There is significant literature on the algorithms of geometric constraint solvers. However, much of it applies to algorithms and systems that are not in wide-spread commercial use in the CAD and related markets.

The D-Cubed 2D DCM and 3D DCM geometric constraint solvers are used in more commercial CAD products by more end-users than any other such solvers by quite some margin. Hence, there is significant interest in their algorithms. However, for reasons of commercial confidentiality the DCM algorithms are largely unpublished. Nevertheless, some important information has been published by John Owen, in some cases working with his academic collaborators.

John Owen founded D-Cubed Limited in 1989, having the role of managing director until its sale to UGS (now Siemens PLM Software) in 2004. He is a key contributor to the innovative core algorithms in the 2D DCM and 3D DCM. John is still with the team he started to set-up back in 1989.

Though mostly not directly relevant to commercial geometric constraint solving issues, for the mathematically-minded the following publications provide interesting insights into some of the wider and more fundamental mathematical issues in this field. They also illustrate some of the overlaps with fields beyond commercial CAD systems. Please note that access to some of the publications requires a subscription with the respective publisher, or a fee payable to the publisher:

Algebraic Solution for Geometry from Dimensional Constraints

J.C. Owen, pages 397-407 of the Symposium on Solid Modelling Foundations and CAD/CAM Applications", 1991, sponsored by ACM SIGGRAPH, Austin, Texas.

http://portal.acm.org/citation.cfm?id=112573&dl=GUIDE&coll=GUIDE

Of these academic references, this is arguably the most wide-ranging, most accessible and most significant for understanding some of the core geometric constraint solving issues in the D-Cubed DCM products. This paper describes some basic graph decompositions and indicates that (for points in 2D) these decompositions are sufficient to solve constraint systems which are generically solvable with ruler and compass constructions.

Constraints on Simple Geometry in Two and Three Dimensions

J.C. Owen, International Journal of Computational Geometry and Applications, pages 421-434, Volume 6, Number 4, 1996.

http://www.worldscinet.com/ijcga/06/0604/S0218195996000265.html

Another relatively accessible and relevant paper for core geometric constraint solving issues in a CAD context. This paper extends the ideas of the preceding paper to include more geometric elements such as lines and circles.

Distance Configurations of Points in a Plane with a Galois Group That Is Not Soluble

J.C. Owen and S.C. Power, arXiv.org, 2005

http://arxiv.org/abs/math/0503717v1

Unpublished conference proceedings which describes the mathematical methods used to prove the completeness of the graph decomposition method.

The Nonsolvability by Radicals of Generic 3-Connected Planar Laman Graphs

J.C. Owen and S.C. Power, Transactions of the American Mathematical Society, Volume 359 (5). pp. 2269-2303, 2007.

http://eprints.lancs.ac.uk/2385/

This paper gives a formal proof for the completeness of the graph decomposition methods described in the preceding references when applied to planar graphs.

Continuous Curves from Infinite Kempe Linkages

J.C. Owen and S.C. Power, Bulletin of the London Mathematical Society, Volume 41, Issue 6, pp. 1105-1111, 2009.

http://blms.oxfordjournals.org/content/41/6/1105.abstract

This paper extends a classical theorem of Kempe (that there is a finite mechanism which traces out any algebraic curve) to show that there is a bounded infinite mechanism which traces out any continuous curve, including space-filling curves. This indicates that as geometric constraint systems become large they may exhibit increasingly exotic behaviours.

Frameworks Symmetry and Rigidity

J.C. Owen and S.C. Power, International Journal of Computational Geometry and Applications, pages 723-750, Volume 20, Number 6, 2010.

http://www.maths.lancs.ac.uk/~power/FSRowen_power.pdf

Symmetries play a key role in determining the singular (non-regular) points for geometric constraint equations. This paper uses techniques recently developed in the theory of rigid structures to predict singular points in geometric constraint equations.

Infinite Bar-Joint Frameworks, Crystals and Operator Theory - preprint

J.C. Owen and S.C. Power, 2010

http://www.maths.lancs.ac.uk/~power/joscpIRFarXiv.pdf

This paper investigates the properties of infinitely large systems of geometric constraint equations! The results are however relevant to the properties of physical materials such as zeolites.

Rigidity of Frameworks Supported on Surfaces - Preprint

A. Nixon, J.C. Owen and S.C. Power, arXiv.org, 2010

http://arxiv.org/PS_cache/arxiv/pdf/1009/1009.3772v1.pdf

This paper investigates the properties of geometric constraint equations where the geometric elements (points) are constrained to lie on an engineering surface such as a sphere, cylinder, cone or torus.