Roulettes of conics, Delaunay surfaces and
applications
Bhagya Athukorallage, Thanuja Paragoda and Magdalena Toda
Abstract. This work represents a survey of results on roulettes of conics and Delaunay surfaces. The differential equations corresponding to roulettes of conics appear
scarcely in the literature, and their real-world applications are also emphasized way too
rarely. The purpose of this survey is to make the connection between all these topics a lot
more obvious and relevant. In the last section, we construct an extension of an example
provided by Thomas Vogel, on the topic of liquid bridges as Delaunay rotational surfaces.
We model this application using numerical methods. 1 2
1
Constant mean curvature surfaces of revolution
(Delaunay surfaces)
Constant mean curvature (CMC) surfaces have played a prominent role in
differential geometry. In 1841, Charles Eugéne Delaunay introduced a way
of constructing rotationally symmetric CMC surfaces in R3 , by proving that
a surface of revolution in R3 is a CMC surface if and only if its profile
curve is the roulette of a conic. A surface of revolution in R3 is generated
by revolving a given profile curve about a line in the plane containing this
given curve [18].
Let us consider a surface of revolution given by the parametrization
σ(u, v) = (u, f (u) cos v, f (u) sin v),
(1.1)
where u belongs to an open interval (α, β) of the real line, f (u) is a realvalued smooth function, and v belongs to the interval (0, 2π).
Definition 1. The first and second fundamental forms [12] of the surface
patch σ(u, v) are defined as (1.1)
I = E du2 + 2F dudv + G dv 2
and
1
II = e du2 + 2f dudv + g dv 2 , (1.2)
2010 Mathematics Subject Classification: 53A10
Keywords: roulettes of conics; Delaunay surfaces; liquid bridges between vertical
plates.
2
1
where the corresponding coefficients
E = σu · σu , F = σu · σv , G = σv · σv ,
e = σuu · N, f = σuv · N, g = σvv · N.
(1.3)
Here, N represents the outward unit normal to the surface, and
N=
σu × σv
.
kσu × σv k
(1.4)
By differentiating the parametric relation σ(u, v), with respect to the parameters u and v, we get
σu (u, v) = (1, f 0 (u) cos v, f 0 (u) sin v),
σv (u, v) = (0, −f (u) sin v, f (u) cos v).
By using the above definition, we obtain the expressions for the coefficients of the First and Second Fundamental Forms (ref (1.3))
E = 1 + f 0 (u)2 ; F = 0; G = f (u)2 ;
00
e = √−f 02 ;
f = 0; g = √ f 02
1+f
1+f
with the unit normal
1
N= p
(f 0 , − cos v, − sin v).
02
1+f
Next, with the aid of the above calculated coefficients values, the mean
curvature H of the surface [12] can be easily calculated using the relation
H =
Eg − 2F f + Ge
.
2(EG − F 2 )
(1.5)
After some simplifications, we obtain
H =
−f f 00 + 1 + f 02
p
.
2f (1 + f 02 )3
(1.6)
Let us assume that H = c/2 is a constant. Then we have the differential
equation
3
1 + f 02 − f f 00 = cf (1 + f 02 ) 2 .
2
Note that for the minimal surfaces of revolution, we obtain
1 + f 02 − f f 00 = 0.
Suppose c = ±1/a with a > 0. Then we get
f2 ± p
2af
1 + f 02
= ±b2 ,
where b is a constant. This represents a well-known result, which can be
stated as
Theorem 2. [19]: A surface of revolution M parametrized by
σ(u, v) = (u, f (u) cos v, f (u) sin v)
has constant mean curvature if and only if the function f (u) satisfies
f2 ± p
2af
1+
f 02
= ±b2 ,
where a and b are constants.
Delaunay introduced the catenoids, unduloids, and nodoids as constant
mean curvature surfaces of revolution, in addition to the previously well
known surfaces spheres and cylinders [2]. He further showed that the profile
curves of these surfaces can be obtained as roulettes of the conics (roulette
of parabola, ellipse, and hyperbola, respectively) - see for example [2, 8, 16].
Consider the path generated by a focus of a conic that is moving along
a straight line (see Fig.1). The trace of the focus is called the roulette of the
corresponding conic. In [8, 16], authors derive the equations for the roulettes,
which are obtained through a parabola, an ellipse, and a hyperbola. The
resulted roulettes are respectively called catenary, undulary, and nodary,
and surfaces that are generated by revolving these curves about a fixed axis
are the catenoid, unduloid, and the nodoid, respectively.
2
Undulary as the roulette of an ellipse, and
the corresponding unduloid
A detailed derivation of the equations of the aforementioned roulettes are
given in [8]. In this section, we provide the actual generation of an undulary
as a roulette of an ellipse. We present the derivation given in [8, 19, 21].
3
Let E be an ellipse with a > b, where a and b are the lengths of the
semi-major and semi-minor axes. Consider the motion of E on a straight
line sl , and let the path of the focus F1 be l. O is the contact point of E
with sl . Let F1 O1 and F2 O2 be perpendicular to line sl . Further, assume F1
and F2 have the Cartesian coordinates (x, y) and (x̃, ỹ), respectively. Let T
be the local tangent line of the curve l that passes through the point (x, y),
and assume φ be the angle between the x−axis and T .
Figure 1: Roulette of an ellipse E. F1 and F2 are the foci of E. l is the locus
of the focus F1 , and T is the local tangent line of l.
By the definition of an ellipse, we have
|F1 O| + |F2 O| = 2a,
(2.1)
|F1 O1 | |F2 O2 | = y ỹ = b2 .
(2.2)
and
The last relation is also known as the pedal property for an ellipse.
Observe that F1 O is perpendicular to the line T . Hence, we have
y = |F1 O| cos φ.
(2.3)
ỹ = |F2 O| cos φ.
(2.4)
Moreover,
Substituting (2.3) and (2.4) into (2.1), results in
y + ỹ = 2a cos φ.
4
(2.5)
Note that, according to (2.2)
b2
.
y
Thus, (2.5) may be expressed in the form
ỹ =
y+
b2
= 2a cos φ.
y
(2.6)
We assume the curve l is parametrized by the arc length s. Then
dx
= cos φ,
ds
(2.7)
and hence, we obtain:
ds
=
dx
s
1+
dy
dx
2
.
(2.8)
Finally, after substituting the previous two results into (2.6), and simplifying, we obtain the equation
2ay
p
1 + y 02
− y 2 = b2 ,
(2.9)
which represents a differential equation corresponding to an undulary.
Note that this equation is a particular case of the general differential
equation obtained at the end of the previous section (see the four differential
equations based on ± signs). This shows that the undulary (as a roulette of
an ellipse) will generate a CMC surface of revolution when rotated around
a line. This surface is called unduloid.
The profile curve of an unduloid (that is, undulary) has a parametrization [6] of the following type
Zt
b2
du
√
x(t) =
,
a
(1 + e cos u) 1 − e2 cos2 u
0
r
1 − e cos t
y(t) = b
,
1 + e cos t
(2.10)
where a and b with a > b, represent the semi-major and semi-minor axes of
an ellipse respectively, and focus and eccentricity of the ellipse are and e.
Moreover, and e are given by the equations:
2 = a2 − b2 ,
e =
.
a
5
Proposition 3. The parametrization of the undulary satisfies the following
differential equation for the case = 1:
2
dy
4a2 y 2
= 2
− 1.
dx
y + b2
Proof. The proof is immediate. Note that the differential equation in the
statement is equivalent to the equation
2ay
p
− y 2 = b2
1 + y 02
(2.11)
which characterizes the undulary. Note that epsilon, hereby was introduced
to allow a comparison with the analogous differential equation describing a
nodary (see Proposition 6).
The surface of revolution described by an undulary profile is called an
unduloid (sometimes spelled ”onduloid”), and has constant nonzero mean
curvature.
Note: In 2010 Ivalo M. Mladenov presented a new class of axially symmetric surfaces, which generalizes Delaunays unduloids and provides solutions
of the shape equation is described in explicit parametric form. This class
provides the first analytical examples of surfaces with periodic curvatures
studied by K. Kenmotsu, and leads to some unexpected relationships among
Jacobian elliptic functions and their integrals [7].
Figure 2: The unduloid
3
Catenaries and catenoids
To generate the catenary as a roulette of a conic, we start from the parabola
x(t) = a sinh−1 (t)
p
y(t) = a 1 + t2
6
(3.1)
(3.2)
where a is the focal length (the distance from the vertex to focus) and the
focus of this parabola is at the point (0, a). Eliminating the parameter t
gives us the usual equation for the catenary
x
y = a cosh
,
a
and using (3.1) and (3.2), we have
dy
dt
dx
dt
at
,
1 + t2
a
√
,
1 + t2
√
=
=
which further implies
dy
= t,
dx
2
dy
= t2 .
dx
(3.3)
Since
y 2 = a2 + a2 t 2 ,
(3.4)
one will obtain
Proposition 4. The parametrization of a standard catenary satisfies the
following differential equation:
2
y2
dy
= 2 − 1.
(3.5)
dx
a
The standard parameterization of a catenoid (as a surface of revolution
whose profile curve is a catenary) is the following:
x(u, v) = a cos u cosh
y(u, v) = a sin u cosh
z(u, v) = v,
7
v va
a
,
,
(3.6)
where u and v are real parameters, and a is a non-zero real constant. The
principal curvatures κ1 , κ2 of the surface are given by
κ1 = −κ2 =
1
.
a cosh2 (v/a)
Note that H = 0, and the Gauss curvature
K=−
1
a2 cosh4 (v/a)
.
The catenoid is a well known minimal surface with many important
applications to the real world. It can be deformation-retracted to a helicoid
(see for example, [13, 1]), meaning that the helicoid and catenoid belong to
the same associated family of surfaces (1-parameter family of deformations).
The following is a well known and very important classical result which
can be found in [11].
Theorem 5. A surface of revolution M, which is a minimal surface, is
contained in either a plane, or a catenoid [11].
Figure 3: The catenoid
As a side note, we would like to mention that in 2010 Masato and Taku
introduced a physics experiment that involves using a soap film to form a
8
catenoid. Using that soap film they created catenoids between two rings
and characterized the catenoid in situ while varying the distance between
the rings [13]. This is an appropriate experiment which combines theory
and experimentation.
4
Nodary and nodoids:
In differential geometry, the locus of a focus of a hyperbola as the point of
contact rolls along a straight line in a plane forms the curve which we call
the nodary. Then a nodoid is a surface of revolution with constant nonzero
mean curvature obtained by rolling a hyperbola along a fixed line, tracing
the focus, and revolving the resulting nodary curve around the line [19].
Figure 4: Cross section of a nodoidal surface.
Let the hyperbola be given by the equation
x2 y 2
− 2
a2
b
= 1,
where a > b > 0. The parametrization of the corresponding nodary [6] is
Zt
b2
cos u du
√
x(t) =
,
a
(e + cos u) e2 − cos2 u
0
r
e − cos t
y(t) = b
.
e + cos t
9
(4.1)
Proposition 6. The parametrization of the nodary satisfies the following
differential equation for the case = −1:
2
dy
4a2 y 2
= 2
− 1.
dx
y + b2
Proof. The proof is immediate.
We are now briefly presenting the standard nodoid (see, e.g., [20]).
Let (x, y, z) be the usual rectangular coordinates for R3 . Consider a
Delaunay nodoid with the x−axis as its rotation axis, and with constant
mean curvature H = 1. Let (x(t), z(t)), t ∈ R be a parametrization of the
profile curve of the nodoid in the xz−plane with z(t) > 0, and so the surface
can now be parametrized by
D(t, θ) = (x(t), z(t) cos θ, z(t) sin θ),
t ∈ R, θ ∈ [0, 2π).
(4.2)
Suppose further that the parameter t is chosen to make the mapping D(t, θ)
conformal with respect to the coordinates (t, θ). Let t = a and t = b be
values at which the nodoid achieves two adjacent necks, that is, z(t) has local
minima at both t = a and t = b equal to the neck radius r. Conformality
implies that the first fundamental form is
ds2 = ((x0 )2 + (z 0 )2 )dt2 + z 2 dθ2 = ρ2 (dt2 + dθ2 ),
with
ρ2 = (x0 )2 + (z 0 )2 = z 2 .
The second fundamental form is then given by
dσ 2 =
1 00 0
(x z − z 00 x0 )dt2 + x0 dθ2 ,
z
and so the coordinates (t, θ) are curvature line coordinates, that is, the
coordinates are isothermic. Furthermore, the mean curvature H = 1 implies
2z 3 − z 0 x00 + x0 z 00 − zx0 = 0.
Note: In 2005, Wayne Rossman gave two numerical methods for computing the first bifurcation point for Delaunay nodoids. With regard to
methods for constructing constant mean curvature surfaces, they concluded
that the bifurcation point in the analytic method of Mazzeo-Pacard is the
same as a limiting point encountered in the integrable systems method of
Dorfmeister-Pedit-Wu [20].
10
5
Application of Delaunay surfaces:
mathematical models for the liquid bridge between two vertical plates
Hereby we model the liquid bridge between two plates using an unduloid.
Thus, the profile curve of the meniscus is considered to be an undulary. We
became interested in this model while investigating the shape of the tear
meniscus that forms around a contact lens. Since the tear film thickness is
much smaller than the radius of the cornea and the contact lens, one may neglect the curvature of both the contact lens and the cornea, and treat them as
flat surfaces. Thus, to make this problem amenable to analysis, we consider
a meniscus of a liquid bridge that forms between two vertical plates. Both
modeling and the stability of this type of problem are extensively studied
in [22, 23, 24]. Authors used the Young-Laplace equation or energy minimization approach to model the liquid meniscus. In this case, both methods
result in a nonlinear differential equation, which describes a constant mean
curvature (CMC) surface. One approach is to model these CMC surfaces as
Delaunay surfaces: catenoids, unduloids, and nodoids. However, due to the
self-intersections, modeling a liquid surface via a nodoid is unphysical [3].
We consider a liquid drop, which is trapped between two vertical plates,
and model the profile curve of the drop using a Calculus of Variations approach. In this method, a formula for the profile curve that formed between
the liquid-air interface is obtained by minimizing the total potential energy
of the drop while imposing a volume constraint [5, 15, 23]. The total potential energy of a liquid is mainly composed of three type of energy forms: (i)
surface energy of a liquid surface, which is proportional to the surface area
of the liquid-air interface (free surface), (ii) wetting energy that arises due
to the contact area of solid-liquid interface, and (iii) gravitation potential
energy, which we will neglect here. A detailed discussion of these three types
of energy terms can be found in [9]. In this study, we consider a rotationallysymmetric liquid drop, and hence the latter energy type is neglected in our
analysis.
Consider a liquid drop which is in between two vertical plates, and the
profile of the drop has the equation z = f (x) with respect to the configuration of the Cartesian coordinate system, which is on the left plate (plate
1) (refer 5). Note that the continuity of the drop implies that f (x) > 0 on
the interval [0, L]. Let the distance between the vertical plates be L, and
assume the shape of the solid-liquid contact area on the plates to be circles
with radii f (0) and f (L), respectively. The relative adhesion coefficients of
11
the liquid with the plates 1 and 2 are β1 and β2 .
Figure 5: Liquid drop in between two parallel plates. Distance between the
two plates is L.
Thus, under the absence of gravity, the total potential energy E of a
rotationally symmetric liquid drop may be written in the following form
[22, 9]:
ZL
E =
p
2πγf (x) 1 + f 02 (x) dx − γβ1 πf (0)2 − γβ2 πf (L)2 .
(5.1)
0
In (5.1), γ denotes the surface energy per unit area of the liquid; βi is the
relative adhesion coefficient between the ith wall and the liquid. The integral
term represents the surface energy of the liquid drop, and the last two terms
denote the wetting energy of the drop. We wish to minimize the energy E
given in (5.1) subjected to the volume constraint
ZL
πf (x)2 dx = V0 ,
(5.2)
0
where V0 denotes the volume of the liquid drop. Thus, the new energy
12
functional Ē that includes the volume constraint (5.2) is
ZL
Ē =
ZL
p
2
2
2
02
2πγf (x) 1 + f (x) dx−γβ1 πf (0) −γβ2 πf (L) +λ
πf (x) dx−V0 .
0
0
(5.3)
Here, the Lagrange multiplier λ is an unknown constant. We consider the
variation of Ē (δ Ē) with respect to the drop radius (capillary surface height)
f (x) and the meniscus height at the end points: f (0) and f (L).
δ Ē = 2πγ
ZL p
1+
f 02 (x) δf
ZL
dx + 2πγ
0
0
f0
δfx dx − 2πγβ1 f (0)δf (L)
fp
1 + f 02
ZL
−2πγβ2 f (L)δf (L) + 2πλ
f (x) δf dx,
(5.4)
0
which simplifies to
f 0 (L)
f 0 (0)
δf (0) + 2πγf (L) p
− β2 δf (L)
δ Ē = −2πγf (0) β1 + p
1 + f 02 (0)
1 + f 02 (L)
ZL p
λ
d
ff0
02
p
f + 1 + f (x) −
δf (x) dx.
(5.5)
+2πγ
γ
dx
1 + f 02 (x)
0
The necessary condition for the energy minimization [10, 17] is that δ Ē = 0.
Thus, we have the following system of equations that reads
p
λ
d
ff0
02
p
f + 1 + f (x) −
= 0 in [0, L],
(5.6)
γ
dx
1 + f 02 (x)
f 0 (0)
β1 + p
= 0 at x = 0,
(5.7)
1 + f 02 (0)
f 0 (L)
p
− β2 = 0 at x = L. (5.8)
1 + f 02 (L)
Let the value of the contact angles of the liquid meniscus with the plates
be θ1 and θ2 , and assume they are rotationally invariant. Hence, we have
the same contact angle values along the periphery of the contact circles. We
observe that f 0 (0) = − cot θ1 and f 0 (L) = cot θ2 . Then the relative adhesion
coefficients β1 and β2 in (5.7) and (5.8) may be expressed as
β1 = cos θ1
and
13
β2 = cos θ2 .
(5.9)
Finally, simplifying (5.6) results in
λ
f f 00 − f 02 − 1
,
=
γ
f (1 + f 02 )3/2
∀ x ∈ [0, L].
(5.10)
Equation (5.10) represents the liquid surface of the drop in terms of its
profile curve f (x), and in the next section, we further observe that the right
hand side of this equation relates to the mean curvature of the liquid surface.
Since Lagrange multiplier λ and the surface energy per unit area of the
liquid γ are constants, (5.10) leads us to an equation of the form:
λ
f f 00 − f 02 − 1
=
= 2H.
γ
f (1 + f 02 )3/2
(5.11)
Note that (1.6) and (5.11) are practically the same equation, up to a sign
that depends on the orientation chosen on the surface.
Remark 7. Multiplying (5.11) through f 0 [19], rearranging the terms, and
integrating with respect to the x variable, one may obtain
f 0 [(1 + f 02 ) − f f 00 ]
p
(1 + f 02 )3
0
f
p
+ H(f 2 )0
02
1+f
0
f
2
p
+ Hf
1 + f 02
f
p
+ Hf 2
1 + f 02
f
f2 + p
H 1 + f 02
2Hf f 0 +
= 0,
= 0,
= 0,
= C1 ,
= C2 ,
(5.12)
where C1 , C2 are constants. Observe that (5.12) has the same form as in
Theorem 2.
5.1
Profile curve for a rotationally symmetric liquid bridge:
undulary
As we observed in Section 1, surfaces that minimize energy under a volume
constraint have a constant mean curvature, and lead to a nonlinear differential equation [4]. In particular, our mathematical model (see (5.10)) has a
14
non-zero mean curvature value for its surface. In [14], the author found and
solved a complex nonlinear differential equation, which describes roulettes
(profile curves of CMC surfaces) of different types of conics. For a nonzero
CMC surface, the author analytically obtained a one-parameter family of
profile curves in terms of the constant mean curvature H, which yields to
[14, 16]
Zs
ϕ(s; H, B) =
1 + B sin(2Ht)
dt ,
1 + B 2 + 2B sin(2Ht)
p
1 + B 2 + 2B sin(2Hs)
(5.13)
.
2|H|
0
Here, B and H are real numbers.
However, as stated in [12], (5.13) does not have a direct geometrical
interpretation of the profile curves. In the same paper, the parametrization
for the profile curve of an unduloid, which is called the undulary, is given
in terms of the elliptic functions and formulas for the length, surface area,
and volume of appropriate parts presented. Here the parameterization for
an undulary is given in terms of two real free parameters: a and c.
µt π
µt π
x(t) = aF
− , k + cE
− ,k ,
(5.14)
2
4
2
4
p
z(t) =
m sin µt + n,
(5.15)
where
µ=
2
,
a+c
k2 =
c2 − a2
,
c2
m=
c2 − a2
,
2
n=
c2 + a2
,
2
(5.16)
and F (ϕ, k) and E(ϕ, k) are the elliptic integrals of the first and second kind,
respectively. Furthermore, authors show that the appropriate variation of
the parameters a and c yield to four Delaunay surfaces: unduloids, nodoids, spheres,
and cylinders. The above parameterizations (5.14) and (5.15) create a rotationally symmetric surface with the mean curvature
H=
1
.
a+c
In the present analysis, we recall the parametrization of an undulary
that was originally introduced by Delaunay (see [2]) and presented as (2.10)
in our paper. Consider again the standard ellipse
x2 y 2
+ 2 = 1,
a2
b
15
(5.17)
where, a and b represent the semi-major and semi-minor axes, respectively.
Assume a > b. Let the focus and the eccentricity of the ellipse be and e.
As in Section 2, and e are given by the following equations:
2 = a2 − b2 ,
e =
.
a
(5.18)
(5.19)
By using (5.18) and (5.19), the parametrization (2.10) can be rewritten
as
a2 − 2
a
Zt
du
q
(1
+
1−
cos
u)
0
a
r
p
a − cos t
.
y(t) =
a2 − 2
a + cos t
x(t) =
2
a2
,
cos2 u
(5.20)
In [23], the author considered an unduloid generated by an ellipse of arc
length 2π. On the other hand, in the present work, we extend Vogel’s study
by considering an unduloid that is generated from an ellipse of arc length 2h,
where h is the distance between the two vertical plates. Thus, the length of
the semi-major axis, a, of the ellipse is expressed as a function of both and
h (see Lemma 8). Then, using (5.20) together with (5.21), we numerically
obtain undulary profiles for different values (see Figures 6(a) and 6(b)).
For the numerical example, the distance between the plates (h) is assumed to be π.
Lemma 8. We may express a = a(, h) [23], which is given by the equation
3 3
h 2 π
π
a(, h) = +
+O
.
(5.21)
π
4 h
h3
√
Here, h is the distance between the vertical parallel plates, and = a2 − b2 .
Proof. Since 2h represents the arc length of the ellipse defined in (5.17), we
have
Z2π p
2h =
a2 − 2 cos2 θ dθ,
0
Z2π
=
s
a
1−
0
16
cos θ
a
2
dθ.
(5.22)
q
θ 2
, (5.22) may be rewritUsing the power series representation of 1 − cos
a
ten in the following form:
s
2 cos2 θ 4 cos4 θ
cos θ 5
cos θ 2
=1−
−
+O
.
1−
a
2a2
8a4
a
By substituting the above relation into (5.22) and integrating the definite
integral, we obtain
h
2
=a−
+ O(4 ).
(5.23)
π
4a
Neglecting the higher order terms and solving the resulted quadratic equation, the length of the semi-major axis a may be expressed as a function of
.
r
3 3
2 π 2
h 2 π
π
h
1+ 1+ 2
= +
+O
(5.24)
a() =
2π
h
π
4 h
h3
In our analysis, we showed that the shape of the liquid meniscus has a
constant mean curvature. This result is due to the neglecting gravitation
potential energy term from the total energy of the liquid drop. This assumption is satisfied, if the volume of the liquid drop is sufficiently small.
Furthermore, if the second variation of the total potential energy is nonnegative, stable liquid bridge is formed. Stability criteria for a liquid bridge in
between two plates are extensively studied in [23, 24]. In [23], the author
proved that for sufficiently large volumes a stable drop exists that has a
profile curve with no inflection point.
Theorem 9. [23] If θ1 + θ2 6= π, then the family of profile curves without
inflection remains stable as volume decrees from infinity, at least until eidV
ther dH
changes sign or an inflection point appears on the boundary. The
latter condition states that all profiles in the family are stable except for the
limiting profile that has f 00 = 0 at an endpoint.
Note that, in general, the profile of an undulary may have zero, one,
two, and four inflection points. For fixed ends, unduloids are stable only if
their critical length equals the single period [3, 15]. In our analysis, we are
interested in tear film whose thickness varies in the interval [40 − 60] µm.
Thus, the parameter h . Hence, the profile of the liquid meniscus is
modeled as a part of an undulary that has no inflection point.
17
1.15
2h
1.1
1.05
y(t)
1
0.95
0.9
0
2
4
6
8
10
12
14
x(t)
(a) h = π and = 0.2
2.5
ǫ=0
ǫ = 0.2
ǫ = 0.4
ǫ = 0.6
ǫ = 0.8
ǫ = 1.0
2
1.5
y(t)
1
0.5
0
0
2
4
6
8
10
12
14
x(t)
(b) h = π and = 0 : 0.2 : 1
Figure 6: Undulary profiles for h = π. Each profile curve is obtained by considering two revolutions of the ellipse, which is defined by the corresponding
value.
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Bhagya Athukorallage
Thanuja Paragoda
Department of Mathematics and Statis-
Department of Mathematics and Statis-
tics,
tics,
Texas Tech University,
Texas Tech University,
Broadway&Boston Ave,
Broadway&Boston Ave,
Lubbock, TX 79409-1042, USA.
Lubbock, TX 79409-1042, USA.
e-mail: bhagya.athukorala@ttu.edu
e-mail: thanuja.paragoda@ttu.edu
Magdalena Toda
Department of Mathematics and Statistics,
Texas Tech University,
Broadway&Boston Ave,
Lubbock, TX 79409-1042, USA.
e-mail: magda.toda@ttu.edu
http://www.math.ttu.edu/~mtoda/
20