Global solutions of well-constrained transcendental systems using expression trees and a single solution test

Maxim Aizenshtein, Michael Barton*, Gershon Elber

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


We present an algorithm which is capable of globally solving a well-constrained transcendental system over some sub-domain D⊂ℝ n, isolating all roots. Such a system consists of n unknowns and n regular functions, where each may contain non-algebraic (transcendental) functions like sin, exp or log. Every equation is considered as a hyper-surface in ℝ n and thus a bounding cone of its normal (gradient) field can be defined over a small enough sub-domain of D. A simple test that checks the mutual configuration of these bounding cones is used that, if satisfied, guarantees at most one zero exists within the given domain. Numerical methods are then used to trace the zero. If the test fails, the domain is subdivided. Every equation is handled as an expression tree, with polynomial functions at the leaves, prescribing the domain. The tree is processed from its leaves, for which simple bounding cones are constructed, to its root, which allows to efficiently build a final bounding cone of the normal field of the whole expression. The algorithm is demonstrated on curve-curve intersection, curve-surface intersection, ray-trap and geometric constraint problems and is compared to interval arithmetic.

Original languageEnglish (US)
Pages (from-to)265-279
Number of pages15
JournalComputer Aided Geometric Design
Issue number5
StatePublished - Jun 1 2012


  • Bounding cone
  • Expression tree
  • Non-algebraic equation system
  • Root solver
  • Single root criteria

ASJC Scopus subject areas

  • Modeling and Simulation
  • Automotive Engineering
  • Aerospace Engineering
  • Computer Graphics and Computer-Aided Design


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