Plane curves of minimal energy

ABSTRACT

The problem of finding the plane curve of minimal elastic energy with prescribed endpoints and end-directions was solved in 1983 by B. K. P. Horn. Here the solution is discussed, given a very short proof, and extended to include a constant on length.

REFERENCES

• 1: B. C. Carlson , Elaine M. Notis, Algorithm 577: Algorithms for Incomplete Elliptic Integrals [S21], ACM Transactions on Mathematical Software (TOMS), v.7 n.3, p.398-403, Sept. 1981 [doi>10.1145/355958.355970]
• 2: FORSYTH, A.R. Calculus of Variations. Dover, New York, 1949.
• 3: B. K. P. Horn, The Curve of Least Energy, ACM Transactions on Mathematical Software (TOMS), v.9 n.4, p.441-460, Dec. 1983 [doi>10.1145/356056.356061]
• 4: M. Kallay, Method to approximate the space curve of least energy and prescribed length, Computer-Aided Design, v.19 n.2, p.73-76, March 1987 [doi>10.1016/S0010-4485(87)80048-9]
• 5: ULLMAN, S. Filling in the gaps: The shape of subjective contours and a model for their generation. Biol. Cybern. 25 (1976), 1-6.

INDEX TERMS

Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.1 Interpolation
          Subjects: Spline and piecewise polynomial interpolation
      G.1.2 Approximation
          Subjects: Spline and piecewise polynomial approximation
      G.1.6 Optimization
          Subjects: Constrained optimization

  I. Computing Methodologies
  I.3 COMPUTER GRAPHICS
      I.3.5 Computational Geometry and Object Modeling
          Subjects: Curve, surface, solid, and object representations

General Terms:
Algorithms, Theory