Skip to main content
SpringerLink
Log in

Two-dimensional offsets and medial axis transform

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

We present a mathematical theory of the two-dimensional offset curves from the viewpoint of medial axis transform. We explore the local geometry of the offset curve in relation with the medial axis transform, culminating in the classification of points on the offset curve. We then study the domain decomposition from the viewpoint of offsets, and in particular introduce the concept of monotonic fundamental domain as a device for detecting the correct topology of offsets as well as for stable numerical computation. The monotonic fundamental domains are joined by peaks or valleys of the medial axis transform, or by what we call the critical horizonal section whose algebro-geometric properties are rigorously treated as well.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ait-Haddou, R., Biard, L., Slawinski, M.A.: Minkowski isoperimetric-hodograph curves. Comput. Aided Geom. Design 17(9), 835–861 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chiang, C.-S., Hoffmann, C.M., Lynch, R.E.: How to compute offsets without self-intersection. Technical Report CSD-TR-91-072. Computer Sciences Department, Purdue University (1991)

  3. Choi, B.K., Park, S.C.: A pair-wise offset algorithm for 2d point-sequence curve. Comput.-Aided Des. 31(12), 735–745 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Choi, H.I., Choi, S.W., Moon, H.P.: Mathematical theory of medial axis tranform. Pacific J. Math. 181(1), 57–88 (1997)

    Article  MathSciNet  Google Scholar 

  5. Choi, H.I., Choi, S.W., Moon, H.P., Wee, N.-S.: New algorithm for medial axis transform of plane domain. Graph. Models Image Process. 59(6), 463–483 (1997)

    Article  Google Scholar 

  6. Chou, J.J., Cohen, E.: Computing offsets and tool paths with Voronoi diagrams. Technical report. Department of Computer Science, University of Utah, Salt Lake City, UT (1990)

  7. Chuang, S.H.F., Kao, C.Z.: One-sided arc approximation of b-spline curves for interference-free offsetting. Comput.-Aided Des. 31(2), 111–118 (1999)

    Article  MATH  Google Scholar 

  8. Coquillart, S.: Computing offsets of B-spline curves. Comput.-Aided Des. 19(6), 305–309 (1987)

    Article  MATH  Google Scholar 

  9. Elber, G., Lee, I.-K., Kim, M.-S.: Comparing offset curve approximation methods. IEEE Comput. Graph. Appl. 17(3), 62–71 (1997)

    Article  Google Scholar 

  10. Farouki, R.T.: Conic approximation of conic offsets. J. Symbolic Comput. 23, 301–313 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  11. Farouki, R.T., Neff, C.A.: Algebraic properties of plane offset curves. Comput. Aided Geom. Design 7, 101–127 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  12. Farouki, R.T., Neff, C.A.: Analytic properties of plane offset curves. Comput. Aided Geom. Design 7, 83–99 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  13. Farouki, R.T., Ramamurthy, R.: Specified-precision computation of curve/curve bisector. Technical Report UM-MEAM-96-10. Dept. of Mechanical Engineering and Applied Mechanics, University of Michigan, Ann Arbor, MI (1998)

  14. Farouki, R.T., S.T.: Pythagorean hodographs. IBM J. Res. Develop. 34, 736–752 (1990)

    Article  MathSciNet  Google Scholar 

  15. Feng, H.-Y., Li, H.: Constant scallop-height tool path generation for three-axis sculptured surface machining. Comput.-Aided Des. 34(9), 647–654 (2002)

    Article  Google Scholar 

  16. Glaeser, G., Wallner, J., Pottmann, H.: Collision-free 3-axis milling and selection of cutting tools. Comput.-Aided Des. 31(3), 225–232 (1999)

    Article  MATH  Google Scholar 

  17. Held, M.: On the Computational Geometry of Pocket Machining. In: Lecture Notes in Computer Science, vol. 500. Springer, Berlin (1991)

    Google Scholar 

  18. Held, M., Lukács, G., Andor, L.: Pocket machining based on coutour-parallel tool paths generated by means of proximity maps. Comput.-Aided Des. 26(3), 189–203 (1994)

    Article  MATH  Google Scholar 

  19. Holla, V.D., Shastry, K.G., Prakash, B.G.: Offset of curves on tessellated surfaces. Comput.-Aided Des. 35(12), 1099–1108 (2003)

    Article  Google Scholar 

  20. Hon-Yuen Tam, H.-W.L., Xu, H.: A geometric approach to the offsetting of profiles on three-dimensional surfaces. Comput.-Aided Des. 36, 887–902 (2004)

    Article  Google Scholar 

  21. Hoschek, J.: Offset curves in the plane. Comput. Aided Des. 17(2), 77–82 (1985)

    Article  Google Scholar 

  22. Hoschek, J.: Spline approximation of offset curves. Comput. Aided Geom. Design 5, 33–40 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  23. Hui, K.C.: Free-form design using axial curve-pairs. Comput.-Aided Des. 34(8), 583–595 (2002)

    Article  MathSciNet  Google Scholar 

  24. Jun, C.-S., Kim, D.-S., Park, S.: A new curve-based approach to polyhedral machining. Comput.-Aided Des. 34(5), 379–389 (2002)

    Article  Google Scholar 

  25. Jüttler, B., Mäurer, C.: Cubic pythagorean hodograph spline curves and applications to sweep surface modeling. Comput.-Aided Des. 31(1), 73–83 (1999)

    Article  MATH  Google Scholar 

  26. Klass, R.: An offset spline approximation for plane cubic splines. Comput. Aided Des. 15(5), 297–299 (1983)

    Article  Google Scholar 

  27. Lartigue, C., Thiebaut, F., Maekawa, T.: Shapes with offsets of nearly constant surface area. Comput. Aided Des. 31(4, 1), 287–296 (1999)

    Google Scholar 

  28. Lartigue, C., Thiebaut, F., Maekawa, T.: Cnc tool path in terms of b-spline curves. Comput.-Aided Des. 33(4, 2), 307–319 (2001)

    Article  Google Scholar 

  29. Lee, E.: Contour offset approach to spiral toolpath generation with constant scallop height. Comput.-Aided Des. 35(6), 511–518 (2003)

    Article  Google Scholar 

  30. Lee, I.-K., Kim, M.-S., Elber, G. Planar curve offset based on circle approximation. Comput.-Aided Des. 28(8), 617–630 (1996)

    Article  MATH  Google Scholar 

  31. Meek, D.S., Walton, D.J.: Offset curves of clothoidal splines. Comput.-Aided Des. 22(4), 199–201 (1990)

    Article  MATH  Google Scholar 

  32. Park, S.C., Choi, B.K.: Uncut free pocketing tool-paths generation using pair-wise offset algorithm. Comput.-Aided Des. 33(10), 739–748 (2001)

    Article  Google Scholar 

  33. Park, S.C., Chung, Y.C.: Offset tool-path linking for pocket machining. Comput.-Aided Des. 34(4), 299–308 (2002)

    Article  Google Scholar 

  34. Park, S.C., Chung, Y.C., Choi, B.K.L: Contour-parallel offset machining without tool-retractions. Comput.-Aided Des. 35(9), 841–849 (2003)

    Article  Google Scholar 

  35. Persson, H.: NC machining of arbitrarily shaped pockets. Comput.-Aided Des. 10(3), 169–174 (1978)

    Article  Google Scholar 

  36. Pham, B.: Offset approximation of uniform B-splines. Comput.-Aided Des. 20(8), 471–474 (1988)

    Article  MATH  Google Scholar 

  37. Pham, B.: Offset curves and surfaces: A brief survey. Comput.-Aided Des. 24(4), 223–229 (1992)

    Article  Google Scholar 

  38. Piegl, L.A., Tiller, W.: Computing offsets of NURBS curves and surfaces1. Comput.-Aided Des. 31(2), 147–156 Feb (1999)

    Article  MATH  Google Scholar 

  39. Pottmann, H.: Curve design with rational Pythagorean-hodograph curves. Adv. Comput. Math. 3, 147–170 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  40. Pottmann, H.: Rational curves and surfaces with rational offsets. Comput. Aided Geom. Design 12(2), 175–192 Mar (1995)

    Article  MATH  MathSciNet  Google Scholar 

  41. Ravi Kumar, G.V.V., Shastry, K.G., Prakash, B.G.: Computing non-self-intersecting offsets of NURBS surfaces. Comput.-Aided Des. 34(3), 209–228 (2002)

    Article  Google Scholar 

  42. Tiller, W., Hanson, E.G.: Offsets of two-dimensional profiles. IEEE Comput. Graph. Appl. 4, 36–46 (1984)

    Article  Google Scholar 

  43. Wang, L.Z., Miura, K.T., Nakamae, E., Yamamoto, T., Wang, T.J.: An approximation approach of the clothoid curve defined in the interval [0, /2] and its offset by free-form curves. Comput.-Aided Des. 33(14), 1049–1058 (2001)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Seoul National University, Seoul, 151-747, Korea

    Hyeong In Choi

  2. Department of Mathematics, Duksung Women’s University, Seoul, 132-714, Korea

    Sung Woo Choi

  3. School of Electronics and Information, Kyung Hee University, Gyeonggi-do, 446-701, Korea

    Chang Yong Han

  4. Department of Naval Architecture and Ocean Engineering, Seoul National University, Seoul, 151-744, Korea

    Tae-wan Kim

  5. Institute of Mathematics and its Application, University of Minnesota, Minneapolis, MN, 55455, USA

    Song-Hwa Kwon

  6. Research Lab., Amisys Corporation, Seoul, 135-080, Korea

    Hwan Pyo Moon

  7. Department of Mathematics and Statistics, Arizona State University, Tempe, AZ, 85287, USA

    Kyeong Hah Roh

  8. Division of Industrial and Mechanical Systems Engineering, Hansung University, Seoul, 136-792, Korea

    Nam-Sook Wee

Authors
  1. Hyeong In Choi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sung Woo Choi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chang Yong Han
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Tae-wan Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Song-Hwa Kwon
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Hwan Pyo Moon
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Kyeong Hah Roh
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Nam-Sook Wee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hyeong In Choi.

Additional information

Communicated by: R.T. Farouki.

First author holds joint appointment in the Research Institute of Mathematics, Seoul National University.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Choi, H.I., Choi, S.W., Han, C.Y. et al. Two-dimensional offsets and medial axis transform. Adv Comput Math 28, 171–199 (2008). https://doi.org/10.1007/s10444-007-9036-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-007-9036-5

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation