Solving global geometric constraints on free-form curves

ABSTRACT

Constraint-based geometric modeling is the standard modeling paradigm in current modern CAD systems. Generally, the user defines constraints on the geometric objects and a solver is applied to find a configuration of the geometry, which satisfies these constraints. Proper application of these constraints allows rapid modification of the geometry without loss of design intent.

However, in current CAD systems, constraint solving for free-form geometric objects is generally limited. In particular, constraining global features such as limits on a curve's curvature values, are not supported.

In this paper we present a general method, within the constraint-based framework, to construct global constraints on free-form curves. The method starts by defining sufficient conditions on the curves in terms of an inequality expression, unlike local constraints the global constraint expression will be defined for all the domain of the curves. We then transform the expression into a symbolic polynomial, whose coefficients are symbolic expressions of the original curves. In the final step, a set of inequality constraints is applied in terms of the symbolic coefficients. These inequality constraints enforce the positivity of the symbolic polynomial.

The final inequality constraints are fed into the solver along with any other local constraints, which the user has provided on the curves. Therefore, the solution returned by the solver satisfies both the global constraints and any other local constraints the user supplies.

We have implemented a prototype of our method using existing commercial constraint solvers. We present results on several problems, which are handled as global geometric constraints using our method.

REFERENCES

• 1: M. P. D. Carmo. Differential Geometry of Curves and Surfaces. Prentice-Hall, 1976.
• 2: Dassault Systemes Catia. http://www.3ds.com/products/catia/catia-discovery.
• 3: E. Cohen, G. Elber, and R. F. Riesenfeld. Geometric Modeling with Splines: An Introduction. A. K. Peters, 2001.
• 4: Siemens PLM Software Components D-Cubed. http://www.plm.automation.siemens.com.
• 5: G. Elber. Trimming local and global self-intersections in offset curves using distance maps. In IMA Conference on the Mathematics of Surfaces, pages 213--222, 2003.
• 6: G. E. Farin. Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide. Academic Press, 4th edition, 1996.
• 7: ALS Geometric Software. http://www.als.gr.
• 8: C. Hoffmann. Constraint-based CAD. JCISE, 5(3):182--187, 2005.
• 9: C. Hoffmann and R. Joan-Arinyo. Parametric modeling. In G. Farin, J. Hoschek, and M.-S. Kim, editors, Handbook of Computer Aided Geometric Design, pages 519--541. Elsevier, 2002.
• 10: AutoDesk Inventor. http://www.autodesk.com.
• 11: B. Juttler. Linear convexity conditions for parametric tensor-product Bézier surface patches. In T. N. T. Goodman and R. R. Martin, editors, Proceedings of the Mathematics of Surfaces VII Conference, pages 189--208, Winchester, UK, 1997. Information Geometers.
• 12: LEDAS Ltd. http://www.ledas.com.
• 13: PTC Pro/Engineer. http://www.ptc.com/products/proengineer.
• 14: J.-K. Seong, G. Elber, and M.-S. Kim. Trimming local and global self-intersections in offset curves/surfaces using distance maps. Computer-Aided Design, 38(3):183--193, 2006.
• 15: Siemens PLM Software NX. http://www.plm.automation.siemens.com.
• 16: Siemens PLM Software Solid Edge. http://www.plm.automation.siemens.com.
• 17: Dassault Systemes SolidWorks Corporation. http://www.solidworks.com.
• 18: S. Wolfram. The Mathematica Book. Wolfram Media, 5th edition, August 2003.
• 19: C. K. Yap. Fundamental problems of algorithmic algebra. Oxford University Press, Inc., New York, NY, USA, 2000.