisation fulfil these conditions. Especially, the concept is designed to enable characterisations under uncertainty with respect to the constraint directions as well as position and orientation of the geometrical elements. The uncertainty information is explicitly modelled in a decidable description logic. Hence, the description logic representation of the geometrical problem may be exploited with deductive reasoning methods in order to assess whether the variational geometry is defined geometrically unique and if the calculated variational geometry corresponds to the modelling intent of the designer.

The main contributions of the dissertation consist in the development of a conceptual, declarative and procedural part. The conditions for a geometrically unique solution selection are formalised in the description logic definition of geometrical uniqueness, building up on the specification of 27 constraint problems, so called basic configurations. The geometrical uniqueness of a given geometrical problem is tested for a corresponding subset of these basic configurations. Additional contributions include the declarative prototype implementation of 32 constraints. A new method for the conversion of ISO 10303 standards into description logics has been developed and applied. For the considered set of geometrical problems upper and lower boundaries are estimated for the number of solutions and the number of solution selectors necessary to attain geometrical uniqueness.

The concept has been implemented, verified and validated. The verification result indicates that a sequential assessment of solution selectors results in linear calculation time with respect to the number of variational geometries. The validation has been performed on a geometry that has been generated with a newly developed algorithm which offers an efficient way to generate variational geometries with specific topological properties.

The validation result demonstrates the completeness and correctness of the characterisation. The transferability of the research results is facilitated by the use of existing standards like ISO 10303-108 and OWL 2.

Finally, opportunities for the further reduction of solution selection information necessary to obtain geometrical unique variational geometries have been identified for future.