Spiral fat arcs - Bounding regions with cubic convergence

Michael Bartoň*, Gershon Elber

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


A bounding region for spiral curve segments shaped by two circular arcs, parts of the osculating circles at the spiral's endpoints, and two lines is introduced. This bounding region, denoted spiral fat arc (SFA) is simple to construct and process, and shows a cubic approximation order to a given spiral curve. Given a general planar parametric curve, it can be split at curvature extrema (and inflection points), solving for the parametric locations for which κ′ = 0 (and κ = 0), κ being the signed curvature field, to yield a set of spiral curves. Each of the spirals is then fitted with a bounding SFA. Finding the intersection locations of two free-form planar curves is a fundamental task in geometric computing and computer aided design, and can immediately benefit from this new SFA bounding region. A recursive curve-curve intersection (CCI) algorithm that efficiently computes the intersection location of two parametric curves using SFAs is also introduced.

Original languageEnglish (US)
Pages (from-to)50-57
Number of pages8
JournalGraphical Models
Issue number2
StatePublished - Mar 2011


  • Bounding regions
  • Curve-curve intersection
  • Fat arcs
  • Monotone curvature
  • Spiral

ASJC Scopus subject areas

  • Software
  • Modeling and Simulation
  • Geometry and Topology
  • Computer Graphics and Computer-Aided Design


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