Discrete Curve Flows in Two-Dimensional Cayley–Klein Geometries
167
α n =
2K
1 + 1 K 2 ,
β n =
2Kκ n
1 + 1 K 2 .
For such an arc-length preserving flow, the evolution of curvature is governed by the
completely integrable differential-difference equation
∂κ n
∂t
= A
x
κ n
2K
1 + 1 K 2
+ A
y
κ n
2Kκ n
1 + 1 K 2
= (1 + 2 κ
2
n+1 )(κ n+1 − κ n−1 ).
References
1. J. Benson, F. Valiquette, Invariant discrete flows. Stud. Appl. Math. 143, 81–119 (2019)
2. A.I. Bobenko, Geometry II: discrete differential geometry. Lecture Notes, TU Berlin (2015),
page.math.tu-berlin.de/bõbenko/Lehre/Skripte/DDG_Lectures.pdf
3. T. Hoffmann, N. Kutz, Discrete curves in CP 1 and the Toda lattice. Stud. Appl. Math. 113,
31–55 (2004)
4. I.M. Yaglom, A Simple Non-Euclidean Geometry and Its Physical Basis (Springer, New York,
1979)
5. R. Yamilov, Symmetries as integrability criteria for differential difference equations. J. Phys. A
Math. Gen. 39, R541–623 (2006)
167
α n =
2K
1 + 1 K 2 ,
β n =
2Kκ n
1 + 1 K 2 .
For such an arc-length preserving flow, the evolution of curvature is governed by the
completely integrable differential-difference equation
∂κ n
∂t
= A
x
κ n
2K
1 + 1 K 2
+ A
y
κ n
2Kκ n
1 + 1 K 2
= (1 + 2 κ
2
n+1 )(κ n+1 − κ n−1 ).
References
1. J. Benson, F. Valiquette, Invariant discrete flows. Stud. Appl. Math. 143, 81–119 (2019)
2. A.I. Bobenko, Geometry II: discrete differential geometry. Lecture Notes, TU Berlin (2015),
page.math.tu-berlin.de/bõbenko/Lehre/Skripte/DDG_Lectures.pdf
3. T. Hoffmann, N. Kutz, Discrete curves in CP 1 and the Toda lattice. Stud. Appl. Math. 113,
31–55 (2004)
4. I.M. Yaglom, A Simple Non-Euclidean Geometry and Its Physical Basis (Springer, New York,
1979)
5. R. Yamilov, Symmetries as integrability criteria for differential difference equations. J. Phys. A
Math. Gen. 39, R541–623 (2006)