# Relieving nematic geometric frustration in the plane

Andrea Pedrini1, Epifanio G Virga1 1 Dipartimento di Matematica, Università di
Pavia, Via Ferrata 5, I-27100 Pavia, Italy<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Frustration in nematic-ordered media (endowed with a director field) is
treated in a purely geometric fashion in a flat, two-dimensional space. We
recall the definition of _quasi-uniform_ distortions and envision these as
viable ways to relieve director fields prescribed on either a straight line or
the unit circle. We prove that using a _planar spiral_ is the only way to fill
the whole plane with a quasi-uniform distortion. Apart from that, all
relieving quasi-uniform distortions can at most be defined in a half-plane;
however, in a generic sense, they are all asymptotically spirals.

,

##  1 Introduction

Nematic liquid crystals are perhaps the most typical example of a soft matter
system whose order parameter is a unit vector $\bm{n}$ enjoying the head-tail
symmetry, which requires all their physical properties to be invariant under
the transformation $\bm{n}\mapsto-\bm{n}$. Such an order parameter is often
called a _director_ to emphasize that only the direction of $\bm{n}$ has
physical significance, whereas in a _spin_ also the sense of orientation is
meaningful. On occasion, a director is said to generate a _line field_ , as
opposed to a vector field.

The microscopic origin of $\bm{n}$ is to be retraced in the orientation of the
elongated molecules that constitute these fascinating, ordered phases: on
average, these molecules tend to be oriented alike, with equally likely
distributions of heads and tails, whenever these can be distinguished from one
another.

Properly speaking, the target of the director mapping $\bm{n}$ is the real
projective plane $\mathbb{RP}^{2}$, although in a number of practical cases
(but not all), $\bm{n}$ is _orientable_ , meaning that it can be lifted into
the unit sphere $\mathbb{S}^{2}$ [1].

Our approach here will be more general than our original liquid crystal
motivation might suggest. The director field $\bm{n}$ will have any possible
interpretation compatible with nematic symmetry. Our focus will be on
_geometric frustration_ and ways of relieving it in two space dimensions. By
geometric frustration we refer to any means capable of preventing a director
field from filling space _uniformly_. Beyond its intuitive meaning, the latter
property is properly defined by identifying scalar measures of distortion,
which we call the _distortion characteristics_ , and requiring them to be
_constant_ throughout space.

This notion was introduced in [2] in three-dimensional Euclidean space and
further extended to the non-Euclidean case in [3, 4]. In flat two-dimensional
space, the request of uniformity confines $\bm{n}$ to be a trivial constant
field [5], and the question then arises as to whether prescribing $\bm{n}$ on
a line in the plane, so as _not_ to be constant there, induces a geometric
frustration that can be relieved via a purely geometric mechanism, with no
reference to any energy consideration.

The purely geometric mechanism we envision was introduced in [6]; it appears
to be the most natural extension of the notion of uniform distortion, one for
which the distortion characteristics instead of being constant are in constant
ratios to one another, so that a single function of position in space is left
free. We call these distortions _quasi-uniform_.

This paper explores the possibility that such an avenue can indeed be taken to
relieve frustration in flat, two-dimensional space. The material presented
here is organized as follows.

In section 2, we recall the definition of distortion characteristics and their
role in defining uniformity. Section 3 is concerned with the definition of
quasi-uniform distortions and their representation in the plane. In section 4,
we show how to fill half a plane with a quasi-uniform distortion relieving a
geometric frustration concentrated on a straight line. It will emerge there
that some frustrations can be relieved globally, but others only locally.
Incidentally, our analysis will prove that no genuine quasi-uniform distortion
can exist in the whole plane. Geometric distortions enforced on a circle will
be considered in section 5. The possibility of relieving them in the whole
plane outside the circle will be related to the topological charge of the
frustration. In section 6, we extract from the families of quasi-uniform
distortions constructed in this paper some features that they have in common.
The notion of frustration adopted in this paper admittedly bears a rather
extended meaning; a more restricted one, which also entails a definition of
_one-dimensional_ uniformity is introduced and discussed in section 7.
Finally, in section 8, we collect our main conclusions and comment on the
general nature of the asymptotic distortion field relieving the two classes of
geometric frustration considered in this work.

A number of appendices close this paper; there we display the mathematical
details of our development that could have hampered our presentation in the
main text.

##  2 Distortion characteristics

A natural measure of distortion for a director field $\bm{n}$ is its spatial
gradient $\nabla\bm{n}$. Were $\nabla\bm{n}\equiv\mathbf{0}$, $\bm{n}$ would
have the same orientation everywhere in the domain in space under
consideration. Being a tensor, $\nabla\bm{n}$ is (covariantly) affected by a
change of frame (observer). We want to extract from it invariant measures of
distortions that bear an intrinsic meaning (independent of observers).

Here this goal is achieved by means of a decomposition of $\nabla\bm{n}$ first
proposed in [7] and then reprised and reinterpreted in [8], where the main
players are the _splay_ scalar $S:=\operatorname{div}\bm{n}$, the _twist_
pseudoscalar $T:=\bm{n}\cdot\operatorname{\operatorname{curl}}\bm{n}$ and the
_bend_ vector $\bm{b}:=\bm{n}\times\operatorname{\operatorname{curl}}\bm{n}$:

$\nabla\bm{n}=-\bm{b}\operatorname{\otimes}\bm{n}+\frac{1}{2}T\mathbf{W}(\bm{n})+\frac{1}{2}S\mathbf{P}(\bm{n})+\mathbf{D}.$
(1)

In (1), $\mathbf{W}(\bm{n})$ denotes the skew-symmetric tensor with axial
vector $\bm{n}$,
$\mathbf{P}(\bm{n}):=\mathbf{I}-\bm{n}\operatorname{\otimes}\bm{n}$ is the
projector on the plane orthogonal to $\bm{n}$ and $\mathbf{D}$ is a traceless
symmetric tensor for which $\mathbf{D}\bm{n}=\bm{0}$. Whenever
$\mathbf{D}\neq\bm{0}$, its properties ensure that it can be represented as

$\mathbf{D}=q(\bm{n}_{1}\operatorname{\otimes}\bm{n}_{1}-\bm{n}_{2}\operatorname{\otimes}\bm{n}_{2})$
(2)

by choosing an orthonormal basis of eigenvectors $(\bm{n}_{1},\bm{n}_{2})$ in
the plane orthogonal to $\bm{n}$, such that the positive scalar $q$ is the
eigenvalue of $\bm{n}_{1}$, and this is oriented so that
$\bm{n}=\bm{n}_{1}\times\bm{n}_{2}$. We call $(\bm{n}_{1},\bm{n}_{2},\bm{n})$
the _distortion frame_ and $q$ the _octupolar splay_.111We prefer the
adjective _octupolar_ to _biaxial_ and _thetrahedral_ , used in [8] and [9],
respectively, in order to stress the connection with an _octupolar tensor_
employed in [6] to represent pictorially $S$, $\bm{b}$, and $q$.

In the distortion frame, $\mathbf{W}(\bm{n})$ is represented as

$\mathbf{W}(\bm{n})=\bm{n}_{2}\operatorname{\otimes}\bm{n}_{1}-\bm{n}_{1}\operatorname{\otimes}\bm{n}_{2}$
(3)

and, by its own definition, the bend vector can always be decomposed as

$\bm{b}:=b_{1}\bm{n}_{1}+b_{2}\bm{n}_{2},$ (4)

for suitable scalars $b_{1}$ and $b_{2}$.222When $\mathbf{D}=\bm{0}$ and
$\bm{b}\neq\bm{0}$, the choice of $\bm{n}_{1}$ and $\bm{n}_{2}$ can be made by
requiring that $\bm{b}=b_{1}\bm{n}_{1}$ and then setting
$\bm{n}_{2}:=\bm{n}\times\bm{n}_{1}$, while for $\mathbf{D}=\bm{0}$ and
$\bm{b}=\bm{0}$ any pair $(\bm{n}_{1},\bm{n}_{2})$ in the plane orthogonal to
$\bm{n}$ could be employed as constituents of a suitable distortion frame. The
following identity, which is a direct consequence of (1), can be used to
determine $q$,333We defer the reader to A for a link between $q^{2}$ and the
classical _saddle-splay_ distortion measure in the Oseen-Frank theory of
liquid crystals.

$2q^{2}=\tr(\nabla\bm{n})^{2}+\frac{1}{2}T^{2}-\frac{1}{2}S^{2}.$ (5)

The decomposition in (1) also suggests a natural question as to the existence
of _uniform_ distortions, for which the director field $\bm{n}$ is not
necessarily constant in space, but $\nabla\bm{n}$ has everywhere the same
representation in the intrinsic distortion frame. This notion was made
formally precise in [2] by prescribing that the _distortion characteristics_
$(S,b_{1},b_{2},T,q)$ remain _constant_ in the whole domain where $\bm{n}$ is
defined. A complete characterization of uniform distortions in the whole
three-dimensional Euclidean space was also obtained in [2]: they are precisely
Meyer’s _heliconical_ distortions [10], for which either

$S=0,\ T=2q,\ b_{1}=b_{2}=b\quad\mathrm{or}\quad S=0,\ T=-2q,\
b_{1}=-b_{2}=b,$ (6)

where $q>0$ and $b$ are arbitrary scalars.

In general, a distortion will be said to be a _twist-bend_ , whenever $S=0$
and $T=\pm 2q$, with distortion characteristics _not_ necessarily constant.
Thus, uniform distortions in three-dimensional space are special twist-bend
distortions.

We find it instructive to comment in A about the bearing that identity (5) has
on the elastic free energy of nematic liquid crystals.444These comments have
no direct impact on our development, which is purely geometric.

In two space dimensions, the existence of uniform distortions depends on the
geometry of the surface $\mathscr{S}$ where the director field $\bm{n}$ lies.
A uniform distortion, with both $S\neq 0$ and $B\neq 0$, where $S$ is to be
interpreted as the _covariant_ divergence of $\bm{n}$ and $B$ as the norm of
the covariant bend vector,555The _covariant_ gradient $\mathsf{D}\bm{n}$ of a
director field $\bm{n}$ on a surface $\mathscr{S}$ can be defined as the
projection onto $\mathscr{S}$ of the surface gradient
$\nabla\\!_{\mathrm{s}}\bm{n}$,
$\mathsf{D}\bm{n}:=\mathbf{P}(\bm{\nu})\nabla\\!_{\mathrm{s}}\bm{n}$, where
$\mathbf{P}(\bm{\nu}):\mathbf{I}-\bm{\nu}\otimes\bm{\nu}$ and $\bm{\nu}$ is an
outer unit normal to $\mathscr{S}$. Here then $S=\tr\mathsf{D}\bm{n}$ and
$B=|\bm{n}\times\mathsf{curl}\,\bm{n}|$, where $\mathsf{curl}\,\bm{n}$ is
twice the axial vector associated with the skew-symmetric part of
$\mathsf{D}\bm{n}$. is possible only if the Gaussian curvature $K$ of
$\mathscr{S}$ is negative and such that $B^{2}+S^{2}=-K$ [5]. Thus, only a
constant field, for which both $S\equiv 0$ and $B\equiv 0$, is an admissible
uniform distortion on a flat surface [10].

The notion of uniform distortion in flat three-dimensional space in [2] has
also been extended to curved spaces in [3] and [4].

In the following sections we study a broader class of distortions in two space
dimensions by relaxing the definition of uniformity; then we present explicit
constructions of two families of such distortions in the plane.

##  3 Quasi-uniform distortions

A _quasi-uniform_ distorsion [6] is represented by a nematic director field
$\bm{n}$ with distortion characteristics in a constant ratio to one another.
More formally, either the distortion characteristics $(S,T,b_{1},b_{2},q)$ are
all zero, but one, or they are all proportional through constants to a nonzero
scalar function $f=f(\bm{x})$ of position $\bm{x}$ in space,

$S=S^{*}f,\quad b_{1}=b^{*}_{1}f,\quad b_{2}=b^{*}_{2}f,\quad T=T^{*}f,\quad
q=q^{*}f,$ (7)

with $S^{*}$, $b^{*}_{1}$, $b^{*}_{2}$, $T^{*}\in\mathbb{R}$ and
$q^{*}\geqslant 0$ all constants. For a quasi-uniform distortion,
decomposition (1) is therefore factorized by $f$ and, by explicitly expanding
$\nabla\bm{n}$ in the distortion frame $(\bm{n}_{1},\bm{n}_{2},\bm{n})$, we
give it the form

$\eqalign{\nabla\bm{n}=f\Bigg{[}&\Bigg{(}\frac{S^{*}}{2}+q^{*}\Bigg{)}\bm{n}_{1}\otimes\bm{n}_{1}-\frac{T^{*}}{2}\bm{n}_{1}\otimes\bm{n}_{2}-b^{*}_{1}\bm{n}_{1}\otimes\bm{n}\\\
&+\frac{T^{*}}{2}\bm{n}_{2}\otimes\bm{n}_{1}+\Bigg{(}\frac{S^{*}}{2}-q^{*}\Bigg{)}\bm{n}_{2}\otimes\bm{n}_{2}-b^{*}_{2}\bm{n}_{2}\otimes\bm{n}\Bigg{]}.}$
(8)

To ease our notation, we shall suppress the superscript ∗ wherever the
distinction between distortion characteristics and their associated constants
need not be specified; in such a case, the distortion characteristics will be
denoted $(fS,fT,fb_{1},fb_{2},fq)$; otherwise, they will be denoted
$(fS^{*},fT^{*},fb^{*}_{1},fb^{*}_{2},fq^{*})$.

In [6], by a direct computation of the distortion characteristics, we showed
the existence of genuine quasi-uniform distortions (which are not uniform).
They are the following _elementary_ fields (see figure 1):

  1. 1.

the _hedgehog_ , represented in a Cartesian frame
$(\bm{e}_{x},\bm{e}_{y},\bm{e}_{z})$ as

$\bm{n}=(x^{2}+y^{2}+z^{2})^{-\frac{1}{2}}(x\bm{e}_{x}+y\bm{e}_{y}+z\bm{e}_{z}),$
(9)

  2. 2.

the _pure bend_
$\bm{n}=(x^{2}+y^{2})^{-\frac{1}{2}}(-y\bm{e}_{x}+x\bm{e}_{y})$,

  3. 3.

the _planar splay_
$\bm{n}=(x^{2}+y^{2})^{-\frac{1}{2}}(x\bm{e}_{x}+y\bm{e}_{y})$, and

  4. 4.

the _planar spiral_

$\bm{n}=(x^{2}+y^{2})^{-\frac{1}{2}}[(x\cos\alpha-y\sin\alpha)\bm{e}_{x}+(x\sin\alpha+y\cos\alpha\bm{e}_{y})],$
(10)

which interpolates between planar splay and pure bend as the parameter
$\alpha$ ranges in the interval $[0,\frac{\pi}{2}]$.

(a) Planar splay. (b) Pure bend. (c) Splay-bend spirals.

Figure 1: Planar quasi-uniform distortions extended to three-dimensional
space. Field lines are depicted in green, while the red $z$-axis is a line
defect.

The question as to the existence of quasi-uniform distortions other than these
can be formally rephrased as the quest for the conditions that a scalar
function $f$ must satisfy for (8) to apply. This formal avenue is undertaken
in B, where we reproduce the approach followed in [2], only requiring quasi-
uniformity, instead of uniformity. Here, we rather prefer a more direct,
constructive approach and start from an instructive somewhat negative result.

###  3.1 Planar quasi-uniform distortions

Consider a planar, non-constant distortion for which

$\bm{n}=\cos\varphi\bm{e}_{x}+\sin\varphi\bm{e}_{y},$ (11)

where $\varphi=\varphi(x,y)$ depends only on the $(x,y)$ coordinates in the
plane, so that $(\nabla\bm{n})\bm{e}_{z}\equiv\bm{0}$. Moreover, since
$\bm{n}\cdot\bm{e}_{z}\equiv 0$, we also have that
$(\nabla\bm{n})^{\mathsf{T}}\bm{e}_{z}\equiv\bm{0}$, and so
$\operatorname{\operatorname{curl}}\bm{n}\parallel\bm{e}_{z}$ and $T\equiv 0$.
By applying both sides of equation (1) to $\bm{e}_{z}$, we then obtain that

$\mathbf{D}\bm{e}_{z}=-\frac{1}{2}S\bm{e}_{z},$ (12)

which by (2) implies that either

$\bm{n}_{2}\equiv\bm{e}_{z}\quad\mathrm{if}\quad
S>0\quad\mathrm{or}\quad\bm{n}_{1}\equiv\bm{e}_{z}\quad\mathrm{if}\quad S<0$
(13)

and, correspondingly,

$S=\pm 2q.$ (14)

In general, we say that a distortion is a _splay-bend_ , whenever $T=0$ and
(14) applies. Thus, any planar distortion is a splay-bend, no matter whether
quasi-uniform or not.

We now show that there is no non-trivial function $\varphi=\varphi(x)$ of a
single variable $x$ that makes $\bm{n}$ in (11) a _modulated_ quasi-uniform
field on the whole plane. This result is obtained by direct inspection of the
distortion characteristics. In fact, by calling $\varphi_{,x}$ the derivative
of $\varphi$, the gradient of the director becomes

$\nabla\bm{n}=\varphi_{,x}(-\sin\varphi\bm{e}_{x}\operatorname{\otimes}\bm{e}_{x}+\cos\varphi\bm{e}_{y}\operatorname{\otimes}\bm{e}_{x})=\varphi_{,x}\bm{n}_{\perp}\operatorname{\otimes}\bm{e}_{x},$
(15)

with
$\bm{n}_{\perp}:=\bm{e}_{z}\times\bm{n}=-\sin\varphi\bm{e}_{x}+\cos\varphi\bm{e}_{y}$.
Since the skew-symmetric part of $\nabla\bm{n}$ is

$\nabla\bm{n}_{\rm
skw}=\frac{\varphi_{,x}\cos\varphi}{2}(\bm{e}_{y}\operatorname{\otimes}\bm{e}_{x}-\bm{e}_{x}\operatorname{\otimes}\bm{e}_{y}),$
(16)

$\operatorname{\operatorname{curl}}\bm{n}=\varphi_{,x}\cos\varphi\bm{e}_{z}$
and $\bm{b}=-\varphi_{,x}\cos\varphi\bm{n}_{\perp}$. Furthermore
$S=-\varphi_{,x}\sin\varphi$ and

$\mathbf{D}=\frac{\varphi_{,x}\sin\varphi}{2}(\bm{e}_{z}\operatorname{\otimes}\bm{e}_{z}-\bm{n}_{\perp}\operatorname{\otimes}\bm{n}_{\perp}),$
(17)

in agreement with (14). As a result, the distortion characteristics
$(fS^{\ast},fT^{\ast},fb_{1}^{\ast},fb_{2}^{\ast},fq^{\ast})$ can be written
by making use of one the following expressions:

$\displaystyle f=-\frac{\varphi_{,x}\sin\varphi}{2},\ S^{*}=2,\
b^{*}_{1}=2\cot\varphi,\ b^{*}_{2}=T^{*}=0,\ q^{*}=1,\qquad$
$\displaystyle\mathrm{if}\ \varphi_{,x}\sin\varphi\leqslant 0,$ (18)
$\displaystyle f=\frac{\varphi_{,x}\sin\varphi}{2},\ S^{*}=-2,\
b^{*}_{1}=T^{*}=0,\ b^{*}_{2}=-2\cot\varphi,\ q^{*}=1,\qquad$
$\displaystyle\mathrm{if}\ \varphi_{,x}\sin\varphi\geqslant 0,$ (19)

and it is clear that $\bm{n}$ is never quasi-uniform, unless $\varphi$ is
constant (which would make $\bm{n}$ constant too).

We note that all planar elementary quasi-uniform distortions, ranging from
pure bend to planar splay, have _defects_ : in the plane, the origin is indeed
a singular point; it extends to an entire line defect in the third dimention
$z$, as illustrated in figure 1.

The lack of modulated quasi-uniform distortions filling the whole plane and
the presence of defects in the most elementary ones suggest that in our quest
we should restrict the domain of $\bm{n}$ and allow for the existence of
defects. In the next two sections, we present the explicit construction of two
families of planar quasi-uniform distortions: one is defined in a half-plane
and prescribed on the boundary, the other includes fields winding outside a
circle with prescribed topological charge. These will identify two separate
mechanisms for relieving frustration in the plane, from a line and from a
circle, respectively. We shall see that not all frustrations can be relieved
in this way; geometric criteria will be given that single out those that can.

##  4 Filling a half-plane

Here we consider the most general family of planar director fields $\bm{n}$ in
(11) by letting $\varphi=\varphi(x,y)$ depend on both coordinates $(x,y)$ in
the plane. The distortion gradient is then given by

$\nabla\bm{n}=-\varphi_{,x}\sin\varphi\bm{e}_{x}\operatorname{\otimes}\bm{e}_{x}-\varphi_{,y}\sin\varphi\bm{e}_{x}\operatorname{\otimes}\bm{e}_{y}+\varphi_{,x}\cos\varphi\bm{e}_{y}\operatorname{\otimes}\bm{e}_{x}+\varphi_{,y}\cos\varphi\bm{e}_{y}\operatorname{\otimes}\bm{e}_{y}.$
(20)

Taking $S=2q$ (for $S=-2q$, we proceed similarly) and letting the distortion
frame comprise the unit vectors (see C.1 for details)

$\bm{n}_{1}=-\sin\varphi\bm{e}_{x}+\cos\varphi\bm{e}_{y}\quad\textrm{and}\quad\bm{n}_{2}=\bm{e}_{z},$
(21)

we express the distortion characteristics as

$S=\varphi_{,y}\cos\varphi-\varphi_{,x}\sin\varphi,\quad
b_{1}=-(\varphi_{,x}\cos\varphi+\varphi_{,y}\sin\varphi),\quad b_{2}=T=0,\quad
q=\frac{S}{2}.\quad$ (22)

Then a quasi-uniform distortion with both $\varphi_{,x}\not\equiv 0$ and
$\varphi_{,y}\not\equiv 0$ must satisfy

$-(\varphi_{,x}\cos\varphi+\varphi_{,y}\sin\varphi)=b^{*}(\varphi_{,y}\cos\varphi-\varphi_{,x}\sin\varphi)$
(23)

for some constant $b^{*}\in\mathbb{R}$, which implies that

$\varphi_{,x}(\cos\varphi-b^{*}\sin\varphi)+\varphi_{,y}(\sin\varphi+b^{*}\cos\varphi)=0.$
(24)

In this way, all distortion characteristics are factorized by the function

${\color[rgb]{0,0,0}{f=\frac{1}{2}(\varphi_{,y}\cos\varphi-\varphi_{,x}\sin\varphi)}}$
(25)

via constants

$S^{*}=2,\quad b^{*}_{1}=2b^{*},\quad b^{*}_{2}=T^{*}=0,\quad q^{*}=1.\quad$
(26)

To solve the quasi-linear partial differential equation (24), we parameterize
$x$, $y$, and $\varphi$ by introducing an auxiliary variable $t\in\mathbb{R}$
and we then follow the propagation of the values $x=x_{0}$, $y=y_{0}$, and
$\varphi_{0}=\varphi(x_{0},y_{0})$ for $t=0$ along the solutions of (24), as
entailed by the classical _method of characteristics_ (see e.g. Sect. 4.8 of
[11], for a fresh didactic exposition666A classical reference is Chapt. 2 of
[12]; for the related Lagrange-Charpit method, the reader is also referred to
[13].), which reduces (24) to the following system of ordinary differential
equations (often referred to as the Lagrange-Charpit system):

$\frac{\operatorname{d}\\!x}{\operatorname{d}\\!t}=\cos\varphi-b^{*}\sin\varphi,\qquad\frac{\operatorname{d}\\!y}{\operatorname{d}\\!t}=\sin\varphi+b^{*}\cos\varphi,\qquad\frac{\operatorname{d}\\!\varphi}{\operatorname{d}\\!t}=0.$
(27)

Therefore, on any characteristic curve

$y=\frac{\sin\varphi_{0}+b^{*}\cos\varphi_{0}}{\cos\varphi_{0}-b^{*}\sin\varphi_{0}}(x-x_{0})\quad\mathrm{and}\quad\varphi(x,y)=\varphi(x_{0},0)=:\varphi_{0}(x_{0}),$
(28)

where $\varphi_{0}(x_{0})$ is the prescribed value of $\varphi$ at $x_{0}$:
all characteristic curves are straight line; they are parallel to $\bm{e}_{y}$
(vertical) for $\tan\varphi_{0}=1/b^{*}$ and parallel to $\bm{e}_{x}$
(horizontal) for $\tan\varphi_{0}=-b^{*}$.

The solution of (24) is uniquely defined only at those points in the plane
belonging precisely to a single characteristic line. Since the slope of a
characteristic line (28) depends on $b^{*}$ (which is constant) and
$\varphi_{0}$ (which propagates unchanged on the entire line), the unique
solution filling the _whole_ plane must have $\varphi$ _constant_ , and so it
generates thorough (20) a constant director field $\bm{n}$.777In three space
dimensions, general compatibility conditions for the function $f$ were given
in [4] for a _regular_ quasi-uniform distortion to fill the whole space (see
their equation (48)). Examples were also given to show that this class in not
empty.

To construct a genuine planar quasi-uniform distortion, we must therefore
prescribe a non-constant $\varphi_{0}$ on the straight line $y=0$, so that
there are no intersections of characteristic lines in the half-plane $y>0$. It
follows from direct inspection of (28) that the only way to achieve this goal
is to ensure that the slope along the line $y=0$,

$M_{\rm
l}:=\frac{\sin\varphi_{0}+b^{*}\cos\varphi_{0}}{\cos\varphi_{0}-b^{*}\sin\varphi_{0}},$
(29)

is a monotonic function of $x_{0}\in\mathbb{R}$. The violation of such a
condition produces a planar domain of non-intersecting characteristic lines
that does _not_ extend to a whole half-plane, as shown in figure 2 where the
boundary condition $\varphi_{0}(x_{0})=\varepsilon\sin(2\pi x_{0}/\ell)$
represents a periodic perturbation of the constant field $\varphi_{0}\equiv
0$, with period $\ell$ and amplitude a (small) parameter $\varepsilon$.

Figure 2: Example of a bounded domain (in white) for a planar and periodic,
quasi-uniform distortion (11). Precisely, in (28), $b^{*}=1$ and
$\varphi_{0}(x_{0})=\pi\sin(\pi x_{0})/10$. The characteristic (straight)
lines where $\varphi_{0}$ propagates unchanged are drawn in blue: they
intersect one another in the (red) region where $\bm{n}$ _cannot_ be uniquely
defined.

Figure 3 illustrates a qualitative study of the slope $M_{\rm l}$ in (29).

(a) Plot of $M_{\rm l}$. (b) Range of corresponding characteristics.

Figure 3: Qualitative study of the slope $M_{\rm l}$ in (29); its graph is
plotted in panel (a). This function vanishes for $\varphi_{0}=-\arctan
b^{*}+k\pi$, with $k\in\mathbb{Z}$, giving rise to horizontal characteristic
lines, while for $\varphi_{0}=-\arctan b^{*}+\pi/2+k\pi$ $M_{\rm l}$ diverges,
jumping from $-\infty$ to $+\infty$ and producing vertical characteristic
lines (red in panel (b)).

It shows how $\varphi_{0}$ is confined to vary in a range of amplitude at most
$\pi$, between $-\arctan b^{*}+k\pi$ and $-\arctan b^{*}+(k-1)\pi$ (for
$k\in\mathbb{Z}$); $\varphi_{0}$ must also be a (not necessarily strictly)
decreasing function of $x_{0}$ in order to select the half-plane $y>0$ as
domain for $\bm{n}$ (an increasing $\varphi_{0}$ would select the half-plane
$y<0$).

These criteria also identify the nematic frustrations localized on the line
$y=0$ that can be relieved quasi-uniformly in the half-plane $y>0$. Thus, a
relivable frustration is, for example,

$\varphi_{0}(x_{0})=-\frac{\pi}{2}(\tanh x_{0}+1)-\arctan b^{*},$ (30)

which propagates to a quasi-uniform distortion whose integral lines are shown
in figure 4(a). Any variation in the rate at which $\varphi_{0}$ varies with
$x_{0}$ would generate a different, but equally relivable quasi-uniform field,
like the one illustrated in figure 4(b).

(a) $\varphi_{0}(x_{0})=-\pi(\tanh x_{0}+1)/2-\arctan 2$. (b)
$\varphi_{0}(x_{0})=-\pi(x_{0}^{5}+1)/2-\arctan 2$.

Figure 4: Integral lines (in green) of planar quasi-uniform distortions in
(11) with $b^{*}=2$ and two choices of relivable frustrations
$\varphi_{0}(x_{0})$. The blue straight lines are the characteristic lines
where $\varphi_{0}$ propagates unchanged: they are vertical at $x_{0}=0$ and
horizontal for $x_{0}\to\pm\infty$. The nematic directors are represented by
short red segments. The field $\bm{n}$ is relieved in the half-plane $y>0$, as
in both panels (a) and (b) $\varphi_{0}$ is a decreasing function of $x_{0}$.

It is interesting to determine the function $f$ that characterizes a
frustration that can be relieved quasi-uniformly in a half-plane. The explicit
expression for $f$ is obtained from (25) in C.1 by a change of variables that
maps $(x,y)$ into $(x_{0},s)$, where $s\in\mathbb{R}$ describes the
characteristic through $(x_{0},0)$, for any given $x_{0}\in\mathbb{R}$. In the
new variables, the line $y=0$ corresponds to $s=0$ and there $f$ is a function
of $x_{0}$ only, which reads as

$f(x_{0},0)=-\frac{1}{2}\frac{\varphi_{0}^{\prime}}{\sin\varphi_{0}+b^{\ast}\cos\varphi_{0}}.$
(31)

On the other hand, along each characteristic,

$\lim_{|s|\to\infty}f(x_{0},s)=0\quad\forall\ x_{0}\in\mathbb{R}.$ (32)

Level sets of $f$ in the plane $(x,y)$ have a clear geometric interpretation:
they represent _restricted loci_ of uniformity, where all distortion
characteristics are constant, although their values can be different on
different level sets. Equation 31 suffices to show that the line of
frustration is _not_ a restricted locus of uniformity. We shall return to this
issue in section 7, in connection with a notion of lower-dimensional
uniformity.

In section 6, we shall provide a further example of relivable frustration and
we shall investigate the behaviour of the corresponding quasi-uniform field
far away from the origin.

##  5 Winding outside a circle

In this section, we present a construction for a class of planar quasi-uniform
distortions winding outside a circle upon which $\bm{n}$ is prescribed. We
consider the $(x,y)$-plane deprived of the region enclosed by the (unit)
circle $\mathbb{S}^{1}$ with center at the origin of a Cartesian frame
$(\bm{e}_{x},\bm{e}_{y})$.888A uniform scaling would map $\mathbb{S}^{1}$ onto
a circle of any selected radius $r_{0}$ with no qualitative consequences on
our development. We imagine $\bm{n}$ prescribed on $\mathbb{S}^{1}$ with a
given _topological charge_ $m$ (also known as the _winding number_) and we ask
whether it can be relieved quasi-uniformly outside $\mathbb{S}^{1}$. Again by
use of the method of characteristics, depending on the value of $m$, we shall
find either the elementary, planar quasi-uniform distortions in figure 1 or
new families of planar quasi-uniform distortions defined outside
$\mathbb{S}^{1}$, some filling the rest of the plane, others living only in a
half-plane.

In the present geometric setting, we find it convenient to rewrite the nematic
field (11) in the local frame $(\bm{e}_{r},\bm{e}_{\vartheta})$ of polar
coordinates $(r,\vartheta)$ as

$\bm{n}=\cos\alpha\bm{e}_{r}+\sin\alpha\bm{e}_{\vartheta},\quad\mathrm{with}\quad\alpha=\alpha(r,\vartheta),\quad\mathrm{for}\quad
r>1,$ (33)

so that the azimuthal angle $\varphi$ between $\bm{n}$ and $\bm{e}_{x}$ is
given by $\varphi=\alpha+\vartheta$, where $\vartheta\in[0,2\pi)$. Here
$\alpha$ represents the _local_ orientation angle of $\bm{n}$ relative to the
radial direction $\bm{e}_{r}$. For the gradient of $\bm{n}$ we then have

$\eqalign{\nabla\bm{n}&=-\alpha_{,r}\sin\alpha\bm{e}_{r}\operatorname{\otimes}\bm{e}_{r}-\frac{1+\alpha_{,\vartheta}}{r}\sin\alpha\bm{e}_{r}\operatorname{\otimes}\bm{e}_{\vartheta}\\\
&\qquad+\alpha_{,r}\cos\alpha\bm{e}_{\vartheta}\operatorname{\otimes}\bm{e}_{r}+\frac{1+\alpha_{,\vartheta}}{r}\cos\alpha\bm{e}_{\vartheta}\operatorname{\otimes}\bm{e}_{\vartheta}}$
(34)

and the distortion characteristics become (see C.2 for details)

$S=\frac{1+\alpha_{,\vartheta}}{r}\cos\alpha-\alpha_{,r}\sin\alpha,\
b_{1}=\alpha_{,r}\cos\alpha+\frac{1+\alpha_{,\vartheta}}{r}\sin\alpha,\
b_{2}=T=0,\ q=\frac{S}{2},$ (35)

valid for $S\geqslant 0$. The natural choice for the factorizing function is
then $f=q$, so that

$S^{*}=2,\quad b^{*}_{2}=T^{*}=0,\quad q^{*}=1.$ (36)

For a genuine quasi-uniform distortion we also require $f$ not to vanish
identically. It follows from (35) that if $f\neq 0$ then

$\frac{1+\alpha_{,\vartheta}}{r}\cos\alpha\neq\alpha_{,r}\sin\alpha$ (37)

and

$\quad
b^{*}_{1}=2\frac{(1+\alpha_{,\vartheta})\sin\alpha+r\alpha_{,r}\cos\alpha}{(1+\alpha_{,\vartheta})\cos\alpha-r\alpha_{,r}\sin\alpha}=2\frac{(1+\alpha_{,\vartheta})\tan\alpha+r\alpha_{,r}}{1+\alpha_{,\vartheta}-r\alpha_{,r}\tan\alpha}.$
(38)

Without loss of generality, we also assume that $b^{*}_{1}\geqslant 0$.111Were
this not the case, we could take $(-\bm{n}_{1},-\bm{n}_{2},\bm{n})$ as
distorsion frame instead of $(\bm{n}_{1},\bm{n}_{2},\bm{n})$, thus recovering
the case $b^{*}_{1}\geqslant 0$.

Quasi-uniformity is then achieved when $b^{*}_{1}=2b^{*}\geqslant 0$ is
constant, that is, when

$r\alpha_{,r}(b^{*}\sin\alpha+\cos\alpha)-\alpha_{,\vartheta}(b^{*}\cos\alpha-\sin\alpha)=b^{*}\cos\alpha-\sin\alpha.$
(39)

By using the method of characteristics, we parametrize $r$, $\vartheta$ and
$\alpha$ in $t$ and we follow the propagation of the values $r=r_{0}$,
$\vartheta=\vartheta_{0}$, and $\alpha_{0}=\alpha(1,\vartheta_{0})$ for $t=0$
along the solutions of (39). The associated Lagrange–Charpit equations are

$\frac{\operatorname{d}\\!r}{\operatorname{d}\\!t}=r(b^{*}\sin\alpha+\cos\alpha),\quad\frac{\operatorname{d}\\!\vartheta}{\operatorname{d}\\!t}=-(b^{*}\cos\alpha-\sin\alpha),\quad\frac{\operatorname{d}\\!\alpha}{\operatorname{d}\\!t}=b^{*}\cos\alpha-\sin\alpha.$
(40)

In the following we focus on the main properties of the characteristic curves
for (40): more details and full computations are collected in D.

By solving the nonparametric equations

$\frac{\operatorname{d}\\!\alpha}{\operatorname{d}\\!\vartheta}=-1\qquad\mathrm{and}\qquad\frac{\operatorname{d}\\!r}{\operatorname{d}\\!\vartheta}=-\frac{r(b^{*}\sin\alpha+\cos\alpha)}{b^{*}\cos\alpha-\sin\alpha}$
(41)

obtained from (40), we arrive at

$\alpha=\varphi_{0}-\vartheta,\quad\mathrm{with}\quad\varphi_{0}(\vartheta_{0}):=\alpha_{0}+\vartheta_{0},\quad\alpha_{0}:=\alpha(1,\vartheta_{0})$
(42)

and

$r=R_{0}|b^{*}\cos(\varphi_{0}-\vartheta)-\sin(\varphi_{0}-\vartheta)|^{-1},\quad\mathrm{with}\quad
R_{0}:=|b^{*}\cos\alpha_{0}-\sin\alpha_{0}|>0.$ (43)

We note that both $\varphi=\alpha+\vartheta$ and
$R:=r|b^{*}\cos\alpha-\sin\alpha|$ are constant along the characteristic
curves. Therefore, by recalling that $x=r\cos\vartheta$ and
$y=r\sin\vartheta$, with the aid of (42), we also see that the quantity

$|(b^{*}\cos\varphi_{0}-\sin\varphi_{0})x+(b^{*}\sin\varphi_{0}+\cos\varphi_{0})y|=R\\\
$ (44)

is constant along a characteristic curve. Thus, any such curve is a straight
line with slope

$M_{\rm
c}:=-\frac{b^{*}\cos\varphi_{0}-\sin\varphi_{0}}{b^{*}\sin\varphi_{0}+\cos\varphi_{0}}$
(45)

through the point $(\cos\vartheta_{0},\sin\vartheta_{0})$ of $\mathbb{S}^{1}$.
Moreover, by (35), the function $f$ can also be written in terms of $\varphi$
and $\alpha$ as

$f=\frac{1}{2}\left(\frac{1}{r}\varphi_{,\vartheta}\cos\alpha-\varphi_{,r}\sin\alpha\right).$
(46)

The domain $\mathcal{D}$ outside $\mathbb{S}^{1}$ where the characteristic
lines do not cross each other is bounded by the characteristic lines tangent
to $\mathbb{S}^{1}$. The latter touch $\mathbb{S}^{1}$ at points
$(\cos\vartheta_{0}^{*},\sin\vartheta_{0}^{*})$, where $\vartheta_{0}^{*}$
solves the equation

$\tan\alpha_{0}(\vartheta_{0}^{*})=-\frac{1}{b^{*}}.$ (47)

Letting $\mathcal{T}(\alpha_{0})$ denote the set of roots of (47) in
$[0,2\pi)$, we can represent $\mathcal{D}$ as

$\mathcal{D}=\Big{\\{}(x,y)\in\mathbb{R}^{2}\,|\,x^{2}+y^{2}>1\Big{\\}}\bigcap_{\vartheta_{0}\in\mathcal{T}(\alpha_{0})}\Big{\\{}(x,y)\in\mathbb{R}^{2}\,|\,x\cos\vartheta_{0}+y\sin\vartheta_{0}<1\Big{\\}}.$
(48)

To provide concrete examples of this construction, we prescribe $\bm{n}$ on
$\mathbb{S}^{1}$ with a given _topological charge_ $m$. Since $\bm{n}$ enjoys
the nematic symmetry, which identifies $\bm{n}$ and $-\bm{n}$, $2m$ is the
integer that counts the number of times the azimuthal angle $\varphi_{0}$
winds around $\mathbb{S}^{1}$ while $\vartheta_{0}$ runs once along it. Thus,
we have that
$\varphi_{0}(\vartheta_{0}+2\pi)=\varphi_{0}(\vartheta_{0})+2m\pi$, and so
$\alpha$ is subject to the following condition on its trace $\alpha_{0}$ on
$\mathbb{S}^{1}$,

$\alpha_{0}(\vartheta_{0}+2\pi)-\alpha_{0}(\vartheta_{0})=2(m-1)\pi.$ (49)

Introducing coordinates $(\vartheta_{0},s)$ in the plane $(x,y)$, with
$s\in\mathbb{R}$ describing the characteristic in (44) with given
$\vartheta_{0}$, we can obtain $f$ from 46 (see C.2); its explicit form is in
general too complicated to be easily interpreted, but on $\mathbb{S}^{1}$ it
simplifies into

$f(\vartheta_{0},0)=\frac{1}{2}\frac{\varphi_{0}^{\prime}}{b^{\ast}\sin\alpha_{0}+\cos\alpha_{0}},$
(50)

which parallels (31), while

$\lim_{|s|\to\infty}f(\vartheta_{0},s)=0\quad\forall\
\vartheta_{0}\in[0,2\pi),$ (51)

which parallels (32). We conclude from (50) that $\mathbb{S}^{1}$ is not a
level set of $f$.

A simple function that satisfies (49) is

$\alpha_{0}(\vartheta_{0})=(m-1)\vartheta_{0}+c_{0}\quad\mathrm{for}\quad\vartheta_{0}\in[0,2\pi),$
(52)

where $c_{0}$ is a constant that for $m\neq 1$ merely affects the field
$\bm{n}$ by a rigid rotation. The corresponding nematic fields $\bm{n}$ are
also known as _Frank’s disclinations_ [14]. For this choice of $\alpha_{0}$,
we have that on each characteristic line,

$\varphi(r,\vartheta)=\alpha(r,\vartheta)+\vartheta=\alpha_{0}(\vartheta_{0})+\vartheta_{0}=\varphi_{0}(\vartheta_{0})=m\vartheta_{0}+c_{0}.$
(53)

It should be noticed that the field $\bm{n}$ generated by the propagation of
$\varphi_{0}$ along characteristics has by continuity the same topological
charge $m$ on any circuit enclosing $\mathbb{S}^{1}$ in $\mathcal{D}$,
insensitive to the fact that the slope of characteristics $M_{\rm c}$ also
depends on $b^{*}$. Indeed, $M_{\rm c}$ takes on any specified value (for
example, $0$) exactly $2m$ times as $\vartheta_{0}$ covers the interval
$[0,2\pi)$, as shown in figure 5.

(a) $m=1/2$.

(b) $m=1$.

(c) $m=3/2$.

Figure 5: Illustrating how the slope $M_{\rm c}$ in (45) depends on $b^{*}$
when $\vartheta_{0}\in[0,2\pi)$ for $\alpha_{0}$ as in (52). In panel (b),
$c_{0}=0$, while in panels (a) and (c) $c_{0}=-3\pi(m-1)/2-\arctan(1/b^{*})$,
so that $\mathcal{T}(\alpha_{0})=\\{3\pi/2\\}$. In all cases, the period of
$M_{\rm c}$ is $2m$, independently on $b^{*}$.

If $m=0$ then $\varphi\equiv c_{0}$ and $\bm{n}$ is everywhere constant and
its domain is the whole plane. If $m=1$ then $\mathcal{D}$ is the whole plane
outside $\mathbb{S}^{1}$, apart from the _resonant_ case where $c_{0}=\arctan
b^{*}$. In this case, characteristics can also be extended inside
$\mathbb{S}^{1}$ and the field lines of $\bm{n}$ are logarithmic spirals
emanating from the origin with constant local angle $\alpha=\arctan b^{*}$;
they range from straight lines (as for the planar splay) when $b^{*}\to 0$ to
concentric circles (as for the pure bend) when $b^{*}\to+\infty$.

To obtain $\mathcal{D}$ when $m\not\in\\{0,1\\}$, we first construct the set
$\mathcal{T}(\alpha_{0})$ for $\alpha_{0}$ as in (52); it comprises $2|m-1|$
elements:

$\mathcal{T}(\alpha_{0})=\Big{\\{}-\frac{\arctan(1/b^{*})+c_{0}}{m-1}+\frac{n\pi}{m-1}\,|\,n=1,\dots,2|m-1|\Big{\\}}.$
(54)

Thus, $\mathcal{D}$ is the (open) regular polyhedron with $2|m-1|$ vertices
circumscribed to $\mathbb{S}^{1}$; figure 6 shows examples of either bounded
and unbounded domains.

(a) $m=-1$.

(b) $m=-1/2$.

(c) $m=0$.

(d) $m=1/2$.

(e) $m=1$.

(f) $m=3/2$.

(g) $m=2$.

(h) $m=5/2$.

(i) $m=3$.

Figure 6: Domain $\mathcal{D}$ in (48) for planar quasi-uniform splay-bend
distortions winding outside $\mathbb{S}^{1}$ for different values of the
topological charge $m$. Directors are represented by red segments, while blue
straight lines are characteristic lines along which $\bm{n}$ propagates
unchanged, starting from the frustration prescribed on $\mathbb{S}^{1}$ with
angle $\alpha$ as in (52): here $c_{0}=0$, while $b^{*}=2$. $\mathcal{D}$ is
the dark blue region bounded by black dashed lines, corresponding to
characteristic lines tangent to $\mathbb{S}^{1}$ at angles
$\vartheta_{0}^{*}\in\mathcal{T}(\alpha_{0})$ in (54). Panels (a), (b), (h),
and (i) provide examples where $\mathcal{D}$ is bounded. In panel (c),
$\bm{n}$ can actually be extended to the whole plane because it is constant,
while in panel (g) this is not possible and the domain is unbounded but in one
direction only. The most interesting cases are shown in panels (d), (e), and
(f): they realize new families of quasi-uniform splay-bend distortions, other
than the well-known quasi-uniform distortions of pure bend, pure splay and
spirals.

The most interesting cases other than $m=1$ are those for $m=1/2$ and $m=3/2$,
where there is only one element in $\mathcal{T}(\alpha_{0})$; the
corresponding nematic fields $\bm{n}$ fill a whole half-plane outside
$\mathbb{S}^{1}$.

To understand better the kind of fields we have thus produced, we study their
extendibility along characteristic lines inside $\mathbb{S}^{1}$ (and possibly
beyond). This construction reveals full extendibility in the case $m=1/2$, but
the extended field $\bm{n}$ turns out to be nothing but a planar spiral as in
(10), only shifted with its defect at the single point on $\mathbb{S}^{1}$
where a characteristic line is tangent (see figure 7).

(a) Characteristic lines. (b) Field lines.

Figure 7: Planar quasi-uniform distortion in (33) winding outside
$\mathbb{S}^{1}$ (black dashed line) with local angle $\alpha$ prescribed as
in (52) with $m=1/2$. Here $b^{*}=2$ and
$c_{0}:=-3\pi(m-1)/2-\arctan(1/b^{*})$, so that
$\mathcal{T}(\alpha_{0})=\\{3\pi/2\\}$. The straight lines are the
characteristics where the azimuthal angle $\varphi$ propagates unchanged. In
panel (a), the characteristic (gray) lines starting from $\mathbb{S}^{1}$ are
extended inside $\mathbb{S}^{1}$ and beyond (blue lines). In panel (b),
directors are represented by short red segments, while the field lines of
$\bm{n}$ are drawn in green: they are logarithmic spirals emanating from the
point of $\mathbb{S}^{1}$ where all the characteristic lines intersect
($\vartheta_{0}=3\pi/2$). The nematic field $\bm{n}$ is the same as the planar
spiral in (10), simply shifted to have its defect at the point $(0,-1)$ in the
$(x,y)$ plane.

A different scenario presents itself for $m=3/2$. Inside $\mathbb{S}^{1}$ the
characteristic lines have plenty of intersections (see figure 8),

(a) Characteristic lines. (b) Field lines.

Figure 8: Planar quasi-uniform distortions as in (33), winding outside
$\mathbb{S}^{1}$ (black dashed line) with local angle $\alpha$ prescribed as
in (52) with $m=3/2$. Here $b^{*}=2$ and
$c_{0}=-3\pi(m-1)/2-\arctan(1/b^{*})$, so that
$\mathcal{T}(\alpha_{0})=\\{3\pi/2\\}$. The straight lines are the
characteristics where the azimuthal angle $\varphi$ propagates unchanged. In
panel (a), the characteristic (gray) lines starting from $\mathbb{S}^{1}$ are
extended inside $\mathbb{S}^{1}$ (blue lines), revealing a large number of
mutual intersections. In panel (b), directors are represented by short red
segments, while the field lines of $\bm{n}$ are drawn in green: they represent
a new family of planar quasi-uniform distortions (parameterized in $b^{*}$)
relieving the frustration of a field with topological charge $m=3/2$ on
$\mathbb{S}^{1}$.

giving rise to a genuinely new family of quasi-uniform distortions
(parameterized in $b^{*}$) relieving in the half-plane $y>-1$ a prescribed
field with charge $m=3/2$ on $\mathbb{S}^{1}$. Similarly, in the case where
$m=1$, a new family of quasi-uniform distortions fill the whole plane outside
$\mathbb{S}^{1}$, as shown in figure 9.

(a) Characteristic lines. (b) Field lines.

Figure 9: Planar quasi-uniform distortions as in (33) winding outside
$\mathbb{S}^{1}$ (black dashed line) with local angle $\alpha$ prescribed as
in (52) with $m=1$. Here $b^{*}=2$ and $c_{0}=0$. The same conventional
representations as in figure 8 apply here for characteristics, field lines,
and directors.

Further examples can be produced by slightly modifying the boundary value
$\alpha_{0}(\vartheta_{0})$, while leaving its winding number unchanged, so
that both the cardinality of $\mathcal{T}(\alpha_{0})$ and the qualitative
appearance of the domain $\mathcal{D}$ remain unaltered. The idea is to
perturb the frustrating boundary condition so as to cause a _crowding_ of the
characteristic lines, making them mutually intersect in more than one point.
An example is provided by the function

$\alpha_{0}(\vartheta_{0})=(m-1)\vartheta_{0}+\frac{m}{3}\sin\vartheta_{0}+c_{0}\quad\mathrm{for}\quad\vartheta_{0}\in[0,2\pi),$
(55)

as illustrated in figure 10.

(a) $m=1/2$.

(b) $m=1$.

(c) $m=3/2$.

Figure 10: Planar quasi-uniform distortions as in (33) winding outside
$\mathbb{S}^{1}$ (black dashed line) with local angle $\alpha_{0}$ as in (55)
and different values of the topological charge $m$. Here $b^{*}=2$ and
$c_{0}=0$ in panel (b), while in both panels (a) and (c) $c_{0}$ has been
tuned to let $\bm{n}$ cover the whole half-plane $y>-1$. The straight (blue)
lines are the characteristics where the azimuthal angle $\varphi$ propagates
unchanged; they intersect in many different points inside $\mathbb{S}^{1}$
(gray lines), thus showing that the corresponding director fields are
different from the spiral in (10).

##  6 Common features

All families of relieving fields $\bm{n}$ found in section 5 tend to be
asymptotically spiraling as in (10) when they can be extended for
$r\to+\infty$. Here, we present some further evidence for this behaviour by
studying the local angle $\alpha$ on different rays emanating from the origin.
In all cases, $\alpha$ tends to a constant determined by $b^{*}$,
independently of the specific ray. A similar conclusion can be drawn for the
families in section 4, thus suggesting a possible universal trait for all
frustrations that can be relieved quasi-uniformly in a half-plane.

Figure 11

(a) $m=1/2$.

(b) $m=1$.

(c) $m=3/2$.

(d) $m=1/2$.

(e) $m=1$.

(f) $m=3/2$.

Figure 11: Behaviour of local angle $\alpha$ for the planar quasi-uniform
distortions in (33) winding outside $\mathbb{S}^{1}$ on different rays
emanating from the origin at different values of $\vartheta$. As the distance
$r$ from the origin diverges to $+\infty$, $\alpha$ tends to $\arctan b^{*}$
(here $b^{*}=2$). In panels (a), (b), and (c), the trace $\alpha_{0}$ of
$\alpha$ on $\mathbb{S}^{1}$ is given by (52), while in panels (d), (e), and
(f), $\alpha_{0}$ is prescribed as in (55). When $m=1/2$ or $m=3/2$, the
(yellow and green) lines corresponding to $\vartheta=7\pi/6$ and
$\vartheta=11\pi/6$ (respectively) stop where the rays intersect the
horizontal line $y=-1$, below which field $\bm{n}$ is not defined. In panel
(b), there is only one (black) line representing $\alpha$, as this is the same
on all rays.

illustrates the behaviour of the local angle $\alpha$: on each ray emanating
from the origin with a fixed angle $\vartheta$ with respect to $\bm{e}_{x}$,
$\alpha$ tends to $\arctan b^{*}$, which is precisely the value constantly
taken on the planar spiral (10).

A similar property is exhibited by the fields constructed in section 4: when
the slope $M_{\rm l}$ in (29) is _strictly_ monotonic and one characteristic
is parallel to the $y$-axis: $\alpha$ tends to $-\arctan b^{*}$, as in the
examples in figure 4. However, whenever the slope is everywhere finite or
somewhere constant, the asymptotic limit of $\alpha$ on rays depends on the
specific ray. An example of this situation is provided by figure 12,

(a) Field lines and characteristics. (b) Local angle $\alpha$ along rays.

Figure 12: Representation of a planar quasi-uniform distortion $\bm{n}$
obtained as a solution of (24) with frustration prescribed on the $x$-axis as
in (56). The characteristic lines (where $\bm{n}$ propagates unaltered) are
parallel to one another for $x_{0}<0$; their inclination on the line $y=0$
(measured by $|M_{\rm l}|$) is first increasing and the decreasing, after
crossing the line $x=0$. As a result, the asymptotic value of the local angle
$\alpha$ depends on the selected ray. In the limit as $r\to+\infty$, $\bm{n}$
reveals a hybrid nature: it is either a constant field or a planar spiral. In
this example, $b^{*}=2$.

where the field $\bm{n}$ is generated by the following frustration prescribed
on the $x$-axis,

$\varphi_{0}(x_{0})=\cases{-\frac{3\pi}{4}\exp(-1/x_{0}^{2})-\arctan
b^{*}+\frac{3\pi}{4}&for $x_{0}\geqslant 0$,\\\ -\arctan
b^{*}+\frac{3\pi}{4}&otherwise.}$ (56)

The quasi-uniform distortions constructed in this paper have another important
feature in common. Quasi-uniformity in the plane has been proved to require
$\bm{n}$ at point $(x,y)$ to make an angle with the characteristic line
passing through $(x,y)$ that remains constant along the characteristic. Such
an inclination represents indeed the constant ratio between splay and bend
along each characteristic. For $\bm{n}$ as in (11), denoting by $\bm{e}_{0}$
the unit vector along the characteristic through $(x_{0},0)$ with slope
$M_{\rm l}$ given by (29), we obtain
$\bm{e}_{0}\cdot\bm{n}=-(1+{b^{*}}^{2})^{-\frac{1}{2}}$, while, for $\bm{n}$
as in (33) and $\bm{e}_{0}$ tangent to the characteristic with slope $M_{\rm
c}$ given by (45), we obtain
$\bm{e}_{0}\cdot\bm{n}=(1+{b^{*}}^{2})^{-\frac{1}{2}}$. Thus, the inclination
of the director field with respect to each characteristic line depends on
$b^{*}$ only.

##  7 One-dimensional uniformity

In this paper, the concept of frustration has been employed with quite an
extensive meaning. Since uniformity for a director field $\bm{n}$ in two space
dimensions is equivalent to have $\bm{n}\equiv\bm{n}_{0}$, we regarded any
director field prescribed on a curve in the plane so as to be incompatible
with a constant extension to the whole plane as a source of frustration. One
may wonder whether there is a sensible, restricted notion of frustration,
which would further illuminate the role of quasi-uniform distortions as means
of frustration relief.

Suppose that $\mathscr{C}$ is a regular curve in the plane and that a director
field $\bm{n}$ is prescribed on it as a function of the arch-length parameter
$s$ for $\mathscr{C}$. Letting $\bm{e}_{z}$ be a unit vector orthogonal to the
plane that contains $\mathscr{C}$, we define
$\bm{n}_{\perp}:=\bm{e}_{z}\times\bm{n}$ as a unit vector field everywhere
orthogonal to $\bm{n}$, so that $(\bm{n},\bm{n}_{\perp})$ plays the role of a
distortion frame in this restricted setting.

Elementary computations show that $\bm{n}^{\prime}=\gamma\bm{n}_{\perp}$,
where a prime denotes differentiation with respect to $s$ and
$\gamma=\gamma(s)$. The intrinsic definition of one-dimensional uniformity
that we introduce requires $\gamma\equiv\gamma_{0}$.

Let now $\bm{t}=\bm{t}(s)$ be the unit tangent to $\mathscr{C}$ and set
$\bm{t}_{\perp}:=\bm{e}_{z}\times\bm{t}$. By representing $\bm{n}$ as

$\bm{n}(s)=\cos\varphi_{0}(s)\bm{t}+\sin\varphi_{0}(s)\bm{t}_{\perp},$ (57)

we easily obtain that

$\bm{n}^{\prime}=(\varphi_{0}^{\prime}+\tau)\bm{n}_{\perp},$ (58)

where $\tau=\tau(s)$ is such that $\bm{t}^{\prime}=\tau\bm{t}_{\perp}$. Thus,
our notion of one-dimensional uniformity reduces to requiring that

$\varphi_{0}^{\prime}+\tau\equiv\gamma_{0}.$ (59)

As expected, this condition puts restrictions on the cases studied above. When
$\mathscr{C}$ is the straight line $y=0$, as in section 4,
$\bm{t}\equiv\bm{e}_{x}$ and $\tau\equiv 0$, so that (59) reduces to
$\varphi_{0}^{\prime}\equiv\gamma_{0}$. When $\mathscr{C}$ is
$\mathbb{S}^{1}$, as in section 5, $\bm{t}=\bm{e}_{\vartheta}$ and $\tau\equiv
1$, so that (59) reduces to
$\varphi_{0}^{\prime}=\alpha_{0}^{\prime}+1\equiv\gamma_{0}$, showing that
whenever (53) applies the frustration imposed on $\mathbb{S}^{1}$ in section 5
can also be interpreted within the restricted meaning introduced here.

For completeness, we now illustrate an example of one-dimensional uniformity
prescribed on a line segment and the ways it has to relax quasi-uniformly in
the plane. We take

$\varphi_{0}=-\frac{\pi}{4}x_{0}\quad\mathrm{for}\quad x_{0}\in[0,1],$ (60)

so that the admissible values for $b^{\ast}$ determined in section 4 obey
$b^{\ast}\geqslant 1$. Figure 13 depicts the outcomes of the analysis
performed following the construction described in section 4.

(a) $b^{\ast}=2$. (b) $b^{\ast}=4$.

Figure 13: Two instances of one-dimensional uniformity on a line segment
relieved in the plane. Characteristics and directors are depicted as customary
in the preceding figures; levels sets of the function $f$ are black curves
with the value they carry marked next to them. As expected from (31), larger
values of $b^{\ast}$ correspond to smaller values of $f$.

Alongside with characteristics and directors, here with the aid of (31) we
also show the level sets of the function $f$. The frustrated field relaxes in
an unbounded quadrilateral based on the interval $[0,1]$. The level sets of
$f$, which represent restricted loci of two-dimensional uniformity, run from
one bounding side to the other and straighten while getting away from the
frustrated base.

##  8 Conclusions

Finding systematic ways for relieving frustration in an ordered medium endowed
with a director field as order descriptor is a topic generally tackled in
energetic terms. Here, we took instead a completely geometric approach to this
problem, identifying quasi-uniformity as a viable notion of frustration
relief. Our study was confined to two space dimensions in flat geometry.

Having first realized that, apart from _planar spirals_ , no non-trivial
quasi-uniform distortion exists in the whole plane, a conclusion that extends
the negative results already known for planar uniform distortions [2, 5], we
endeavoured to find planar quasi-uniform distortions in half a plane.

Two general settings were considered: in one, the director field was
prescribed along a straight line, in the other it was prescribed on the unit
circle. In both cases, we provided a method to construct quasi-uniform
distortions obeying the prescribed frustration. We also identified the
frustrations that can be relieved in a half-plane and those that cannot, thus
turning our method into a selection criterion for relievable frustrations. To
show how such a criterion works, we considered the case where a classical
Frank’s disclination field is prescribed on the unit circle; we proved that
such frustrations relax quasi-uniformly in a half-plane only if the
topological charge is either $m=1$ or $m=3/2$ (the case $m=1/2$ being
trivially a planar spiral).

Quasi-uniform distortions in a half-plane are plenty, but a universal feature
emerged which is common to all the quasi-uniform distortions that we
constructed in this paper: away from the generating frustration, they all tend
to become a planar spiral (being exactly so only in very selected cases).

A number of issues remained unresolved; we mention just two. First, we have
not classified all possible quasi-uniform distortions in half a plane. Second,
we have not attempted to characterize quasi-uniform distortions on curved
surfaces. Accomplishing these tasks would possibly extend to the realm of
quasi-uniformity in two space dimensions results already achieved for
uniformity in dimensions three [2, 3, 4] and two [5], respectively.

This research was performed under the auspices of the Italian _INdAM (GNFM)_.
A.P. wishes to acknowledge financial support from the Italian _MIUR_ through
European Programmes _REACT EU 2014-2020_ and _PON 2014-2020
CCI2014IT16M2OP005_. Both authors are grateful to the anonymous Reviewers of
this paper for their stimulating insights and suggestions.

##  Appendix A The Oseen-Frank energy

For ordinary nematic liquid crystals, the undistorted ground state coincides
with a constant director field $\bm{n}$: the molecular orientation is the same
at each point occupied by the material, with no distortion.

The elastic free energy penalizes any distortion that would make
$\nabla\bm{n}\neq\mathbf{0}$. In the classical elastic theory formulated by
Oseen [15] and Frank [14], the cost for the deviation from the ground state is
measured by the free energy density

$\eqalign{W_{\mathrm{OF}}(\bm{n},\nabla\bm{n})&:=\frac{1}{2}K_{11}(\operatorname{div}\bm{n})^{2}+\frac{1}{2}K_{22}(\bm{n}\cdot\operatorname{\operatorname{curl}}\bm{n})^{2}\\\
&\qquad+\frac{1}{2}K_{33}\big{|}\bm{n}\times\operatorname{\operatorname{curl}}\bm{n}\big{|}^{2}+K_{24}\Big{[}\tr(\nabla\bm{n})^{2}-(\operatorname{div}\bm{n})^{2}\Big{]},}$
(61)

where each term corresponds to a specific elastic distortion and the
nonnegative coefficients $K_{11}$, $K_{22}$, $K_{33}$ and $K_{24}$ are Frank’s
elastic constants of _splay_ , _bend_ , _twist_ and _saddle-splay_ ,
respectively. Equation (61) delivers the most general frame-indifferent
function $W(\bm{n},\nabla\bm{n})$, at most quadratic in $\nabla\bm{n}$, which
enjoys the nematic symmetry (see, for example, Chapt. 3 of [16]).

By use of (5) and denoting $B:=|\bm{b}|$, $W_{\mathrm{OF}}$ takes the
following equivalent form (see [8]),

$W_{\mathrm{OF}}=\frac{1}{2}(K_{11}-K_{24})S^{2}+\frac{1}{2}(K_{22}-K_{24})T^{2}+\frac{1}{2}K_{33}B^{2}+K_{24}(2q^{2}),$
(62)

which is semipositive definite whenever the elastic constants satisfy
_Ericksen’s inequalities_ [17],

$K_{11}\geqslant K_{24}\geqslant 0,\quad K_{22}\geqslant K_{24}\geqslant
0,\quad K_{33}\geqslant 0.$ (63)

##  Appendix B Compatibility conditions

In this Appendix we provide a set of compatibility conditions that a function
$f$ must satisfy in order to be the common factor in (8) for a quasi-uniform
distortion. We first arrive at a set of nine conditions for a general field
$\bm{n}$ in three-dimensional space, then we specialize them to the case of a
planar quasi-uniform distortion.

###  B.1 General three-dimensional case

The orthonormality of the distortion frame $(\bm{n}_{1},\bm{n}_{2},\bm{n})$
requires the gradients $\nabla\bm{n}_{1}$, $\nabla\bm{n}_{2}$, and
$\nabla\bm{n}$ to be represented by three vectors $\bm{c}_{1}$, $\bm{c}_{2}$,
and $\bm{d}$ (called the _connectors_ in [2]) through the equations999The
method of connectors is in a way intermediate between Cartan’s method of
moving frames [18] employed in [3, 4] and classical vector calculus employed
in [19], being perhaps closer in spirit to the latter.

$\nabla\bm{n}=\bm{n}_{1}\operatorname{\otimes}\bm{c}_{1}+\bm{n}_{2}\operatorname{\otimes}\bm{c}_{2},\
\nabla\bm{n}_{1}=-\bm{n}\operatorname{\otimes}\bm{c}_{1}+\bm{n}_{2}\operatorname{\otimes}\bm{d},\
\nabla\bm{n}_{2}=-\bm{n}\operatorname{\otimes}\bm{c}_{2}-\bm{n}_{1}\operatorname{\otimes}\bm{d}.$
(64)

Connectors $\bm{c}_{1}$ and $\bm{c}_{2}$ are completely determined by
demanding that $\nabla\bm{n}$ be given as in (8),

$\bm{c}_{1}=f\Bigg{[}\Bigg{(}\frac{S^{*}}{2}+q^{*}\Bigg{)}\bm{n}_{1}-\frac{T^{*}}{2}\bm{n}_{2}-b^{*}_{1}\bm{n}\Bigg{]},\quad\bm{c}_{2}=f\Bigg{[}\frac{T^{*}}{2}\bm{n}_{1}+\Bigg{(}\frac{S^{*}}{2}-q^{*}\Bigg{)}\bm{n}_{2}-b^{*}_{2}\bm{n}\Bigg{]},$
(65)

while the third connector, which we decompose as
$\bm{d}=d_{1}\bm{n}_{1}+d_{2}\bm{n}_{2}+d_{3}\bm{n}$, remains undetermined.

The existence of a quasi-uniform distortion filling the whole space requires
that both fields $\bm{n}$ and $\bm{n}_{1}$ (and so also
$\bm{n}_{2}=\bm{n}\times\bm{n}_{1}$) are uniquely defined everywhere, which,
under smoothness assumptions, amounts to require that both gradients
$\nabla\bm{n}$ and $\nabla\bm{n}_{1}$ be integrable. In a flat space, the
latter request is equivalent to the symmetry in the last two legs of the
second gradients

$\displaystyle\eqalign{\nabla^{2}\bm{n}&=\bm{n}_{1}\operatorname{\otimes}\nabla\bm{c}_{1}-\bm{n}\operatorname{\otimes}\bm{c}_{1}\operatorname{\otimes}\bm{c}_{1}+\bm{n}_{2}\operatorname{\otimes}\bm{c}_{1}\operatorname{\otimes}\bm{d}+\bm{n}_{2}\operatorname{\otimes}\nabla\bm{c}_{2}\\\
&\qquad-\bm{n}\operatorname{\otimes}\bm{c}_{2}\operatorname{\otimes}\bm{c}_{2}-\bm{n}_{1}\operatorname{\otimes}\bm{c}_{2}\operatorname{\otimes}\bm{d},}$
(68)
$\displaystyle\eqalign{\nabla^{2}\bm{n}_{1}&=-\bm{n}\operatorname{\otimes}\nabla\bm{c}_{1}-\bm{n}_{1}\operatorname{\otimes}\bm{c}_{1}\operatorname{\otimes}\bm{c}_{1}-\bm{n}_{2}\operatorname{\otimes}\bm{c}_{1}\operatorname{\otimes}\bm{c}_{2}+\bm{n}_{2}\operatorname{\otimes}\nabla\bm{d}\\\
&\qquad-\bm{n}\operatorname{\otimes}\bm{d}\operatorname{\otimes}\bm{c}_{2}-\bm{n}_{1}\operatorname{\otimes}\bm{d}\operatorname{\otimes}\bm{d}.}$
(71)

Such integrability conditions reduce to the symmetry of the following six
second-order tensors obtained by contraction of the first leg of the third-
order tensors in (68) with the vectors of the frame
$(\bm{n}_{1},\bm{n}_{2},\bm{n})$,

$\displaystyle\bm{n}_{1}\cdot\nabla^{2}\bm{n}$
$\displaystyle=\nabla\bm{c}_{1}-\bm{c}_{2}\operatorname{\otimes}\bm{d},$ (72)
$\displaystyle\bm{n}_{2}\cdot\nabla^{2}\bm{n}$
$\displaystyle=\bm{c}_{1}\operatorname{\otimes}\bm{d}+\nabla\bm{c}_{2},$ (73)
$\displaystyle\bm{n}\cdot\nabla^{2}\bm{n}$
$\displaystyle=-\bm{c}_{1}\operatorname{\otimes}\bm{c}_{1}-\bm{c}_{2}\operatorname{\otimes}\bm{c}_{2},$
(74) $\displaystyle\bm{n}_{1}\cdot\nabla^{2}\bm{n}_{1}$
$\displaystyle=-\bm{c}_{1}\operatorname{\otimes}\bm{c}_{1}-\bm{d}\operatorname{\otimes}\bm{d},$
(75) $\displaystyle\bm{n}_{2}\cdot\nabla^{2}\bm{n}_{1}$
$\displaystyle=-\bm{c}_{1}\operatorname{\otimes}\bm{c}_{2}+\nabla\bm{d},$ (76)
$\displaystyle\bm{n}\cdot\nabla^{2}\bm{n}_{1}$
$\displaystyle=-\nabla\bm{c}_{1}-\bm{d}\operatorname{\otimes}\bm{c}_{2}.$ (77)

Both tensors in (74) and (75) are already symmetric; the symmetry requirements
for the remaining tensors amount to $12$ scalar equations, not all of which
turn out to be independent.

Keeping in mind that $f$ depends on position $\bm{x}$, we write $\nabla
f=f_{1}\bm{n}_{1}+f_{2}\bm{n}_{2}+f_{3}\bm{n}$ and set $\nabla
d_{1}=d_{11}\bm{n}_{1}+d_{12}\bm{n}_{2}+d_{13}\bm{n}$, $\nabla
d_{2}=d_{21}\bm{n}_{1}+d_{22}\bm{n}_{2}+d_{23}\bm{n}$, and $\nabla
d_{3}=d_{31}\bm{n}_{1}+d_{32}\bm{n}_{2}+d_{33}\bm{n}$, finally arriving at the
following set of nine compatibility conditions,

$\displaystyle\frac{T}{2}f_{1}+\Bigg{(}\frac{S}{2}+q\Bigg{)}f_{2}=-f^{2}b_{1}T+2fqd_{1},$
(78) $\displaystyle
b_{1}f_{1}+\Bigg{(}\frac{S}{2}+q\Bigg{)}f_{3}=f^{2}\Bigg{[}\frac{T^{2}}{4}-\Bigg{(}\frac{S}{2}+q\Bigg{)}^{2}-b_{1}^{2}\Bigg{]}+fb_{2}d_{1},$
(79) $\displaystyle
b_{1}f_{2}-\frac{T}{2}f_{3}=f^{2}\Bigg{(}\frac{ST}{2}-b_{1}b_{2}\Bigg{)}+f(b_{2}d_{2}-2qd_{3}),$
(80)
$\displaystyle\Bigg{(}\frac{S}{2}-q\Bigg{)}f_{1}-\frac{T}{2}f_{2}=f^{2}b_{2}T+2fqd_{2},$
(81) $\displaystyle
b_{2}f_{1}+\frac{T}{2}f_{3}=-f^{2}\Bigg{(}\frac{ST}{2}+b_{1}b_{2}\Bigg{)}-f(b_{1}d_{1}+2qd_{3}),$
(82) $\displaystyle
b_{2}f_{2}+\Bigg{(}\frac{S}{2}-q\Bigg{)}f_{3}=f^{2}\Bigg{[}\frac{T^{2}}{4}-\Bigg{(}\frac{S}{2}-q\Bigg{)}^{2}-b_{2}^{2}\Bigg{]}-fb_{1}d_{2},$
(83) $\displaystyle
f^{2}\Bigg{(}\frac{S^{2}}{4}-q^{2}+\frac{T^{2}}{4}\Bigg{)}=-fTd_{3}-d_{1}^{2}-d_{2}^{2}+d_{12}-d_{21},$
(84) $\displaystyle
f^{2}\Bigg{[}b_{1}\frac{T}{2}-b_{2}\Bigg{(}\frac{S}{2}+q\Bigg{)}\Bigg{]}=f\Bigg{(}\frac{S}{2}+q\Bigg{)}d_{1}+f\frac{T}{2}d_{2}-fb_{1}d_{3}-d_{2}d_{3}+d_{13}-d_{31},$
(85) $\displaystyle
f^{2}\Bigg{[}b_{1}\Bigg{(}\frac{S}{2}-q\Bigg{)}+b_{2}\frac{T}{2}\Bigg{]}=f\Bigg{(}\frac{S}{2}-q\Bigg{)}d_{2}-f\frac{T}{2}d_{1}-fb_{2}d_{3}+d_{1}d_{3}+d_{23}-d_{32},$
(86)

where $(fS,fT,fb_{1},fb_{2},fq)$, with $(S,T,b_{1},b_{2},q)$ all _constants_ ,
are the corresponding distortion characteristics. These equations are
equivalent versions, fit to the present purpose, of the compatibility
conditions considered in [19]. As for using them as sufficient conditions to
determine quasi-uniform distortions, the same caveats issued in [19] also
apply here, as witnessed by the limited use we make of them below.

###  B.2 Spatial twist-bend

Here we show that conditions (78)–(86) are consistent with the family of
_heliconical_ quasi-uniform distortions constructed in [4], which in a
Cartesian frame $(\bm{e}_{x},\bm{e}_{y},\bm{e}_{z})$ can be described as

$\bm{n}=\sin\alpha\cos g\bm{e}_{x}\pm\sin\alpha\sin
g\bm{e}_{y}+\cos\alpha\bm{e}_{z},$ (87)

with $g=g(z)$ and $\alpha\in\mathbb{R}$ a constant. By writing

$\nabla\bm{n}=g_{,z}\sin\alpha[-\sin
g\bm{e}_{x}\operatorname{\otimes}\bm{e}_{z}\pm\cos
g\bm{e}_{y}\operatorname{\otimes}\bm{e}_{z}]$ (88)

and by identifying the distortion frame $(\bm{n}_{1},\bm{n}_{2},\bm{n})$ with

$\eqalign{\bm{n}_{1}&=\frac{1}{\sqrt{2}}[(\cos\alpha\cos g\pm\sin
g)\bm{e}_{x}+(\cos\alpha\sin g\mp\cos g)\bm{e}_{y}-\sin\alpha\bm{e}_{z}],\\\
\bm{n}_{2}&=\frac{1}{\sqrt{2}}[(\cos\alpha\cos g\mp\sin
g)\bm{e}_{x}+(\cos\alpha\sin g\pm\cos g)\bm{e}_{y}-\sin\alpha\bm{e}_{z}],}$
(89)

it becomes a simple matter to check by direct inspection that the field in
(87) is quasi-uniform, as its distortion characteristics can be written as
$(fS^{\ast},fT^{\ast},fb_{1}^{\ast},fb_{2}^{\ast},fq^{\ast})$ with

$S^{*}=0,\quad T^{*}=\mp\sin^{2}\alpha,\quad
q^{*}=\frac{\sin^{2}\alpha}{2}=\mp\frac{T^{*}}{2},\quad
b^{*}_{1}=-b^{*}_{2}=\frac{\sin\alpha\cos\alpha}{\sqrt{2}},f=g_{,z}.$ (90)

Since $S=0$ and $T=\pm 2q$, the field in (87) is a twist-bend quasi-uniform
distortion. By extracting the connector $\bm{d}$ from (89) and projecting
$\nabla f$ on the distortion frame $(\bm{n}_{1},\bm{n}_{2},\bm{n})$, we arrive
at the following equations,

$\displaystyle\nabla
f=g_{,zz}\bm{e}_{z}=\frac{g_{,zz}}{\sqrt{2}}[-\sin\alpha(\bm{n}_{1}+\bm{n}_{2})+{\sqrt{2}}\cos\alpha\bm{n}],$
(91) $\displaystyle f_{1}=f_{2}=-\frac{g_{,zz}}{\sqrt{2}}\sin\alpha,\quad
f_{3}=g_{,zz}\cos\alpha,$ (92)
$\displaystyle\bm{d}=g_{,z}\cos\alpha\bm{e}_{z}=\frac{g_{,z}}{\sqrt{2}}\cos\alpha[-\sin\alpha(\bm{n}_{1}+\bm{n}_{2})+{\sqrt{2}}\cos\alpha\bm{n}],$
(93) $\displaystyle\nabla d_{1}=\nabla
d_{2}=-\frac{\sin\alpha\cos\alpha}{\sqrt{2}}\nabla f,\quad\nabla
d_{3}=\cos^{2}\alpha\nabla f,$ (94)

whose use in the above compatibility conditions shows that they are
identically satisfied.

###  B.3 Planar splay-bend

In general, a planar distortion is a splay-bend (with $S=0$ and $T=\pm 2q$).
For it to be quasi-uniform, it must also satisfy the compatibility conditions
(78)–(86), which reduce to

$\displaystyle\Bigg{(}\frac{S}{2}+q\Bigg{)}f_{2}=2fqd_{1},$ (95)
$\displaystyle
b_{1}f_{1}+\Bigg{(}\frac{S}{2}+q\Bigg{)}f_{3}=-f^{2}\Bigg{[}\Bigg{(}\frac{S}{2}+q\Bigg{)}^{2}+b_{1}^{2}\Bigg{]}+fb_{2}d_{1},$
(96) $\displaystyle b_{1}f_{2}=-f^{2}b_{1}b_{2}+f(b_{2}d_{2}-2qd_{3}),$ (97)
$\displaystyle\Bigg{(}\frac{S}{2}-q\Bigg{)}f_{1}=2fqd_{2},$ (98)
$\displaystyle b_{2}f_{1}=-f^{2}b_{1}b_{2}-f(b_{1}d_{1}+2qd_{3}),$ (99)
$\displaystyle
b_{2}f_{2}+\Bigg{(}\frac{S}{2}-q\Bigg{)}f_{3}=-f^{2}\Bigg{[}\Bigg{(}\frac{S}{2}-q\Bigg{)}^{2}+b_{2}^{2}\Bigg{]}-fb_{1}d_{2},$
(100) $\displaystyle
f^{2}\Bigg{(}\frac{S^{2}}{4}-q^{2}\Bigg{)}=-d_{1}^{2}-d_{2}^{2}+d_{12}-d_{21},$
(101)
$\displaystyle-f^{2}b_{2}\Bigg{(}\frac{S}{2}+q\Bigg{)}=f\Bigg{(}\frac{S}{2}+q\Bigg{)}d_{1}-fb_{1}d_{3}-d_{2}d_{3}+d_{13}-d_{31},$
(102) $\displaystyle
f^{2}b_{1}\Bigg{(}\frac{S}{2}-q\Bigg{)}=f\Bigg{(}\frac{S}{2}-q\Bigg{)}d_{2}-fb_{2}d_{3}+d_{1}d_{3}+d_{23}-d_{32}.$
(103)

Since for a planar field either $\bm{n}_{1}$ or $\bm{n}_{2}$ is constant, by
(64), the connector $\bm{d}$ vanishes identically. Furthermore, letting $f$ be
different from zero, we see from (14) that (101) is identically satisfied,
while (102) and (103) require that either $b^{*}_{2}$ or $b^{*}_{1}$ vanishes.
More specifically, if for $\bm{n}_{1}\equiv\bm{e}_{z}$ then by (64)
$\bm{c}_{1}=\bm{0}$, which by (65) implies that $b^{*}_{1}=0$ and
$S^{*}=-2q^{*}$, coherently with (14). Conditions (98) and (99) then read as

$2q^{*}f_{1}=0\quad\mathrm{and}\quad b^{*}_{2}f_{1}=0,$ (104)

respectively, whence $f_{1}=0$ (otherwise $\bm{n}$ is constant), and by (100)

$b^{*}_{2}f_{2}-2q^{*}f_{3}=-f^{2}\big{[}4(q^{*})^{2}+(b^{*}_{2})^{2}\big{]}.$
(105)

On the other hand, if $\bm{n}_{2}\equiv\bm{e}_{z}$ then
$\nabla\bm{n}_{2}=\bm{0}$ and $\bm{c}_{2}=\bm{0}$, again by (64), whence it
follows from (65) that $b^{*}_{2}=0$ and $S^{*}=2q^{*}$. In this case, the
nontrivial compatibility conditions are (95) and (97), which imply $f_{2}=0$,
and also (96), which reads as

$b^{*}_{1}f_{1}+2q^{*}f_{3}=-f^{2}\big{[}4(q^{*})^{2}+(b^{*}_{1})^{2}\big{]}.$
(106)

The difference between (105) and (106) only lies in the conventional (but
useful) requirement that $\bm{n}_{1}$ is the eigenvector of $\mathbf{D}$ with
_positive_ eigenvalue $q$; we shall consider these two cases as essentially
the same: in both of them, the quasi-uniformity requirement translates into
basically the same equation, of which we study now two simple consequences.

For $q=0$, by (14), the field $\bm{n}$ is a pure bend distortion and, by
letting (without loss of generality) $\bm{b}=b\bm{n}_{1}$ and setting $f=b$
(whence $b^{*}=1$), we obtain from (106) that $f_{1}=-f^{2}$, together with
$f_{2}=0$.

For $q>0$, we can let $f=q$, so that $q^{*}=1$. Then, assuming that
$\bm{b}\neq\bm{0}$, we arrive at the following dichotomy,

$\displaystyle\mathrm{either}\quad S^{*}=-2,\quad b^{*}_{1}=f_{1}=0,\quad
b^{*}_{2}f_{2}-2f_{3}=-f^{2}\big{[}4+(b^{*}_{2})^{2}\big{]}$ (107)
$\displaystyle\mathrm{or}\quad S^{*}=2,\quad b^{*}_{2}=f_{2}=0,\quad
b^{*}_{1}f_{1}+2f_{3}=-f^{2}\big{[}4+(b^{*}_{1})^{2}\big{]}.$ (108)

Spirals as in (10) fall into (107) when $\cos\alpha>0$ and into (108) when
$\cos\alpha<0$.

##  Appendix C Planar distortion characteristics

In this Appendix, we collect a number of details concerning the distortion
characteristics of the planar fields considered in sections 4 and 5 of the
main text.

###  C.1 Frustrated line

We start from the gradient decomposition in (1) for the director field
$\bm{n}$ represented by (11). It readily follows from the latter that

$\eqalign{\mathbf{P}(\bm{n})=\mathbf{I}-\bm{n}\operatorname{\otimes}\bm{n}&=\sin^{2}\varphi\bm{e}_{x}\operatorname{\otimes}\bm{e}_{x}-\sin\varphi\cos\varphi(\bm{e}_{x}\operatorname{\otimes}\bm{e}_{y}+\bm{e}_{y}\operatorname{\otimes}\bm{e}_{x})\\\
&\qquad+\cos^{2}\varphi\bm{e}_{y}\operatorname{\otimes}\bm{e}_{y}+\bm{e}_{z}\operatorname{\otimes}\bm{e}_{z}}$
(109)

and from (20) that

$S=\operatorname{div}\bm{n}=-(\varphi_{,x}\sin\varphi-\varphi_{,y}\cos\varphi)$
(110)

and

$\bm{b}=-(\nabla\bm{n})\bm{n}=-(\varphi_{,x}\cos\varphi+\varphi_{,y}\sin\varphi)(-\sin\varphi\bm{e}_{x}+\cos\varphi\bm{e}_{y}).$
(111)

Hence

$\eqalign{\bm{b}\operatorname{\otimes}\bm{n}=(\varphi_{,x}\cos\varphi+\varphi_{,y}\sin\varphi)&\big{[}\sin\varphi\cos\varphi(\bm{e}_{x}\operatorname{\otimes}\bm{e}_{x}-\bm{e}_{y}\operatorname{\otimes}\bm{e}_{y})\\\
&\quad+\sin^{2}\varphi\bm{e}_{x}\operatorname{\otimes}\bm{e}_{y}-\cos^{2}\varphi\bm{e}_{y}\operatorname{\otimes}\bm{e}_{x}\big{]}.}$
(112)

Moreover, the skew-symmetric part $\nabla\bm{n}_{\rm skw}$ of $\nabla\bm{n}$
is given by

$2\nabla\bm{n}_{\rm
skw}:=\nabla\bm{n}-\nabla\bm{n}^{\top}=(\varphi_{,x}\cos\varphi+\varphi_{,y}\sin\varphi)(\bm{e}_{y}\operatorname{\otimes}\bm{e}_{x}-\bm{e}_{x}\operatorname{\otimes}\bm{e}_{y}),$
(113)

which delivers
$\operatorname{\operatorname{curl}}\bm{n}=(\varphi_{,x}\cos\varphi+\varphi_{,y}\sin\varphi)\bm{e}_{z}$
(and $T=0$).

We can now use (1) to derive

$\eqalign{\mathbf{D}&=\nabla\bm{n}+\bm{b}\operatorname{\otimes}\bm{n}-\frac{S}{2}\mathbf{P}(\bm{n})\\\
&=\frac{\varphi_{,x}\sin\varphi-\varphi_{,y}\cos\varphi}{2}\big{[}\sin\varphi\cos\varphi(\bm{e}_{x}\operatorname{\otimes}\bm{e}_{y}+\bm{e}_{y}\operatorname{\otimes}\bm{e}_{x})\\\
&\qquad-\sin^{2}\varphi\bm{e}_{x}\operatorname{\otimes}\bm{e}_{x}-\cos^{2}\varphi\bm{e}_{y}\operatorname{\otimes}\bm{e}_{y}+\bm{e}_{z}\operatorname{\otimes}\bm{e}_{z}\big{]},}$
(114)

whose eigenvalues are
$\pm\frac{1}{2}(\varphi_{,x}\sin\varphi-\varphi_{,y}\cos\varphi)$. Therefore,
when $\varphi_{,x}\sin\varphi>\varphi_{,y}\cos\varphi$

$q=-\frac{S}{2}=\frac{\varphi_{,x}\sin\varphi-\varphi_{,y}\cos\varphi}{2}$
(115)

and the distortion frame $(\bm{n}_{1},\bm{n}_{2},\bm{n})$ is given by

$\bm{n}_{1}:=\bm{e}_{z},\quad\bm{n}_{2}:=\sin\varphi\bm{e}_{x}-\cos\varphi\bm{e}_{y},\quad\bm{n}:=\cos\varphi\bm{e}_{x}+\sin\varphi\bm{e}_{y},$
(116)

while for $\varphi_{,x}\sin\varphi<\varphi_{,y}\cos\varphi$

$q=\frac{S}{2}=-\frac{\varphi_{,x}\sin\varphi-\varphi_{,y}\cos\varphi}{2}$
(117)

and

$\bm{n}_{1}:=-\sin\varphi\bm{e}_{x}+\cos\varphi\bm{e}_{y},\quad\bm{n}_{2}:=\bm{e}_{z},\quad\bm{n}:=\cos\varphi\bm{e}_{x}+\sin\varphi\bm{e}_{y}.$
(118)

In the very special case where
$\varphi_{,x}\sin\varphi=\varphi_{,y}\cos\varphi$ both $S$ and $q$ vanish, and
the field carries a pure bend distortion (which is of course quasi-uniform)
with

$b_{1}=-(\varphi_{,x}\cos\varphi+\varphi_{,y}\sin\varphi),\qquad b_{2}=0$
(119)

and distortion frame as in (118).

To obtain $f$ from (25), we first parameterize the characteristic described by
(28) as follows101010It should be noted that neither here nor in (132) $s$
represents the arc-length along characteristics.

$\cases{x=x_{0}+s=:\xi(x_{0},s),\\\
y=\frac{\sin\varphi_{0}+b^{\ast}\cos\varphi_{0}}{\cos\varphi_{0}-b^{\ast}\sin\varphi_{0}}s=:\eta(x_{0},s),}$
(120)

and then use the identity
$\varphi(\xi(x_{0},s),\eta(x_{0},s))=\varphi_{0}(x_{0})$ to derive the linear
system

$\cases{\varphi_{,x}\xi_{,s}+\varphi_{,y}\eta_{,s}=0,\\\
\varphi_{,x}\xi_{,x_{0}}+\varphi_{,y}\eta_{,x_{0}}=\varphi_{0}^{\prime},}$
(121)

in $(\varphi_{,x},\varphi_{,y})$. Solving (121), we readily arrive at

$f(x_{0},s)=\frac{1}{2}\frac{\varphi_{0}^{\prime}}{\frac{1+(b^{\ast})^{2}}{\cos\varphi_{0}-b^{\ast}\sin\varphi_{0}}s\varphi_{0}^{\prime}-(\sin\varphi_{0}+b^{\ast}\cos\varphi_{0})},$
(122)

from which both (31) and (32) in the main text follow.

###  C.2 Frustrated circle

For the director field in (33), we easily see that

$\mathbf{P}(\bm{n})=\sin^{2}\alpha\bm{e}_{r}\operatorname{\otimes}\bm{e}_{r}-\sin\alpha\cos\alpha(\bm{e}_{r}\operatorname{\otimes}\bm{e}_{\vartheta}+\bm{e}_{\vartheta}\operatorname{\otimes}\bm{e}_{r})+\cos^{2}\alpha\bm{e}_{\vartheta}\operatorname{\otimes}\bm{e}_{\vartheta}+\bm{e}_{z}\operatorname{\otimes}\bm{e}_{z}.$
(123)

It also follows from (34) that

$\bm{b}=\Bigg{(}\alpha_{,r}\cos\alpha+\frac{1+\alpha_{,\vartheta}}{r}\sin\alpha\Bigg{)}(\sin\alpha\bm{e}_{r}-\cos\alpha\bm{e}_{\vartheta})$
(124)

and

$\eqalign{\bm{b}\operatorname{\otimes}\bm{n}=\Bigg{(}\alpha_{,r}\cos\alpha+\frac{1+\alpha_{,\vartheta}}{r}\sin\alpha\Bigg{)}&\big{(}\sin\alpha\cos\alpha\bm{e}_{r}\operatorname{\otimes}\bm{e}_{r}-\cos^{2}\alpha\bm{e}_{\vartheta}\operatorname{\otimes}\bm{e}_{r}\\\
&\quad+\sin^{2}\alpha\bm{e}_{r}\operatorname{\otimes}\bm{e}_{\vartheta}-\sin\alpha\cos\alpha\bm{e}_{\vartheta}\operatorname{\otimes}\bm{e}_{\vartheta}\big{)}.}$
(125)

Moreover,

$2\nabla\bm{n}_{\rm
skw}=\Bigg{(}\alpha_{,r}\cos\alpha+\frac{1+\alpha_{,\vartheta}}{r}\sin\alpha\Bigg{)}(\bm{e}_{\vartheta}\operatorname{\otimes}\bm{e}_{r}-\bm{e}_{r}\operatorname{\otimes}\bm{e}_{\vartheta})$
(126)

and

$\operatorname{\operatorname{curl}}\bm{n}=\Bigg{(}\alpha_{,r}\cos\alpha+\frac{1+\alpha_{,\vartheta}}{r}\sin\alpha\Bigg{)}\bm{e}_{z}.$
(127)

Bu use of (1), we then arrive at

$\eqalign{\mathbf{D}=\frac{1}{2}\Bigg{(}\frac{1+\alpha_{,\vartheta}}{r}\cos\alpha&-\alpha_{,r}\sin\alpha\Bigg{)}\big{(}\sin^{2}\alpha\bm{e}_{r}\operatorname{\otimes}\bm{e}_{r}-\sin\alpha\cos\alpha\bm{e}_{r}\operatorname{\otimes}\bm{e}_{\vartheta}\\\
&\quad-\sin\alpha\cos\alpha\bm{e}_{\vartheta}\operatorname{\otimes}\bm{e}_{r}+\cos^{2}\alpha\bm{e}_{\vartheta}\operatorname{\otimes}\bm{e}_{\vartheta}-\bm{e}_{z}\operatorname{\otimes}\bm{e}_{z}\big{)},}$
(128)

whose eigenvalues are

$\pm\frac{S}{2}=\pm\frac{1}{2}\Bigg{(}\frac{1+\alpha_{,\vartheta}}{r}\cos\alpha-\alpha_{,r}\sin\alpha\Bigg{)}.$
(129)

Thus, for $S\geqslant 0$, the distortion frame
$(\bm{n}_{1},\bm{n}_{2},\bm{n})$ has

$\bm{n}_{1}=\sin\alpha\bm{e}_{r}-\cos\alpha\bm{e}_{\vartheta},\qquad\bm{n}_{2}=-\bm{e}_{z},$
(130)

while for $S<0$

$\bm{n}_{1}=\bm{e}_{z},\qquad\bm{n}_{2}=\sin\alpha\bm{e}_{r}-\cos\alpha\bm{e}_{\vartheta}.$
(131)

A quick comparison with (124) shows that $\bm{b}$ and $\bm{n}_{1}$ are
collinear in the former case, while $\bm{b}$ and $\bm{n}_{2}$ are collinear in
the latter.

It follows from (44) and (45) that coordinates $(\vartheta_{0},s)$ along
characteristics can be related to the Cartesian coordinates $(x,y)$ through
the change of variables

$\cases{x=\cos\vartheta_{0}+s=:\xi(\vartheta_{0},s),\\\
y=\sin\vartheta_{0}-\frac{b^{\ast}\cos\varphi_{0}-\sin\varphi_{0}}{b^{\ast}\sin\varphi_{0}+\cos\varphi_{0}}s=:\eta(\vartheta_{0},s),}$
(132)

which here replaces (120). Use of (132) in the identity
$\varphi(\xi(\vartheta_{0},s),\eta(\vartheta_{0},s))=\varphi_{0}(\vartheta_{0})$
and resort to the relations

$\cases{\frac{1}{r}\varphi_{,\vartheta}=-\sin\vartheta\varphi_{,x}+\cos\varphi_{,y},\\\
\varphi_{,r}=\cos\vartheta\varphi_{,x}+\sin\vartheta\varphi_{,y},}$ (133)

where $\vartheta$ is to be related to $\vartheta_{0}$ through (132) and the
identities $\cos\vartheta=x/r$ and $\sin\vartheta=y/r$, lead us from (46) to
$f$ expressed as a function of $(\vartheta_{0},s)$. The resulting expression
is rather complicated and inexpressive, but it easily reduces to (50), as can
also be checked directly by letting $\vartheta=\vartheta_{0}$ in (133). Also,
for large $s$, $f=O(1/s^{2})$, which justifies (51).

##  Appendix D Characteristic lines

We collect here details of computations needed in section 5.

Equation (43) follows from (41), as

$\int\frac{1}{r}\operatorname{d}\\!r=-\int\frac{b^{*}\sin(\varphi_{0}-\vartheta)+\cos(\varphi_{0}-\vartheta)}{b^{*}\cos(\varphi_{0}-\vartheta)-\sin(\varphi_{0}-\vartheta)}\operatorname{d}\\!\vartheta$
(134)

and then

$r=R_{0}|b^{*}\cos(\varphi_{0}-\vartheta)-\sin(\varphi_{0}-\vartheta)|^{-1},\quad\mathrm{with}\quad
R_{0}:=r_{0}|b^{*}\cos\alpha_{0}-\sin\alpha_{0}|>0.$ (135)

Both $\varphi:=\alpha+\vartheta$ and $R:=r|b^{*}\cos\alpha-\sin\alpha|$ are
constant along the characteristic curves since

$\cases{\frac{\operatorname{d}\\!\varphi}{\operatorname{d}\\!t}=\frac{\operatorname{d}\\!\alpha}{\operatorname{d}\\!t}+\frac{\operatorname{d}\\!\vartheta}{\operatorname{d}\\!t}=0,&\\\
\eqalign{\frac{\operatorname{d}\\!R}{\operatorname{d}\\!t}&=\frac{\operatorname{d}\\!r}{\operatorname{d}\\!t}|b^{*}\cos\alpha-\sin\alpha|+r\frac{\operatorname{d}\\!}{\operatorname{d}\\!t}|b^{*}\cos\alpha-\sin\alpha|\\\
&=r(b^{*}\sin\alpha+\cos\alpha)|b^{*}\cos\alpha-\sin\alpha|\\\
&\qquad-r\frac{(b^{*}\cos\alpha-\sin\alpha)^{2}}{|b^{*}\cos\alpha-\sin\alpha|}(b^{*}\sin\alpha+\cos\alpha)=0.}&\\\
}$ (136)

Outside $\mathbb{S}^{1}$, that is for $r>1$, we thus obtain that

$\cases{r=\Bigg{|}\frac{b^{*}\cos\alpha_{0}-\sin\alpha_{0}}{b^{*}\cos(\alpha_{0}+\vartheta_{0}-\vartheta)-\sin(\alpha_{0}+\vartheta_{0}-\vartheta)}\Bigg{|}=\Bigg{|}\frac{b^{*}\cos\alpha_{0}-\sin\alpha_{0}}{b^{*}\cos\alpha-\sin\alpha}\Bigg{|},&\\\
\alpha=\alpha_{0}+\vartheta_{0}-\vartheta.&\\\ }$ (137)

Letting in the $(x,y)$ plane be $x=r\cos\vartheta$ and $y=r\sin\vartheta$,
since the function

$\eqalign{R&=|b^{*}r\cos\alpha-r\sin\alpha|\\\
&=|b^{*}r[\cos(\alpha_{0}+\vartheta_{0})\cos\vartheta+\sin(\alpha_{0}+\vartheta_{0})\sin\vartheta]\\\
&\qquad-r[\sin(\alpha_{0}+\vartheta_{0})\cos\vartheta-\cos(\alpha_{0}+\vartheta_{0})\sin\vartheta]|\\\
&=|(b^{*}\cos\varphi_{0}-\sin\varphi_{0})x+(b^{*}\sin\varphi_{0}+\cos\varphi_{0})y|}$
(138)

is also constant, we derive the expression in (45) for the slope $M_{\rm c}$
of the characteristic lines.

By remarking that
$(b^{*}\cos\varphi_{0}-\sin\varphi_{0})^{2}+(b^{*}\sin\varphi_{0}+\cos\varphi_{0})^{2}=(b^{*})^{2}+1$
and requiring $x(s)^{2}+y(s)^{2}>1$, where $s$ is the arc-length parameter
along characteristics, we arrive at

$\cases{x(s)=\frac{|b^{*}\sin\varphi_{0}+\cos\varphi_{0}|}{\sqrt{(b^{*})^{2}+1}}s+\cos\vartheta_{0},&\\\
y(s)=-\operatorname{sign}(b^{*}\sin\varphi_{0}+\cos\varphi_{0})\frac{b^{*}\cos\varphi_{0}-\sin\varphi_{0}}{\sqrt{(b^{*})^{2}+1}}s+\sin\vartheta_{0},&\\\
}$ (139)

or

$\cases{x(s)=\cos\vartheta_{0},&\\\ y(s)=s+\sin\vartheta_{0},&\\\
}\qquad\mathrm{if}\ \tan(\varphi_{0})=-\frac{1}{b^{*}}$ (140)

for

$\cases{\eqalign{s\in[0,+\infty)&\quad\mathrm{if}\
(b^{*}\sin\varphi_{0}+\cos\varphi_{0})(b^{*}\sin\alpha_{0}+\cos\alpha_{0})>0\\\
&\qquad\mathrm{or}\ b^{*}\sin\alpha_{0}+\cos\alpha_{0}\neq
b^{*}\sin\varphi_{0}+\cos\varphi_{0}=0\ \mathrm{and}\
\vartheta_{0}\in(0,\pi),}\\\ \eqalign{s\in(-\infty,0]&\quad\mathrm{if}\
(b^{*}\sin\varphi_{0}+\cos\varphi_{0})(b^{*}\sin\alpha_{0}+\cos\alpha_{0})<0\\\
&\qquad\mathrm{or}\ b^{*}\sin\alpha_{0}+\cos\alpha_{0}\neq
b^{*}\sin\varphi_{0}+\cos\varphi_{0}=0\ \mathrm{and}\
\vartheta_{0}\in(\pi,2\pi),}\\\ \eqalign{s\in\mathbb{R}&\quad\quad\quad\ \
\mathrm{if}\ b^{*}\sin\alpha_{0}+\cos\alpha_{0}=0.}}$ (141)

The last case in (141) represents the characteristic lines tangent to
$\mathbb{S}^{1}$ that delimit the domain $\mathcal{D}$ in (48).

## References

## References

  * [1] Ball J M and Zarnescu A 2008 Mol. Cryst. Liq. Cryst. 495 221/[573]–233/[585] URL https://doi.org/10.1080/15421400802430067
  * [2] Virga E G 2019 Phys. Rev. E 100(5) 052701 URL https://link.aps.org/doi/10.1103/PhysRevE.100.052701
  * [3] da Silva L C B and Efrati E 2021 New J. Phys. 23 063016 URL https://dx.doi.org/10.1088/1367-2630/abfdf6
  * [4] Pollard J and Alexander G P 2021 New J. Phys. 23 063006 URL https://doi.org/10.1088/1367-2630/abfdf4
  * [5] Niv I and Efrati E 2018 Soft Matter 14(3) 424–431 URL http://dx.doi.org/10.1039/C7SM01672G
  * [6] Pedrini A and Virga E G 2020 Phys. Rev. E 101(1) 012703 URL https://link.aps.org/doi/10.1103/PhysRevE.101.012703
  * [7] Machon T and Alexander G P 2016 Phys. Rev. X 6(1) 011033 URL https://link.aps.org/doi/10.1103/PhysRevX.6.011033
  * [8] Selinger J V 2018 Liq. Cryst. Rev. 6 129–142 URL https://doi.org/10.1080/21680396.2019.1581103
  * [9] Selinger J V 2022 Ann. Rev. Condens. Matter Phys. 13 49–71 URL https://doi.org/10.1146/annurev-conmatphys-031620-105712
  * [10] Meyer R B 1976 Molecular Fluids. Les Houches Summer School in Theoretical Physics, 1973
  * [11] Salsa S 2016 Partial Differential Equations in Action. From Modelling to Theory 3rd ed (UNITEXT vol 99) (Cham: Springer)
  * [12] Courant R and Hilbert D 1989 Methods of Mathematical Physics vol 2 (New York: John Wiley & Sons)
  * [13] Delgado M 1997 SIAM Review 39 298–304 URL http://www.jstor.org/stable/2133111
  * [14] Frank F C 1958 Discuss. Faraday Soc. 25(0) 19–28 URL http://dx.doi.org/10.1039/DF9582500019
  * [15] Oseen C W 1933 Trans. Faraday Soc. 29(140) 883–899 URL http://dx.doi.org/10.1039/TF9332900883
  * [16] Virga E G 1994 Variational Theories for Liquid Crystals (Applied Mathematics and Mathematical Computation vol 8) (London: Chapman & Hall)
  * [17] Ericksen J L 1966 Phys. Fluids 9 1205–1207 URL https://aip.scitation.org/doi/abs/10.1063/1.1761821
  * [18] Clelland J N 2017 From Frenet to Cartan: The Method of Moving Frames (Graduate Studies in Mathematics vol 178) (Providence, Rhode Island: American Mathematical Society)
  * [19] da Silva L C B, Bar T and Efrati E 2023 J. Elast. URL https://doi.org/10.1007/s10659-023-09988-7