Homework 1

Solution:

Recall the Bishop frame:

With Frenet Serret Equations:

In the Bishop frame ${\gamma }_t={\gamma }_s\times {\gamma }_{ss}+\alpha [{\gamma }_{sss}+\frac{3}{2}{\gamma }_{ss}\times ({\gamma }_s\times {\gamma }_{ss})]$ for $\alpha \in \mathbb{R}$ is rewritten to:

with ${\gamma }_{ss}={\kappa }_{1}U+{\kappa }_{2}V$.

We note that $\gamma$ has to satisfy the compatibility conditions $({\gamma }_t{)}_{ss}=({\gamma }_{ss}{)}_t$.

We begin by computing $T_t=({\gamma }_s{)}_t=({\gamma }_t{)}_s$:

Since $V\cdot T=0$ (as we are working in an orthonormal frame) we obtain that $\frac{\partial }{\partial t}(V\cdot T)=0$ thus:

Similarly, we get that $U\cdot V_t=-U_t\cdot V$.
We are interested in computing $V_t\cdot U$ and in order to do so we compute $(V_t\cdot U{)}_s$ instead:

Thus $V_t\cdot U=\frac{1}{2}{\kappa }^2+\alpha ({\kappa }_1{{\kappa }_2}_s-{\kappa }_2{{\kappa }_1}_s)+C(t)$, where $C$ is a real valued function in $t$.

We now note that $\gamma$ has to satisfy the compatibility conditions $({\gamma }_t{)}_{ss}=({\gamma }_{ss}{)}_t$ and we use the fact that since $U\cdot U=1$ we get that $U_t\cdot U=0$. Thus:

Hence:

We observe that $\kappa {\kappa }_s={\kappa }_1{{\kappa }_1}_s+{\kappa }_2{{\kappa }_2}_s$, ${{\kappa }_1}_s=k_{s}cos(\theta )-{\kappa }_2\tau$ and ${{\kappa }_2}_s=k_{s}sin(\theta )+{\kappa }_1\tau$, thus as a consequence we compute the following identities:

Thus we finally obtain:

Similarly one can also show that:

Now using the Hasimoto map $\psi =\kappa e^{{\int }^s\tau }={\kappa }_1+i{\kappa }_2$ with (1) and (2) we thus get afforemention equation that we wanted to show: