The Visual Computer

, Volume 35, Issue 6–8, pp 921–933 | Cite as

Minkowski sum computation for planar freeform geometric models using \(G^1\)-biarc approximation and interior disk culling

  • Sangjun Han
  • Seung-Hyun Yoon
  • Myung-Soo KimEmail author
  • Gershon Elber
Original Article


We present an efficient algorithm for computing the Minkowski sum of two planar geometric models bounded by B-spline curves. The boundary curves are first approximated by \(G^1\)-biarc splines within a given error bound \(\epsilon > 0\). A superset of Minkowski sum boundary is then generated using the biarc approximations. For non-convex models, the superset contains redundant arcs. An efficient and robust elimination of the redundancies is the main challenge of Minkowski sum computation. For this purpose, we use the Minkowski sum of interior disks of the two input models, which are again disks in the Minkowski sum interior. The majority of redundant arcs are eliminated by testing each against a small number of interior disks selected for efficiency. From the planar arrangement of remaining arcs, we construct the Minkowski sum boundary in a correct topology. We demonstrate a real-time performance and the stability of circle-based Minkowski sum computation using a large set of test data.


Minkowski sum B-spline curves Biarc splines Interior disks Trimming 



This study was funded by the MSIT/IITP of Korea (No. 2017-0-00367), by the National Research Foundation of Korea (No. NRF-2018R1D1A1B07048036 and NRF-2019R1A2C1003490), and by the ISRAEL SCIENCE FOUNDATION (Grant No. 597/18).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Sangjun Han
    • 1
  • Seung-Hyun Yoon
    • 2
  • Myung-Soo Kim
    • 3
    Email author
  • Gershon Elber
    • 4
  1. 1.Department of Mechanical and Aerospace EngineeringSeoul National UniversitySeoulKorea
  2. 2.Department of Multimedia EngineeringDongguk UniversitySeoulKorea
  3. 3.Department of Computer Science and EngineeringSeoul National UniversitySeoulKorea
  4. 4.Computer Science DepartmentTechnion, Israel Institute of TechnologyHaifaIsrael

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