Leading a continuation method by geometry for solving geometric constraints

Geometric constraint problems arise in domains such as CAD, Robotics, Molecular Chemistry, whenever one expects 2D or 3D configurations of some geometric primitives fulfilling some geometric constraints. Most well-constrained 3D problems are resistant to geometric knowledge based systems. They are often solved by purely numerical methods that are efficient but provide only one solution. Finding all the solutions can be achieved by using, among others, generic homotopy methods, that become costly when the number of constraints grows. This paper focuses on using geometric knowledges to specialize a so-called coefficient parameter continuation to 3D geometric constraint systems. Even if the proposed method does not ensure obtaining all the solutions, it provides several real ones. Geometric knowledges are used to justify it and lead the search of new solutions.