Discrete Curve Flows in
Two-Dimensional Cayley–Klein
Geometries
Joseph Benson and Francis Valiquette
Abstract Using the method of equivariant moving frames, we study geometric
flows of discrete curves in the nine Cayley–Klein planes. We show that, under a
certain arc-length preserving flow, the curvature invariant κ n evolves according to
the differential-difference equation
∂κ n
∂t = (1 + 2
n+1 )(κ n+1 − κ n−1 ), where the
value of ∈ {−1, 0, 1} is linked to the geometry of the Cayley–Klein plane.
Keywords Discrete curve flows · Cayley–Klein geometry
1 Introduction
Invariant submanifold flows, particularly curve and surface flows, arise in a wide
range of applications, including geometric optics, computer vision, visual tracking
and control, and much more. Given a geometric submanifold flow, one of the typical
problems consists of determining the induced evolution on the geometric invariants
of the submanifold. For many geometric flows, this leads to completely integrable
evolution equations.
A similar phenomenon occurs in discrete geometry. For example, discrete
geometric curve flows in the Euclidean plane have been considered in [1–3], and
it was shown that the curvature evolves, under a certain arc-length preserving
flow, according to the differential-difference mKdV equation, which is completely
integrable. In this paper, we extend some of the work done in [1–3] by considering
discrete geometric curve flows in all nine 2-dimensional Cayley–Klein geometries,
[4]. Using the method of equivariant moving frames, our computations are performed symbolically, which allows us to tackle the nine geometries simultaneously.
J. Benson
Macalester College, Saint Paul, MN, USA
e-mail: jbenson4@macalester.edu
F. Valiquette ()
Monmouth University, West Long Branch, NJ, USA
e-mail: fvalique@monmouth.edu
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_15
157
Two-Dimensional Cayley–Klein
Geometries
Joseph Benson and Francis Valiquette
Abstract Using the method of equivariant moving frames, we study geometric
flows of discrete curves in the nine Cayley–Klein planes. We show that, under a
certain arc-length preserving flow, the curvature invariant κ n evolves according to
the differential-difference equation
∂κ n
∂t = (1 + 2
n+1 )(κ n+1 − κ n−1 ), where the
value of ∈ {−1, 0, 1} is linked to the geometry of the Cayley–Klein plane.
Keywords Discrete curve flows · Cayley–Klein geometry
1 Introduction
Invariant submanifold flows, particularly curve and surface flows, arise in a wide
range of applications, including geometric optics, computer vision, visual tracking
and control, and much more. Given a geometric submanifold flow, one of the typical
problems consists of determining the induced evolution on the geometric invariants
of the submanifold. For many geometric flows, this leads to completely integrable
evolution equations.
A similar phenomenon occurs in discrete geometry. For example, discrete
geometric curve flows in the Euclidean plane have been considered in [1–3], and
it was shown that the curvature evolves, under a certain arc-length preserving
flow, according to the differential-difference mKdV equation, which is completely
integrable. In this paper, we extend some of the work done in [1–3] by considering
discrete geometric curve flows in all nine 2-dimensional Cayley–Klein geometries,
[4]. Using the method of equivariant moving frames, our computations are performed symbolically, which allows us to tackle the nine geometries simultaneously.
J. Benson
Macalester College, Saint Paul, MN, USA
e-mail: jbenson4@macalester.edu
F. Valiquette ()
Monmouth University, West Long Branch, NJ, USA
e-mail: fvalique@monmouth.edu
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_15
157