# A new approach to
rotational Weingarten surfaces

Paula Carretero  Departamento de Matemáticas
Universidad de Jaén
23071 Jaén, Spain<EMAIL_ADDRESS>and  Ildefonso Castro  Departamento
de Matemáticas
Universidad de Jaén
23071 Jaén, Spain and IMAG, Instituto de Matemáticas de la Universidad de
Granada<EMAIL_ADDRESS>
###### Abstract.

Weingarten surfaces are those whose principal curvatures satisfy a functional
relation, whose set of solutions is called the curvature diagram or the
W-diagram of the surface. Making use of the notion of geometric linear
momentum of a plane curve, we propose a new approach to the study of
rotational Weingarten surfaces in Euclidean 3-space. Our contribution consists
of reducing any type of Weingarten condition on a rotational surface to a
first order differential equation on the momentum of the generatrix curve. In
this line, we provide two new classification results involving a cubic and an
hyperbola in the W-diagram of the surface characterizing, respectively, the
non-degenerated quadric surfaces of revolution and the elasticoids, defined as
the rotational surfaces generated by the rotation of the Euler elastic curves
around their directrix line.

As another application of our approach, we deal with the problem of
prescribing mean or Gauss curvature on rotational surfaces in terms of
arbitrary continuous functions depending on distance from the surface to the
axis of revolution. As a consequence, we provide simple new proofs of some
classical results concerning rotational surfaces, like Euler’s theorem about
minimal ones, Delaunay’s theorem on constant mean curvature ones, and
Darboux’s theorem about constant Gauss curvature ones.

###### Key words and phrases:

Weingarten surfaces, rotational surfaces, principal curvatures, quadric
surfaces of revolution, elasticoids

###### 2010 Mathematics Subject Classification:

Primary 53A04, 53A05; Secondary 74B20

##  1\. Introduction

Following [Ch45], the Weingarten surfaces are those whose principal curvatures
$\kappa_{1}$ and $\kappa_{2}$ satisfy a certain functional relation
$\Phi(\kappa_{1},\kappa_{2})=0$. The set of solutions of this equation is also
called the curvature diagram or the $W$-diagram of the surface (see [H51]).
These surfaces were introduced by Weingarten in [W61] and their study plays an
important role in classical Differential Geometry (see e.g. [H51] and [V59]).
Applications of Weingarten surfaces on computer aided design can be found in
[BG96].

In particular, the ones satisfying a linear relation
$a\kappa_{1}+b\kappa_{2}=c$, $a^{2}+b^{2}\neq 0$, $c\in\mathbb{R}$, are called
linear Weingarten surfaces. In this case, the W-diagram is contained in a
straight line or degenerates to one point. This class of Weingarten surfaces
include umbilical surfaces, isoparametric surfaces, constant mean curvature
and minimal surfaces or those surfaces where one of the principal curvatures
is constant. On the other hand, some types of thin axial symmetric shells
subjected to uniform normal pressure are modelled on rotational surfaces whose
principal curvatures obey some specific quadratic relations (see e.g. [PHM19]
and [PM20]). Concretely, the W-diagram is contained in a certain parabola in
this case. The general case that the curvature diagram is a standard parabola
was studied in [KS05].

Among the invariant surfaces, precisely the rotational ones are probably the
most studied surfaces in Euclidean 3-space. Perhaps the main reason could be
they are a nice family where considering firstly interesting geometric
properties to attach on any surface, because its geometry can be controlled by
the geometry of the generatrix curve. As an illustration, we can mention the
classical theorems of Euler [E44], Delaunay [D41] and Darboux [D90]
classifying minimal, constant mean curvature and constant Gauss curvature
rotational surfaces in 1744, 1841 and 1890 respectively. However, the complete
classification of rotational linear Weingarten surfaces has not been achieved
surprisingly until 2020 in [LP20a]. Some other interesting rotational
Weingarten surfaces have been also recently studied in [LP20b] and [LP20c]. We
refer to [KS05] (and references therein) for the study of closed rotational
Weingarten surfaces satisfying $\kappa_{1}=c\,\kappa_{2}^{2n+1}$, generalizing
results of Hopf [H51] when $n=0$ and the case of ellipsoids of revolution when
$n=1$. These may be good reasons why rotational surfaces are probably one of
the main classes of Weingarten surfaces and continue to deserve attention. We
propose in this paper a new approach for their study, inspired mainly by
[CCI16].

In order to explain our focus, we start recalling that plane curves are
uniquely determined, up to rigid motions, by its intrinsic equation giving its
curvature $\kappa$ as a function of its arc-length. It is well known that one
needs three quadratures in the integration process. In [S99], David A. Singer
considered a different sort of problem, proposing to determine a plane curve
if its curvature is given in terms of its position. Probably, the most
interesting solved case in this setting corresponds to the Euler elastic
curves (see [S08]), whose curvature is proportional to one of the coordinate
functions, e.g. $\kappa(x,z)=c\,x$ for curves in the $xz$-plane. Motivated by
the above question and by the classical elasticae, the authors studied in
[CCI16] the plane curves whose curvature depends on the distance to a line
(say the $z$-axis and so $\kappa=\kappa(x)$) and in [CCIs17] the plane curves
whose curvature depends on the distance from a point (say the origin, and so
$\kappa=\kappa(r)$, $r=\sqrt{x^{2}+z^{2}}$) requiring in both cases the
computation of three quadratures too. They also considered the analogous
problems in Lorentz-Minkowski plane in [CCIs18] and [CCIs20a].

The key tool for the study of plane curves whose curvature depends on distance
to a line is the notion of geometric linear momentum of a plane curve (see
[CCIs20b] or Section 2). It is a smooth function associated to any plane
curve, which completely determines it (up to a family of distinguished
translations) in relation with its relative position with respect to a fixed
line (see Corollary 2.4). Geometrically, the geometric linear momentum
controls the angle of the Frenet frame of the curve with this fixed line and
receives that name because, in physical terms, it can be described as the
linear momentum (with respect to the fixed line) of a particle of unit mass
with unit speed and trajectory the path of the plane curve. Moreover, it can
be interpreted as an anti-derivative of the curvature of the curve when this
is expressed as a function of the distance to the fixed line.

Therefore it seems to be natural that when one deals with rotational surfaces,
generated by the rotation of a plane curve (the generatrix curve) around a
coplanar fixed line (the axis of revolution) the geometric linear momentum of
the generatrix with respect to the axis of revolution plays a predominant role
to control the geometry of the rotational surface. We show it in Corollary
3.1, proving that any rotational surface is uniquely determined, up to
translations along the axis of revolution, by the geometric linear momentum of
the generatrix curve. This main result is confirmed when we study the geometry
of a rotational surface since both its first and second fundamental forms can
be expressed only in terms of the geometric linear momentum and, of course,
the (non-constant) distance from the surface to the axis of revolution. This
allows our main contribution in the paper which consists of reducing any type
of Weingarten condition on a rotational surface to a first order differential
equation on the momentum of the generatrix curve (see Section 3). We
illustrate this procedure analysing under our optics the two types of linear
Weingarten surfaces one can find in literature (see Subsections 3.2 and 3.3)
and we emphasize two new classification results involving a cubic and an
hyperbola in the W-diagram of the rotational Weingarten surface (see Theorems
3.4 and 3.6 respectively) characterizing, respectively, the non-degenerated
quadric surfaces of revolution and the elasticoids (defined as the rotational
surfaces generated by the rotation of the Euler elastic curves around their
directrix line).

As another application, we deal in Section 4 with the problem of prescribing
mean or Gauss curvature on rotational surfaces in terms of arbitrary
continuous functions depending on distance from the surface to the axis of
revolution, providing in Theorem 4.1 one-parameter families of rotational
surfaces with prescribed mean or Gaussian curvature. We point out that
Kenmotsu [K80] constructed a 3-parameter family of surfaces of revolution
admitting $H=H(s)$ as the mean curvature for a given continuous function
$H(s)$, $s$ being the arc parameter of the generatrix curve.

As a consequence of Theorem 4.1, we provide simple new proofs of the above
mentioned classical results concerning rotational surfaces like Euler’s
theorem about minimal ones (see Corollary 4.3), Delaunay’s theorem on constant
mean curvature ones (see Corollary 4.4), and Darboux’s theorem about constant
Gauss curvature ones (see Corollary 4.5).

##  2\. The geometric linear momentum of a plane curve

We introduce a smooth function associated to any plane curve, which completely
determines it (up to a family of distinguished isometries) in relation with
its relative position with respect to a fixed line.

Let $\alpha:I\subseteq\mathbb{R}\rightarrow\mathbb{R}^{2}$ be a regular plane
curve parametrized by arc-length; that is, $|\dot{\alpha}(s)|=1,\,\forall s\in
I$, where $I$ is some interval in $\mathbb{R}$. Here $\,\dot{}\,$ means
derivation with respect to $s$. We denote by $\langle\cdot,\cdot\rangle$ the
usual inner product in $\mathbb{R}^{2}$. Let $T:=\dot{\alpha}$ be the unit
tangent vector to the curve $\alpha$ and let $N:=i\dot{\alpha}$ be the vector
orthogonal to $T$ such that the frame $(T,N)$ is positively oriented. The
Frenet equations of $\alpha$ are given by

(2.1) $\dot{T}(s)=\kappa(s)N(s),\quad\dot{N}(s)=-\kappa(s)T(s),$

where $\kappa=\kappa(s)=\det(\dot{\alpha}(s),\ddot{\alpha}(s))$ is the
(signed) curvature of $\alpha$. Up to sign, $\kappa=\pm\dot{\theta}$, where
$\theta$ can be chosen as the angle $\angle(T,(1,0))$.

We are interested in the geometric condition that the curvature of $\alpha$
depends on the distance to a fixed line of $\mathbb{R}^{2}$. We choose
Cartesian coordinates in $\mathbb{R}^{2}$ and we write $\alpha=(x,z)$. Thus it
is enough to study the condition $\kappa=\kappa(x)$, since $x$ represents the
signed distance to the $z$-axis.

A key role is played then by the geometric linear momentum of the plane curve
$\alpha$ (with respect to the $z$-axis). At a given point $\alpha(s)$ on the
curve, it is defined by

$\mathcal{K}(s):=\langle T(s),(0,1)\rangle=\sin\theta(s).$

Geometrically, $\mathcal{K}$ controls the angle of the Frenet frame of the
curve with the coordinate axes.

We can also observe a physical interpretation if we write $\mathcal{K}$ in
Cartesian coordinates. Since $\alpha=(x,z)$, then $T=(\dot{x},\dot{z})$ and so
the linear momentum $\mathcal{K}$ at $\alpha(s)$ is given by

(2.2) $\mathcal{K}(s)=\dot{z}(s).$

Hence, in physical terms, $\mathcal{K}=\mathcal{K}(s)$ may be described as the
linear momentum (with respect to the $z$-axis) of a particle of unit mass with
unit speed and trajectory $\alpha(s)$. We point out that $\mathcal{K}$ assumes
values in $[-1,1]$ and it is well defined, up to the sign, depending on the
orientation of $\alpha$.

######  Remark 2.1.

If the plane curve $\alpha=(x,z)$ is not necessarily parametrized by arc
length, i.e. $\alpha=\alpha(t)$, $t$ being any parameter, one can computes the
geometric linear momentum $\mathcal{K}=\mathcal{K}(t)$ by means of

$\mathcal{K}(t)=\frac{z^{\prime}(t)}{|\alpha^{\prime}(t)|},$

where ′ denotes derivation respect to $t$.

It is an easy exercise the computation of the geometric linear momenta of some
distinguish plane curves for our purposes, using Remark 2.1 if necessary, some
of them located in a determinate position with respect to $z$-axis, that we
collect in the following list (see Figure 1):

######  Example 2.2.

  1. (1)

Vertical lines: $\mathcal{K}\equiv\pm 1$.

  2. (2)

Line with slope $\tan\theta_{0}$, $\theta_{0}\in(-\pi/2,\pi/2)$, given by
$z=\tan\theta_{0}\,x$: $\mathcal{K}\equiv\sin\theta_{0}\in(-1,1)$.

  3. (3)

Circle centred at $(a,0)$, $a\in\mathbb{R}$, and radius $R>0$:
$\mathcal{K}(s)=\cos(s/R)=(x-a)/R=\mathcal{K}(x)$.

  4. (4)

Catenary $x=a\cosh(z/a)$, $a>0$: $\mathcal{K}(x)=a/x$.

  5. (5)

Cycloid of radius $R>0$ given by $x=R(1-\cos t)$, $z=R(t-\sin t)$,
$t\in\mathbb{R}$:
$\mathcal{K}(t)=\sin(t/2)=\sqrt{x}/\sqrt{2R}=\mathcal{K}(x)$.

  6. (6)

Tractrix of height $a>0$ given by $x=a\operatorname{sech}t$, $z=a(t-\tanh t)$,
$t\in\mathbb{R}$: $\mathcal{K}(t)=\tanh
t=\sqrt{1-x^{2}/a^{2}}=\mathcal{K}(x)$.

Figure 1. From left to right: catenary, cycloid, tractrix.

On the other hand, when $\alpha$ is unit-speed we have that

(2.3) $\dot{x}^{2}+\dot{z}^{2}=1.$

Using (2.2) and (2.3), assuming that $x$ is non constant, we get:

(2.4)
$ds=\frac{dx}{\sqrt{1-\dot{z}^{2}}}=\frac{dx}{\sqrt{1-\mathcal{K}^{2}}},\quad
dz=\mathcal{K}\,ds.$

Thus, given $\mathcal{K}=\mathcal{K}(x)$ as an explicit function, looking at
(2.4) one may attempt to compute $x(s)$ and $z(s)$ in three steps: integrate
to get $s=s(x)$, invert to get $x=x(s)$ and integrate to get $z=z(s)$.

In addition, (2.1) implies that $\ddot{z}=\kappa\dot{x}$. In this way,
assuming that $\kappa=\kappa(x)$ and using (2.2), we obtain:

(2.5)
$d\mathcal{K}=\dot{\mathcal{K}}(s)ds=\ddot{z}(s)ds=\kappa(x)\dot{x}(s)ds=\kappa(x)dx.$

Therefore we deduce that $\mathcal{K}$ can be interpreted as an anti-
derivative of $\kappa=\kappa(x)$. As a summary, we can determine by
quadratures in a constructive explicit way the plane curves such that
$\kappa=\kappa(x)$, in the spirit of [S99, Theorem 3.1].

######  Theorem 2.3.

Let $\kappa=\kappa(x)$ be a continuous function. Then the problem of
determining a curve $(x(s),z(s))$, with $s$ the arc length parameter, whose
curvature is $\kappa(x)$ with geometric linear momentum $\mathcal{K}(x)$
satisfying (2.5) —$x$ representing the (non constant) signed distance to the
$z$-axis— is solvable by quadratures where $x(s)$ and $z(s)$ are obtained
through (2.4) after inverting $s=s(x)$.

Moreover, such a curve is uniquely determined, up to translations in the
$z$-direction (and a translation of the arc parameter $s$), by the geometric
linear momentum $\mathcal{K}(x)$.

If we focus on the determining role of the geometric linear momentum, we can
rephrase the above result simply as follows:

######  Corollary 2.4.

Any plane curve $\alpha=(x,z)$, with $x$ non-constant, is uniquely determined
by its geometric linear momentum $\mathcal{K}$ as a function of its distance
to $z$-axis, that is, by $\mathcal{K}=\mathcal{K}(x)$. The uniqueness is
modulo translations in the $z$-direction. Moreover, the curvature of $\alpha$
is given by $\kappa(x)=\mathcal{K}^{\prime}(x)$.

######  Remark 2.5.

If we prescribe a continuous function $\kappa=\kappa(x)$ as curvature of a
plane curve, the proof of Theorem 2.3 offers an algorithm to recover the curve
through the computation of three quadratures, following the sequence:

  1. (i)

A one-parameter family of anti-derivatives of $\kappa(x)$:

$\int\\!\kappa(x)dz=\mathcal{K}(x).$

  2. (ii)

Arc-length parameter $s$ of $\alpha=(x,z)$ in terms of $x$, defined —up to
translations of the parameter— by the integral:

$s=s(x)=\int\\!\frac{dx}{\sqrt{1-\mathcal{K}(x)^{2}}},$

where $-1<\mathcal{K}(x)<1$, and inverting $s=s(x)$ to get $x=x(s)$.

  3. (iii)

$z$-coordinate of the curve —up to translations along $z$-axis— by the
integral:

$z(s)=\int\\!\mathcal{K}(x(s))\,ds.$

We note that we get a one-parameter family of plane curves satisfying
$\kappa=\kappa(x)$ according to the linear momentum $\mathcal{K}(x)$ chosen in
(i) and verifying $\mathcal{K}(x)^{2}<1$. It will distinguish geometrically
the curves inside a same family by their relative position with respect to
$z$-axis. We recall that we can easily recover $\kappa$ from $\mathcal{K}$
simply by means of $\kappa=d\mathcal{K}/dx$. In some sense,
$\mathcal{K}=\mathcal{K}(x)$ can be interpreted as the extrinsic equation of
$\alpha$ in this setting, in contrast to the classical intrinsic equation
$\kappa=\kappa(s)$.

We now show two illustrative examples applying steps (i)-(iii) of the
algorithm described in Remark 2.5:

######  Example 2.6 ($\kappa\\!\equiv\\!0$).

Then $\mathcal{K}\equiv c\in(-1,1)$, and easily $x(s)=\sqrt{1-c^{2}}\,s$,
$z(s)=c\,s$. Writing $c=\sin\theta_{0}$, $\theta_{0}\in(-\pi/2,\pi/2)$, we
arrive at Example 2.2(2). Up to $z$-translations, we get all the (non
vertical) lines in the plane.

######  Example 2.7 ($\kappa\equiv k_{0}>0$).

Then $\mathcal{K}(x)=k_{0}x+c$, and it is not difficult to obtain that
$x(s)=\frac{1}{k_{0}}\left(\sin(k_{0}s)-c\right)$,
$z(s)=-\frac{1}{k_{0}}\cos(k_{0}s)$. Writing $k_{0}=1/R$ we arrive at Example
2.2(3). Up to $z$-translations, we get all the circles with radius $R=1/k_{0}$
in the plane.

######  Remark 2.8.

The main difficulties one can find carrying on the strategy described in
Remark 2.5 (or in Theorem 2.3) to determine a plane curve whose curvature is
$\kappa=\kappa(x)$ are the following:

  1. (1)

The integration of $s=s(x)$: Even in the case that $\mathcal{K}(x)$ were
polynomial, the corresponding integral is not necessarily elementary. For
example, when $\mathcal{K}(x)$ is a quadratic polynomial, it can be solved
using Jacobian elliptic functions (see e.g. [BF71]). This last case is
equivalent to $\kappa(x)$ be linear, i.e. $\kappa(x)=2ax+b$, $a\neq 0$,
$b\in\mathbb{R}$. These are precisely the Euler elastic curves (see [CCI16,
Section 3]).

  2. (2)

The previous integration gives us $s=s(x)$; it is not always possible to
obtain explicitly $x=x(s)$, what is necessary to determine the curve.

  3. (3)

Even knowing explicitly $x=x(s)$, the integration to get $z(s)$ may be
impossible to perform using elementary or known functions.

We finish this section with an example illustrating the difficulty (2) in
Remark 2.8.

######  Example 2.9.

Consider $\mathcal{K}(x)=c/\sqrt{x}$, $x>c^{2}>0$. Then
$s=s(x)=\sqrt{x}\sqrt{x-c^{2}}+c^{2}\ln(\sqrt{x}\sqrt{x-c^{2}})$ and it is not
possible to invert to get $x=x(s)$. However, eliminating $ds$ in Remark
2.5(ii) and (iii), we obtain:

(2.6) $z=z(x)=\pm\int\frac{\mathcal{K}(x)dx}{\sqrt{1-\mathcal{K}(x)^{2}}}.$

Applying the above formula in this example, we arrive at
$z^{2}=4c^{2}(x-c^{2})$ that corresponds with the parabola with vertex
$(c^{2},0)$ and focus $(2c^{2},0)$ (see Figure 2).

Figure 2. Parabola of Example 2.9.

##  3\. Rotational Weingarten surfaces

We start this section considering rotational surfaces, also called surfaces of
revolution. They are surfaces globally invariant under the action of any
rotation around a fixed line called axis of revolution. The rotation of a
curve (called generatrix) around a fixed line generates a surface of
revolution. The sections of a surface of revolution by half-planes delimited
by the axis of revolution, called meridians, are special generatrices. The
sections by planes perpendicular to the axis are circles called parallels of
the surface.

We denote $S_{\alpha}$ the rotational surface in $\mathbb{R}^{3}$ generated by
the rotation around $z$-axis of a plane curve $\alpha$ in the $xz$-plane (see
Figure 3). That is, $\alpha$ is the generatrix curve that we consider
parametrized by arc-length, with parametric equations given by $x=x(s)>0$,
$y=0$, $z=z(s)$, $s\in I\subseteq\mathbb{R}$. Then $S_{\alpha}$ is
parametrized by

$S_{\alpha}\equiv
X(s,\theta)=\left(x(s)\cos\theta,x(s)\sin\theta,z(s)\right),\ (s,\theta)\in
I\times(-\pi,\pi).$

It is obvious that if we translate the generatrix curve $\alpha$ of a
rotational surface $S_{\alpha}$ along $z$-axis, we obtain a congruent surface
to $S_{\alpha}$. Then, as an immediate consequence of Corollary 2.4, we deduce
the following key result:

Figure 3. $S_{\alpha}$: rotational surface generated by $\alpha=(x,z)$.

######  Corollary 3.1.

Any rotational surface $S_{\alpha}$, with generatrix curve $\alpha=(x,z)$, is
uniquely determined, up to $z$-translations, by the geometric linear momentum
$\mathcal{K}=\mathcal{K}(x)$ of its generatrix curve, being $x$ non-constant.

######  Remark 3.2.

The only rotational surface excluded in Corollary 3.1 is the right circular
cylinder, corresponding to $x$ being constant. We recall that it is a flat
rotational surface and its principal curvatures are $0$ (along generatrices)
and $1/a$ (along parallel circles), $a>0$ being the radius of the cylinder.

Looking at Examples 2.2 and 2.9 and making use of Corollary 3.1, we can list
the following classical and conventional rotational surfaces joint with its
determining geometric linear momentum. See Figure 4.

######  Proposition 3.3.

Up to translations in $z$-direction:

  1. (1)

Any horizontal plane is uniquely determined by the geometric linear momentum
$\mathcal{K}\equiv 0$.

  2. (2)

The circular cone with opening $\theta_{0}\in(-\pi/2,\pi/2)$, given by
$x^{2}+y^{2}=\cot^{2}\theta_{0}\,z^{2}$, is uniquely determined by the
geometric linear momentum $\mathcal{K}\equiv\sin\theta_{0}$.

  3. (3)

The sphere of radius $R>0$, given by $x^{2}+y^{2}+z^{2}=R^{2}$, is uniquely
determined by the geometric linear momentum $\mathcal{K}(x)=x/R$.

  4. (4)

The torus of revolution with major radius $|a|\neq 0$ and minor radius $R>0$,
given by $(\sqrt{x^{2}+y^{2}}-a)^{2}+z^{2}=R^{2}$, is uniquely determined by
the geometric linear momentum $\mathcal{K}(x)=(x-a)/R$.

  5. (5)

The catenoid of chord $a>0$, given by $x^{2}+y^{2}=a^{2}\cosh^{2}(z/a)$, is
uniquely determined by the geometric linear momentum $\mathcal{K}(x)=a/x$.

  6. (6)

The onducycloid of radius $R>0$, defined as the surface generated by rotation
of the cycloid in Example 2.2(5) around its base, is uniquely determined by
the geometric linear momentum $\mathcal{K}(x)=\sqrt{x}/\sqrt{2R}$.

  7. (7)

The pseudosphere of pseudoradius $a>0$ (cf. [F93]), defined as the surface
generated by the rotation of the tractrix in Example 2.2(6) around its
asymptote, is uniquely determined by the geometric linear momentum
$\mathcal{K}(x)=\sqrt{1-x^{2}/a^{2}}$.

  8. (8)

The antiparaboloid of radius $c^{2}>0$, defined as the surface generated by
the rotation of the parabola with vertex $(c^{2},0)$ and focus $(2c^{2},0)$,
$c>0$, around its directrix line, given by
$x^{2}+y^{2}=(c^{2}+z^{2}/4c^{2})^{2}$, is uniquely determined by the
geometric linear momentum $\mathcal{K}(x)=c/\sqrt{x}$.

Figure 4. Rotational surfaces. From left to right: catenoid, onducycloid,
pseudosphere, antiparaboloid.

We can confirm the result established in Corollary 3.1 when we study the
geometry of $S_{\alpha}$ through its first and second fundamental forms, $I$
and $II$, since a direct computation, using that
$\kappa(x)=\mathcal{K}^{\prime}(x)$, shows that both can be expressed only in
terms of the geometric linear momentum $\mathcal{K}$ and, of course, the non
constant distance $x$ from the surface to the axis of revolution:

$I\equiv ds^{2}+x^{2}d\theta^{2},\quad
II\equiv\mathcal{K}^{\prime}(x)ds^{2}+x\mathcal{K}(x)d\theta^{2}.$

Therefore we get the following expressions for the principal curvatures
$\kappa_{1}$ and $\kappa_{2}$, whose curvature lines are the meridians and
parallels respectively of the rotational surface $S_{\alpha}$:

(3.1) $\kappa_{1}\equiv
k_{\text{m}}=\mathcal{K}^{\prime}(x),\quad\kappa_{2}\equiv
k_{\text{p}}=\frac{\mathcal{K}(x)}{x}.$

Thus, the mean curvature $H$ of $S_{\alpha}$ is given by

(3.2) $2H=\mathcal{K}^{\prime}(x)+\frac{\mathcal{K}(x)}{x},$

and the Gauss curvature $K_{\text{G}}$ of $S_{\alpha}$ is given by

(3.3) $K_{\text{G}}=\frac{\mathcal{K}(x)\mathcal{K}^{\prime}(x)}{x}.$

Now we can pay attention to rotational Weingarten surfaces. Weingarten
surfaces must satisfy a functional relation between their principal
curvatures. In the case of rotational surfaces, the principal curvatures are
reached along meridians and parallel of $S_{\alpha}$ and, from (3.1), it is
clear that rotational surfaces constitute a distinguish class of Weingarten
surfaces. For example, if $k_{\text{m}}=\mathcal{K}^{\prime}(x)=\kappa(x)$ is
invertible, from (3.1) we arrive at
$k_{\text{p}}=\mathcal{K}(\kappa^{-1}(k_{\text{m}}))/\kappa^{-1}(k_{\text{m}})$.
In general, we just simply write $\Phi(k_{\text{m}},k_{\text{p}})=0$. But,
taking into account (3.1), we easily deduce that the above functional relation
translates into a first order differential equation for the geometric linear
momentum $\mathcal{K}=\mathcal{K}(x)$ determining $S_{\alpha}$ according
Corollary 3.1:

$\hat{\Phi}(x,\mathcal{K}(x),\mathcal{K}^{\prime}(x))=0.$

If the above equation can be rewritten as
$\mathcal{K}^{\prime}(x)=F(x,\mathcal{K}(x))$ or, equivalently
$k_{\text{m}}=\hat{F}(k_{\text{p}})$, it follows that locally any equation of
this type with an arbitrary continuous function $F$ (or $\hat{F}$) admits a
solution.

We can illustrate the above simple reasoning analysing some interesting
Weingarten-type conditions.

###  3.1. Rotational surfaces with some constant principal curvature

We distinguish two cases. First, if some principal curvature of $S_{\alpha}$
is null, we have that $k_{\text{m}}k_{\text{p}}=0$. If $k_{\text{m}}=0$, using
(3.1), we get that $\mathcal{K}\equiv c$, $c\in(-1,1)$. Then, using Corollary
3.1, Remark 3.2 and Proposition 3.3, we conclude that $S_{\alpha}$ is a
circular cylinder, a plane (when $c=0$) or a circular cone (when $c\neq 0$).
And if $k_{\text{p}}=0$, using (3.1), we get that $\mathcal{K}\equiv 0$ and
$S_{\alpha}$ is a plane.

We consider now the case that some principal curvature is a nonzero constant.
If $k_{\text{m}}=a\neq 0$, using (3.1), we get that $\mathcal{K}(x)=ax+c$,
$c\in\mathbb{R}$. Then, using Proposition 3.3, we conclude that $S_{\alpha}$
is a sphere when $c=0$ or a torus of revolution when $c\neq 0$. And if
$k_{\text{p}}=a\neq 0$, using (3.1), we get that $\mathcal{K}(x)=ax$ and
$S_{\alpha}$ is a sphere or a circular cylinder (see Remark 3.2).

###  3.2. Linear rotational Weingarten surfaces

They are defined by the linear relation $a\,k_{\text{m}}+b\,k_{\text{p}}=c,\
a,b,c\in\mathbb{R},\ a^{2}+b^{2}\neq 0.$ We can assume $a\neq 0$ (see
Subsection 3.1 if $a=0$) and we just write

(3.4) $k_{\text{m}}=p\,k_{\text{p}}+\,q,\,p,q\in\mathbb{R},\,p\neq 0.$

These surfaces have been recently classified in [LP20a] by means of a
qualitative study, providing closed (embedded and not embedded) surfaces and
periodic (embedded and not embedded) surfaces with a geometric behaviour
similar to Delaunay surfaces ([D41]). In [LP20a] there is a necessary
distinction of cases according to $p=1$ or $p\neq 1$. In fact, when $p\neq 1$,
the generatrix curves are $\frac{p}{p-1}$\- elastic curves (see [LP21] for a
description of them) generalizing classical elastic curves corresponding to
$p=2$ (see [LS84] and [MO03]). Under our optics, (3.4) translates into the
linear o.d.e. $\mathcal{K}^{\prime}(x)=p\,\mathcal{K}(x)/x+q$, that it is easy
to solve. Using Corollary 3.1, both above mentioned families are uniquely
determined, up to $z$-translations, by the following geometric linear momenta:

$p\neq 1:\quad\mathcal{K}(x)=\frac{q\,x}{1-p}+c\,x^{p},\,c\in\mathbb{R},$

and

$p=1:\quad\mathcal{K}(x)=q\,x\ln x+c\,x,\,c\in\mathbb{R}.$

This can be a reasonable explanation of the commented distinction of cases in
[LP20a].

The case $p=1$, $q=0$ in (3.4), i.e. $k_{\text{m}}=k_{\text{p}}$, leads
obviously to the only umbilical surfaces (planes and spheres) since then
$\mathcal{K}(x)=c\,x$, $c\in\mathbb{R}$ (see Proposition 3.3(1),(3)).

We pay now attention to the case $q=0$ in (3.4), i.e.
$k_{\text{m}}=p\,k_{\text{p}}$, which leads to $\mathcal{K}(x)=c\,x^{p}$.
Under our optics, we can localize distinguished rotational surfaces satisfying
$k_{\text{m}}=p\,k_{\text{p}}$ in this subclass of linear Weingarten surfaces.
Using Proposition 3.3, we get catenoids if $p=-1$, onducycloids if $p=1/2$ and
antiparaboloids if $p=-1/2$.

###  3.3. “Linear” rotational Weingarten surfaces

Some authors define linear Weingarten surfaces as those ones such that a
linear combination of its mean curvature $H$ and Gauss curvature
$K_{\text{G}}$ is constant (see e.g. [GMM03]):

(3.5) $a\,K_{\text{G}}+2b\,H+c=0,\ a^{2}+b^{2}\neq 0,c\in\mathbb{R}.$

They are also referred as special Weingarten surfaces in [L08]. They include
CMC (constant mean curvature) surfaces when $a=0$ and CGC (constant Gauss
curvature) surfaces when $b=0$. The closed ones were studied in [Ch45] and
[H51] among others. In [RS94], the authors studied properly embedded surfaces
in $\mathbb{R}^{3}$ satisfying $a\,K_{\text{G}}+2b\,H=1$, where $a$ and $b$
are positive. For $a\neq 0$, the curvature diagram corresponding to (3.5) is
given by the rectangular hyperbola

(3.6)
$\left(k_{\text{m}}+\frac{b}{a}\right)\left(k_{\text{p}}+\frac{b}{a}\right)=\frac{b^{2}-ac}{a^{2}}.$

Using (3.1), (3.6) translates into the o.d.e.

$(c\,x+b\,\mathcal{K})dx+(b\,x+a\,\mathcal{K})d\mathcal{K}=0,$

that it is exact. Its implicit solution is given by

$a\,\mathcal{K}(x)^{2}+2\,b\,x\,\mathcal{K}(x)+c\,x^{2}=d\in\mathbb{R},$

which leads to

$\mathcal{K}(x)=\frac{1}{a}\left(-b\,x\pm\sqrt{(b^{2}-ac)x^{2}+ad}\right)$

when $a\neq 0$. If $c=d=0$, we recover the trivial cases corresponding to the
plane (when $b=0$) and the sphere of radius $\left|\frac{-a}{2b}\right|$ (when
$b\neq 0$).

###  3.4. Cubic rotational Weingarten surfaces

It is known (see [BG94]) that the ellipsoid of revolution

(3.7) $\frac{x^{2}+y^{2}}{a^{2}}+\frac{z^{2}}{b^{2}}=1$

satisfies the relation

(3.8) $k_{\text{m}}=\frac{a^{4}}{b^{2}}\,k_{\text{p}}^{3}.$

In [KS05], it is also proved that any closed surface of revolution satisfying
$k_{\text{m}}=c\,k_{\text{p}}^{3}$, for any positive constant $c>0$, is
congruent to some ellipsoid of revolution. The aim of this section is to
generalize this result using our local approach to the study of rotational
Weingarten surfaces.

For our purposes, recalling Corollary 3.1, we need to compute the geometric
linear momentum of the ellipsoid (3.7). We parametrize the generatrix
semiellipse by $x=a\cos t$, $z=b\sin t$, $t\in[-\pi/2,\pi/2]$, and using
Remark 2.1, it is not difficult to get that

(3.9) $\mathcal{K}(x)=\frac{bx}{\sqrt{a^{4}-(a^{2}-b^{2})x^{2}}}.$

Then, using (3.1), we can check (3.8) easily.

We proceed in the same way with the one-sheet hyperboloid of revolution

(3.10) $\frac{x^{2}+y^{2}}{a^{2}}-\frac{z^{2}}{b^{2}}=1,$

obtaining, from the generatrix hyperbola $x=a\cosh t$, $z=b\sinh t$,
$t\in\mathbb{R}$, and Remark 2.1, that

(3.11) $\mathcal{K}(x)=\frac{bx}{\sqrt{(a^{2}-b^{2})x^{2}-a^{4}}}$

and now we can check that the one-sheet hyperboloid of revolution satisfies
the relation

(3.12) $k_{\text{m}}=-\frac{a^{4}}{b^{2}}k_{\text{p}}^{3}.$

However, for the two-sheets hyperboloid of revolution

(3.13) $-\frac{x^{2}+y^{2}}{a^{2}}+\frac{z^{2}}{b^{2}}=1,$

as now the generatrix hyperbola is $x=a\sinh t$, $z=\pm b\cosh t$, $t\geq 0$,
we get from Remark 2.1 that

(3.14) $\mathcal{K}(x)=\frac{\pm bx}{\sqrt{a^{4}+(a^{2}-b^{2})x^{2}}}$

and so we arrive at the following relation satisfied by the two-sheets
hyperboloid of revolution:

(3.15) $k_{\text{m}}=\frac{a^{4}}{b^{2}}\,k_{\text{p}}^{3},$

that it is formally the same than the one (3.8) of the ellipsoid of
revolution.

Finally, for the paraboloid of revolution

(3.16) $z=\frac{x^{2}+y^{2}}{2a},$

we use the generatrix parabola $x=t$, $z=\frac{t^{2}}{2a}$, $t\geq 0$, and
Remark 2.1 gives that

(3.17) $\mathcal{K}(x)=\frac{x}{\sqrt{a^{2}+x^{2}}}$

and then the paraboloid of revolution satisfies the relation

(3.18) $k_{\text{m}}=a^{2}k_{\text{p}}^{3}.$

Now we are in a position to state our main result in this section
characterizing all the quadric surfaces of revolution (see Figure 5) in terms
of a cubic Weingarten relation.

Figure 5. Quadric surfaces of revolution, From left to right: Ellipsoid, one-
sheet hyperboloid, two-sheets hyperboloid, paraboloid.

######  Theorem 3.4.

The only rotational surfaces satisfying $k_{\text{m}}=\mu\,k_{\text{p}}^{3}$,
$\mu\neq 0$, are the plane and the non-degenerate quadric surfaces of
revolution.

###### Proof.

Using (3.1), the cubic Weingarten relation
$k_{\text{m}}=\mu\,k_{\text{p}}^{3}$ translates into the separable o.d.e.

$\mathcal{K}^{\prime}(x)=\mu\,\mathcal{K}(x)^{3}/x^{3}.$

Its constant solution $\mathcal{K}\equiv 0$ leads to the plane (see
Proposition 3.3(1)). Its non constant solution is given by

(3.19) $\mathcal{K}(x)=\pm\frac{x}{\sqrt{\mu+c\,x^{2}}},\,c\in\mathbb{R}.$

We are going to identify the rotational surfaces uniquely determined, up to
$z$-translations, by the one parameter family of geometric linear momenta
(depending on $c$) given in (3.19) (see Corollary 3.1). There is no
restriction if we only consider plus sign in (3.19).

We distinguish two cases according to the sign of $\mu$.

  * •

$\mu>0$. We separate in turn three possibilities:

    * (i)

$c<1$: Then $a^{2}=\frac{\mu}{1-c}$ is well defined, and putting
$b^{2}=\frac{a^{4}}{\mu}$, we get that (3.19) is exactly (3.9) and Corollary
3.1 gives that we arrive at the ellipsoid of revolution (3.7). In particular,
if $c=0$, we obtain the sphere of radius $\sqrt{\mu}$.

    * (ii)

$c>1$: We now define $a^{2}=\frac{\mu}{c-1}$ and $b^{2}=\frac{a^{4}}{\mu}$.
Then we obtain that (3.19) is exactly (3.14) and Corollary 3.1 concludes that
we arrive at the two-sheets hyperboloid of revolution (3.13).

    * (iii)

$c=1$: We define $a^{2}=\mu$ and we have that (3.19) is exactly (3.17). We
deduce from Corollary 3.1 that we arrive at the paraboloid of revolution
(3.16).

  * •

$\mu<0$. Since $\mu+c\,x^{2}>0$ from (3.19) and taking into account that
always $\mathcal{K}(x)^{2}<1$, we deduce that $(1-c)x^{2}<\mu<0$ and so $c>1$.
Then $a^{2}=\frac{\mu}{1-c}$ is well defined, and putting
$b^{2}=-\frac{a^{4}}{\mu}$, we get that (3.19) is exactly (3.11) and Corollary
3.1 gives that we arrive at the one-sheet hyperboloid of revolution (3.10).

This proves the result. ∎

######  Remark 3.5.

We point out that, although it is not explicitly established in [KS05] because
the authors were interested in closed Weingarten surfaces, the case $\mu>0$ in
Theorem 3.4 was considered in [KS05].

###  3.5. Rotational Weingarten surfaces generated by elastic curves

The elastic curves are those plane curves whose curvature is, at all points,
proportional to the distance to a fixed line, called the directrix. They were
studied by Jacques Bernoulli in 1691 who named it elastica, by Euler in 1744,
and Poisson in 1833 (cf. [F93]). With the elastica in the $xz$-plane and the
directrix as the $z$ axis, the above condition can be written as
$\kappa(x)=2ax$, $a>0$, and then we have a one-parameter family of elastic
curves determined, up to $z$-translations, by the geometric linear momenta

(3.20) $\mathcal{K}(x)=ax^{2}-k,\ k>-1,$

since $\mathcal{K}(x)^{2}<1$ (see Remark 2.5). We define the elasticoids as
the rotational surfaces generated by the rotation of an elastic curve around
its directrix. Using Corollary 3.1, they are uniquely determined, up to
translations along $z$-axis, by the geometric linear momenta given in (3.20).
Following [F93] or [S08] for example, we can distinguish seven types of
elasticoids according to the seven types of elastic curves depending on the
possible ranges of values of the modulus $k\in(-1,+\infty)$ of the elliptic
functions which appear in the parametrizations of the elastic curves
generating the elasticoids. See Figures 6, 7, 8 and 9 for a description of
them.

Figure 6. Elasticoids I: surfaces of revolution generated by, from left to
right, the pseudo-sinusoids ($-1<k<0$) and the right lintearia ($k=0$). Figure
7. Elasticoids II: surfaces of revolution generated by, from left to right,
the elastic curves with $0<k<k_{1}$ and the pseudolemniscate
($k=k_{1}:=0.65222\dots$). Figure 8. Elasticoids III: surfaces of revolution
generated by, from left to right, the elastic curves with $k_{1}<k<1$ and the
convict curve ($k=1$). Figure 9. Elasticoids IV: surfaces of revolution
generated by the pseudotrochoids ($k>1$).

Using (3.1) and (3.20), we have that the principal curvatures for the
elasticoids are given by $k_{\text{m}}=2ax$ and $k_{\text{p}}=ax-k/x$. We can
eliminate $x$ since $a\neq 0$ obtaining the W-diagram of the elasticoids:

(3.21) $k_{\text{m}}^{2}-2k_{\text{p}}\,k_{\text{m}}=4ak.$

If $k\neq 0$, they are hyperbolae with asymptotes $k_{\text{m}}=0$ and
$k_{\text{m}}=2k_{\text{p}}$ (see Figure 10), which are obtained in (3.21)
when $k=0$.

Figure 10. W-diagram of the elasticoids.

The elasticoids with null modulus, i.e. $k=0$, are linear Weingarten surfaces
since they satisfy $k_{\text{m}}=2k_{\text{p}}$ (see Subsection 3.2).
Otherwise, we provide the following uniqueness result for Weingarten surfaces
involving hyperbolae in the curvature diagram.

######  Theorem 3.6.

The only rotational surfaces satisfying
$k_{\text{m}}^{2}-2k_{\text{p}}\,k_{\text{m}}+\mu=0$, $\mu\neq 0$, are the
sphere of radius $R=1/\mu$ ($\mu>0$) and the elasticoids with nonzero modulus.

###### Proof.

Using (3.1), the Weingarten relation
$k_{\text{m}}^{2}-2k_{\text{p}}\,k_{\text{m}}+\mu=0$ translates into the
o.d.e.

(3.22)
$\mathcal{K}^{\prime}(x)^{2}-2\mathcal{K}(x)\mathcal{K}^{\prime}(x)/x+\mu=0.$

We make the change of variable $\lambda(x)=\mathcal{K}(x)/x$ and (3.22)
becomes into the separable o.d.e. for $\lambda$:

(3.23) $\lambda^{\prime}(x)=\frac{\sqrt{\lambda^{2}(x)-\mu}}{x}.$

The constant solution $\lambda^{2}\equiv\mu>0$ leads to
$\mathcal{K}(x)=\sqrt{\mu}\,x$ and Proposition 3.3(3) gives the sphere of
radius $R=1/\mu$. Otherwise, by integrating (3.23), we get
$\lambda(x)+\sqrt{\lambda^{2}(x)-\mu}=2ax$, for some constant $a>0$. From here
we easily deduce that $\mathcal{K}(x)=ax^{2}+\frac{\mu}{4a}$. Looking at
(3.20), using Corollary 3.1, we arrive at the elasticoid of modulus
$k=-\frac{\mu}{4a}$, which is consistent with (3.21). This finishes the proof.
∎

##  4\. Prescribing curvature on a rotational surface

Making use of Corollary 3.1, we present in this section an existence and
uniqueness result on prescribing mean or Gauss curvature for a rotational
surface with an arbitrary continuous function depending on the distance from
the surface to the axis of revolution.

######  Theorem 4.1.

  1. (a)

Let $H=H(x)$, $x>0$, be a continuous function. Then there exists a one-
parameter family of rotational surfaces with mean curvature $H(x)$, $x$ being
the distance from the surface to the axis of revolution. The surfaces in the
family are uniquely determined, up to translations along $z$-axis, by the
geometric linear momenta of their generatrix curves given by

(4.1) $x\,\mathcal{K}(x)=2\\!\int\\!xH(x)dx.$

  2. (b)

Let $K_{\text{G}}=K_{\text{G}}(x)$, $x>0$, be a continuous function. Then
there exists a one-parameter family of rotational surfaces with Gauss
curvature $K_{\text{G}}(z)$, $x$ being the distance from the surface to the
axis of revolution. The surfaces in the family are uniquely determined, up to
translations along $z$-axis, by the geometric linear momenta of their
generatrix curves given by

(4.2) $\mathcal{K}(x)^{2}=2\\!\int\\!xK_{\text{G}}(x)dx.$

######  Remark 4.2.

The parameter in both uniparametric families described in Theorem 4.1 comes
from the integration constant in (4.1) and (4.2).

###### Proof.

The proof of part (a) simply relies on solving the linear equation

$\mathcal{K}^{\prime}(x)+\frac{\mathcal{K}(x)}{x}=2H(x)$

for unknown $\mathcal{K}$, coming from (3.2), and applying Corollary 3.1.

The same reasoning gives part (b) solving now the immediate equation

$\mathcal{K}^{\prime}(x)\mathcal{K}(x)=xK_{\text{G}}(x)$

coming from (3.3). ∎

As a first application of part (a) in Theorem 4.1, we provide a very short
simple proof of a classical result of Euler (cf. [E44]) concerning rotational
minimal surfaces, i.e. those with vanishing mean curvature.

######  Corollary 4.3.

The only minimal rotational surfaces are the plane and the catenoid.

###### Proof.

If $H=0$, then (4.1) leads to $\mathcal{K}(x)=c/x$, $c\in\mathbb{R}$. When
$c=0$, Proposition 3.3(1) gives the plane, and when $c\neq 0$, Proposition
3.3(5) leads to the catenoid. ∎

Applying again part(a) in Theorem 4.1, we now deal with a new shorter proof of
the classification of rotational CMC surfaces given by Delaunay (cf. [D41],
see also [PM19]). We recall (see [F93]) that the Delaunay surfaces are the
surfaces of revolution generated by the rotation of the Delaunay roulettes
around their base. According to [F93], the differential equation of these
curves in the $xz$-plane is given by

(4.3) $\left(\frac{dx}{dz}\right)^{2}=\frac{4a^{2}x^{2}-(x^{2}+\epsilon
b^{2})^{2}}{(x^{2}+\epsilon b^{2})^{2}},$

with $\epsilon=1$ for the elliptic roulette (ellipse with semi-axes $a$ and
$b$, $a>b$), and $\epsilon=-1$ for the hyperbolic roulette (hyperbola with
semi-axes $a$ and $b$). The surfaces associated to an elliptic roulette are
called onduloids and those ones associated to a hyperbolic roulette nodoids.
Since the parabolic roulette is the catenary, the associated surface is none
other than the catenoid (case of zero mean curvature). The Delaunay surfaces
have the remarkable property, apart from the circular cylinder (corresponding
to a circular roulette), of being the only surfaces of revolution with nonzero
constant mean curvature (if we include the sphere, which is a limit case). See
Figure 11.

Figure 11. Delaunay roulettes (the profile curves of the Delaunay surfaces)
generated by rolling the conics listed below them; this is Figure 1 in [PM19].

######  Corollary 4.4.

The only nonzero constant mean curvature rotational surfaces are the circular
cylinder, the sphere and the Delaunay surfaces.

###### Proof.

Assume, without restriction, $H\equiv H_{0}>0$. Then (4.1) leads to

(4.4) $\mathcal{K}(x)=H_{0}x+c/x,\ c\in\mathbb{R}.$

If $c=0$, $\mathcal{K}(x)=H_{0}x$ and we get the sphere of radius
$\frac{1}{H_{0}}$ from Proposition 3.3(3). We must also take into account the
case $x$ constant, because it produces the circular cylinder of radius
$\frac{1}{2H_{0}}$ (see Remark 3.2). Being $x$ non constant, we look for the
generatrix curve determined by (4.4) applying Remark 2.5, and obtain:

$s=s(x)=\int\frac{xdx}{\sqrt{x^{2}-(H_{0}x^{2}+c)^{2}}}$

and

$z=z(s)=\int\left(H_{0}x(s)+\frac{c}{x(s)}\right)ds.$

From the above two expressions, we deduce that

$\left(\frac{dz}{dx}\right)^{2}=\frac{(H_{0}x^{2}+c)^{2}}{x^{2}-(H_{0}x^{2}+c)^{2}}$

Comparing with (4.3), we arrive at Delaunay roulettes considering
$a=\frac{1}{2H_{0}}$ and $\epsilon b^{2}=\frac{c}{H_{0}}$, that is, if $c>0$,
taking $\epsilon=1$ and so $b^{2}=c/H_{0}$ and if $c<0$, taking $\epsilon=-1$
and so $b^{2}=-c/H_{0}$. This finishes the proof. ∎

Now we deal with some applications of part (b) in Theorem 4.1. It is immediate
to get the flat rotational surfaces, since $K_{\text{G}}\equiv 0$ implies that
$\mathcal{K}$ must be constant, and we arrive at planes, cones and cylinders
(see Proposition 3.3(1),(2) and Remark 3.2). More interesting is the study
(initiated by Darboux in 1890) of rotational surfaces with nonzero constant
Gauss curvature. They are called Darboux surfaces and we present them
following the description in [F93] and references therein.

Let $K_{\text{G}}\equiv K_{0}\neq 0$.

  * (i)

Positive Gaussian curvature $K_{0}=1/a^{2}$, $a>0$.

In cylindrical coordinates $(r,\theta,z)$, the surfaces are defined by

(4.5) $r=ak\cos t,\quad z=aE(k,t):=a\int_{0}^{t}\sqrt{1-k^{2}\sin^{2}u}\,du,\
k>0,$

where $E(k,\cdot)$ denotes the elliptic integral of second kind and modulus
$k>0$ ([BF71]). The case $k=1$ correspond to the sphere and the shapes are
different according to $0<k<1$ or $k>1$. See Figure 12.

Figure 12. Rotational surfaces with $K_{\text{G}}\equiv K_{0}>0$. From left to
right: $0<k<1$, $k=1$ (sphere), $k>1$.

  * (ii)

Negative Gaussian curvature $K_{0}=-1/a^{2}$, $a>0$.

There are also three kinds of surfaces, but this time with three different
parametrizations. See Figure 13.

    * –

First type: surfaces with a conical point. In cylindrical coordinates, the
surfaces are given by

(4.6) $r=ak\sinh t,\quad z=a\int_{0}^{t}\sqrt{1-k^{2}\cosh^{2}u}\,du,\ 0<k<1.$

    * –

Second type: surfaces that look like a hyperboloid. In cylindrical
coordinates, the surfaces are described by

(4.7) $r=ak\cosh t,\quad z=a\int_{0}^{t}\sqrt{1-k^{2}\sinh^{2}u}\,du,\ k\neq
0.$

    * –

Third type: the pseudosphere. The pseudosphere is the surface of revolution
generated by the rotation of a tractrix around its asymptote (see Proposition
3.3(7)). It was studied by Ferdinand Minding (1806-1885) and Eugene Beltrami
in 1868.

Figure 13. Rotational surfaces with $K_{\text{G}}\equiv K_{0}<0$. From left to
right: surface with a conical point, surface of hyperboloid-type,
pseudosphere.

Applying part(b) in Theorem 4.1, we provide a new proof of the classification
of rotational surfaces with nonzero constant Gaussian curvature.

######  Corollary 4.5.

The only nonzero constant Gauss curvature rotational surfaces are the Darboux
surfaces.

###### Proof.

Assume $K_{\text{G}}\equiv K_{0}\neq 0$. Then (4.2) leads to

(4.8) $\mathcal{K}(x)^{2}=K_{0}x^{2}+c,\ c\in\mathbb{R}.$

If $c=0$, $\mathcal{K}(x)^{2}=K_{0}x^{2}$, $K_{0}>0$, and we get the sphere of
radius $\frac{1}{\sqrt{K_{0}}}$ from Proposition 3.3(3).

Now we look for the rotational surface determined by (4.8) (see Corollary
3.1). Applying (2.6), we obtain:

(4.9) $z=z(x)=\pm\int\frac{\sqrt{c+K_{0}x^{2}}}{\sqrt{1-c-K_{0}x^{2}}}.$

We must distinguish two cases according to the sign of $K_{0}$. If
$K_{0}=1/a^{2}>0$, we have that

(4.10) $z=z(x)=\pm\int\frac{\sqrt{a^{2}c+x^{2}}}{\sqrt{a^{2}(1-c)-x^{2}}},$

with $c<1$ in this case. We put $c=1-k^{2}$, $k\in\mathbb{R}$, and make the
change of variable $x=ak\cos t$ in (4.10). Then we arrive at $z=\mp
a\int\sqrt{1-k^{2}\sin^{2}t}\,dt$. Looking at (4.5), we recover the Darboux
surfaces with positive constant Gauss curvature. We point out that $k=1$ (and
then $c=0$) gives again the sphere of radius $a>0$ as then
$z(x)=\mp\sqrt{a^{2}-x^{2}}$.

If $K_{0}=-1/a^{2}$, from (4.9) we get that

(4.11) $z=z(x)=\pm\int\frac{\sqrt{a^{2}c-x^{2}}}{\sqrt{a^{2}(1-c)+x^{2}}},$

with $c>0$ in this case.

When $0<c<1$, we write $c=1-k^{2}$, $0<k<1$. Then we make the change of
variable $x=ak\sinh t$ in (4.11) and so we arrive at $z=\pm
a\int\sqrt{1-k^{2}\cosh^{2}t}\,dt$. Looking at (4.6), we recover the first
type of Darboux surfaces with negative constant Gauss curvature.

When $c>1$, we can set $c=1+k^{2}$, $k\neq 0$. Now we make the change of
variable $x=ak\cosh t$ in (4.11) and arrive at $z=\pm
a\int\sqrt{1-k^{2}\sinh^{2}t}\,dt$. Looking at (4.7), we recover the second
type of Darboux surfaces with negative constant Gauss curvature.

When $c=1$, we simply have that

$z=z(x)=\pm\int\frac{\sqrt{a^{2}-x^{2}}}{x}dx=\pm\left(\sqrt{a^{2}-x^{2}}-a\ln\left(\frac{a+\sqrt{a^{2}-x^{2}}}{x}\right)\right),$

that is nothing but the Cartesian equation of the tractrix (cf. [F93]), the
generatrix curve of the pseudosphere. Anyway, from (4.8) we deduce that
$\mathcal{K}(x)=\sqrt{1-x^{2}/a^{2}}$ in this case. Then Proposition 3.3(7)
also leads to the pseudosphere. This finishes the proof.

∎

## Acknowledgements

This publication is part of the R+D+i project PID2020-117868GB-I00, financed
by MCIN / AEI / 10.13039 / 501100011033 /. The second author is also supported
by the EBM / FEDER UJA 2020 project 1380860.

## References

  * [BG94] B. van-Brunt and K. Grant. Hyperbolic Weingarten surfaces. Math. Proc. Cambridge Philos. Soc. 116 (1994), 489–504.
  * [BG96] B. van-Brunt and K. Grant. Potential applications of Weingarten surfaces in CAGD, I: Weingarten surfaces and surface shape investigation. Comput. Aided Geom. Design 13 (1996), 569–582.
  * [BF71] P.F. Byrd & M.D. Friedman. Handbook of elliptic integrals for engineers and physicists. 2nd edition. Springer Verlag, 1971.
  * [CCI16] I. Castro and I. Castro-Infantes. Plane curves with curvature depending on distance to a line. Diff. Geom. Appl. 44 (2016), 77–97.
  * [CCIs17] I. Castro, I. Castro-Infantes and J. Castro-Infantes. New plane curves with curvature depending on distance from the origin. Mediterr. J. Math. (2017) 14: 108.
  * [CCIs18] I. Castro, I. Castro-Infantes and J. Castro-Infantes. Curves in Lorentz-Minkowski plane: elasticae, catenaries and grim-reapers. Open Math. 16 (2018), 747–766.
  * [CCIs20a] I. Castro, I. Castro-Infantes and J. Castro-Infantes. Curves in Lorentz-Minkowski plane with curvature depending on their position. Open Math. 18 (2020), 749–770.
  * [CCIs20b] I. Castro, I. Castro-Infantes and J. Castro-Infantes. On a Problem of David Singer about Prescribing Curvature for Curves. Geom. Integrability & Quantization 21 (2020), 100–117.
  * [Ch45] S.S Chern. Some new characterizations of the Euclidean sphere. Duke Math. J. 12 (1945), 279–290.
  * [D90] G. Darboux. Sur la surface dont la courbure totale est constante. Ann. Sci. Éc. Norm. Supér. (3) 7 (1890), 9–18.
  * [D41] C. Delaunay. Sur la surface de revolution dont la courbure moyenne est constante. J. Math. Pures Appl. 6 (1841), 309–320.
  * [E44] L. Euler. Methodus inveniendi lineas curvas: maximi minimive proprietate gaudentessive solutio problematis isoperimetrici latissimo sensu accepti (in Latin). Opera Omnia: Series 1, Volume 24 (1744).
  * [F93] R. Ferréol. Encyclopédie des formes mathématiques remarquables. www. mathcurve. com
  * [GMM03] J.A. Gálvez, A. Martínez and F. Milán. Linear Weingarten surfaces in $\mathbb{R}^{3}$. Monatsh. Math. 138 (2003), 133–144.
  * [H51] H. Hopf. Über Flächen mit einer Relation zwischen den Hauptkrümmungen. Math. Nachr. 4 (1951), 232–249.
  * [K80] K. Kenmotsu. Surfaces of revolution with prescribed mean curvature. Tôhoku Math. J. 32 (1980), 147–153.
  * [KS05] W. Kühnel and M. Steller. On closed Weingarten surfaces. Monatsh. Math. 146 (2005), 113–126.
  * [LS84] J. Langer and D. Singer. The total squared curvature of closed curves. J. Diff. Geom. 20 (1984), 1–22.
  * [L08] R. López. Special Weingarten surfaces foliated by circles. Monatsh. Math. 154 (2008), 289–302.
  * [LP20a] R. López and A. Pámpano. Clasification of rotational surfaces in Euclidean space satisfying a lineal relation between their principal curvatures. Math. Nachr. 293(4) (2020), 735–753.
  * [LP20b] R. López and A. Pámpano. Rotational surfaces of constant astigmatism in space forms. J. Math. Anal. Appl. 483(1) (2020), 123602, 21 pp.
  * [LP20c] R. López and A. Pámpano. Classification of rotational surfaces with constant skew curvature in 3-space forms. J. Math. Anal. Appl. 489(2) (2020), 124195, 19 pp.
  * [LP21] R. López and A. Pámpano. Stationary soap films with vertical potentials. Preprint 2021, arXiv:2111.03293[math.DG]
  * [MO03] I.V. Mladenov and J. Oprea. The Mylar balloon revisited. Amer. Math. Monthly 110 (2003), 761–784.
  * [PHM19] V.I. Pulov, M.T. Hadzhilazova and I.V. Mladenov. Deformations Without Bending: Explicit Examples. Geom. Integrability & Quantization 20 (2019), 246–254.
  * [PM19] V.I. Pulov and I.V. Mladenov. Rotating liquid drops and Delaunay surfaces. J. Geom. Symmetry Phys. 54 (2019), 55–78.
  * [PM20] V.I. Pulov and I.V. Mladenov. Further deformations of the axially-symmetric non-bending surfaces. J. Geom. Symmetry Phys. 55 (2020), 51–73.
  * [RS94] H. Rosenberg and R. Sa Earp. The geometry of properly embedded special surfaces in $\mathbb{R}^{3}$; e.g., surfaces satisfying $aH+bK=1$, where $a$ and $b$ are positive. Duke Math. J. 73 (1994), 291–306.
  * [V59] K. Voss. (1959) Über geschlossene Weingartensche Flächen. Math. Annalen 138 (1959), 42–54.
  * [W61] J. Weingarten. Ueber eine Klasse auf einander abwickelbarer Flächen. J. Reine Angew. Math. 59 (1861) 382–393.
  * [S99] D. Singer. Curves whose curvature depends on distance from the origin. Amer. Math. Monthly 106 (1999), 835–841.
  * [S08] D. Singer. Lectures on elastic curves and rods. Curvature and Variational Modelling in Physics and Biophysics, AIP Conf. Proc. vol. 1002 (2008), 3–32.