The goal of this post is to implement certain time-continuous flows on discrete space curves. By a time-continuous flow of a closed discrete space curve
In this post we will focus on the Möbius tangent flow, which is a discrete analogue of the continuous tangent flow
While in the continuous setup the tangent flow is trivial (it just acts by reparametrization), the discrete tangent flow will change the shape of the polygon.
Given a closed discrete space curve with
E.g. one reasonable choice for
Mainly the flow
We observe loops traveling along the curve, interacting when passing each other while basically keeping their shape. This kind of phenomenon hints at a connection to soliton theory.
Another flow related to soliton theory is the vortex filament flow. It describes how vortex filaments move and is related to the cubic non-linear Schrödinger equation – an equation well-known to soliton theorists.
To motivate the discrete vortex filament flow let us look at the continuous equation and assume for a second that we are dealing with a Frenet curve, i.e.
Certainly a discrete curve is not a Frenet curve. Though unless
The surface swept out by the vortex filament flow are called Hashimoto surfaces. They can be easily visualized with help of Houdini’s trail node. The point attribute @Alpha
controls the alpha channel of the point color and can be used to fade out the trail. Here is a picture how such a surface can look like.
Homework (due 29 May). Use your Runge-Kutta solver to compute the Möbius tangent and vortex filament flow of a prescribed closed discrete space curve.
Update: Albert has written a digital asset which makes it possible to plot how certain quantities (stored as detail attribute) evolve under the flow.
Homework (due 5 June). Plot the length, the area (projected to a plane) and the energy of a curve (with constant edge length) moving under the tangent and the vortex filament flow. Here energy is defined by