Chapter

Advances in Visual Computing

Volume 5875 of the series Lecture Notes in Computer Science pp 608-619

Visualizing Arcs of Implicit Algebraic Curves, Exactly and Fast

  • Pavel EmeliyanenkoAffiliated withMax-Planck-Institut für Informatik
  • , Eric BerberichAffiliated withSchool of Computer Science, Tel-Aviv University
  • , Michael SagraloffAffiliated withMax-Planck-Institut für Informatik

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Abstract

Given a Cylindrical Algebraic Decomposition [2] of an implicitly defined algebraic curve, visualizing distinct curve arcs is not as easy as it stands because, despite the absence of singularities in the interior, the arcs can pass arbitrary close to each other. We present an algorithm to visualize distinct arcs of algebraic curves efficiently and precise (at a given resolution), irrespective of how close to each other they actually pass. Our hybrid method inherits the ideas of subdivision and curve-tracking methods. With an adaptive mixed-precision model we can render the majority of curves using machine arithmetic without sacrificing the exactness of the final result. The correctness and applicability of our algorithm is borne out by the success of our web-demo presented in [11].

Keywords

Algebraic curves geometric computing curve rendering visualization exact computation