Research

My research focuses on robust and adaptive methods for the solution of problems in computer graphics and geometric modeling. Robustness means that I am interested in using computers to prove properties of curves and surfaces. This usually takes the form of solving equations in several variables. The main tools for achieving robustness are interval computation methods using interval arithmetic and affine arithmetic. Interval methods provide guaranteed numerical results that are not affected by rounding errors in floating-point computations. More importantly, interval methods allows us to analyse the global behaviour of functions over whole regions of the space without sampling it. Adaptiveness means that I want to concentrate the computational effort near interesting regions of the space, such as near a solution curve or in regions where the surface curvature is high. Global analysis with interval methods leads naturally to adaptive methods.

 

I am also interested in using computers to prove properties of non-linear dynamical systems. Again, interval methods are the main tools. One result of this research is a computer-aided proof that the Jouanolou foliation of low degree admits no nontrivial minimal sets. Recently, I have been working on adaptive algorithms for generating guaranteed images of Julia sets and fractal basins for Newton's method.

 

I am also interested in programming languages and I am one of the designers of the Lua language.

Selected publications

  1. Oliveira, J. B., Figueiredo, L. H., Robust approximation of offsets, bisectors, and medial axes of plane curves, Reliable Computing 9 #2 (2003) 161-175.
  2. Lopes, H., Oliveira, J. B., Figueiredo, L. H., Robust adaptive polygonal approximation of implicit curves, Computers & Graphics 26 #6 (2002) 841-852.
  3. Camacho, C., Figueiredo, L. H., The dynamics of the Jouanolou foliation on the complex projective 2-space, Ergodic Theory and Dynamical Systems 21 #3 (2001) 757-766.
  4. Coelho, L. C. G., Gattass, M., Figueiredo, L. H., Intersecting and trimming parametric meshes on finite-element shells, International Journal for Numerical Methods in Engineering 47 #4 (2000) 777-800.
  5. Velho, L., Figueiredo, L. H., Gomes, J., A unified approach for hierarchical adaptive tesselation of surfaces, ACM Transactions on Graphics 18 #4 (1999) 329-360.
  6. Velho, L., Figueiredo, L. H., Gomes, J., A methodology for piecewise linear approximation of surfaces, Journal of the Brazilian Computer Society 3 #3 (1997) 30-42.
  7. Figueiredo, L. H., Stolfi, J., Adaptive enumeration of implicit surfaces with affine arithmetic, Computer Graphics Forum 15 #5 (1996) 287-296.
  8. Ierusalimschy, R., Figueiredo, L. H., Celes, W., Lua: an extensible extension language, Software: Practice & Experience 26 #6 (1996) 635-652.
    First prize (technological category) in the II Compaq Award for Research and Development in Computer Science.
  9. Figueiredo, L. H., Gomes, J., Sampling implicit objects with physically-based particle systems, Computers & Graphics 20 #3 (1996) 365-375.
  10. Figueiredo, L. H., Gomes, J., Computational morphology of curves, The Visual Computer 11 #2 (1995) 105-112.
  11. Figueiredo, L. H., Adaptive sampling of parametric curves, in A. Paeth (ed.), Graphics Gems V, Academic Press, 1995, 173-178.


Last update: Fri May 15 08:11:46 BRST 2009 by lhf.