Acta Polytechnica


DOI:10.14311/AP.2020.60.0313
Acta Polytechnica 60(4):313–317, 2020 © Czech Technical University in Prague, 2020

available online at https://ojs.cvut.cz/ojs/index.php/ap

EXTERNAL ROLLING OF A POLYGON ON CLOSED
CURVILINEAR PROFILE

Tetiana Kresan, Serhii Pylypaka, Zinovii Ruzhylo, Ivan Rogovskii,
Oleksandra Trokhaniak∗

National University of Life and Environmental Sciences of Ukraine, Heroiv Oborony Str., 15, Kyiv, Ukraine
∗ corresponding author: klendii_o@ukr.net

Abstract. The rolling of a flat figure in the form of an equilateral polygon on a curvilinear profile
is considered. The profile is periodic. It is formed by a series connection of an arc of a symmetrical
curve. The ends of the arc rely on a circle of a given radius. The equation of the curve, from which the
curvilinear profile is constructed, is found. This is done provided that the centre of the polygon, when
it rolls in profile, must also move in a circle. Rolling occurs in the absence of sliding. Therefore, the
length of the arc of the curve is equal to the length of the side of the polygon. To find the equations
of the curve of the profile, a first-order differential equation is constructed. Its analytical solution is
obtained. The parametric equations of the curve are obtained in the polar coordinate system. The
limits of the change of an angular parameter for the construction of a profile element are found. It is a
part of the arc of the curve. According to the obtained equations, curvilinear profiles with different
numbers of their elements are constructed.

Keywords: Equilateral polygon, curvilinear profile, external rolling, differential equation, centroids.

1. Introduction
Some flat figures, including polygons that can be rolled
on a curvilinear periodic profile without sliding, are
considered in [1]. The profile is formed by equal sym-
metrical curvilinear elements of their serial connection
so that the ends of the elements abut on a straight
line. When rolling a polygon on such profile, its centre
moves in a straight line. Constructing a closed profile
in which curvilinear elements touch a circle is impor-
tant for the design of centroids of non-circular wheels.
When rolling a polygon on such profile, its centre
moves in a circle. If both centres (centre of the curvi-
linear profile and centre of the polygon) are stationary,
then you can roll these figures while rotating around
their centres. One centroid will be a polygon, the
other will be a closed profile. Many works are devoted
to the study of the rolling of flat figures one by one.
There are common examples of rolling a straight line
segment along a curve and vice versa - rolling a curvi-
linear profile along a straight line. For the first case,
the rolling of a straight line in a circle is the classic
one, as a result of which the point of the line describes
the evolvent of a circle, and for the second case, the
rolling of a circle along a straight line, as a result of
which a circle point describes a cycloid [2]. In [3], in-
formation on the rolling of second-order curves along
a straight line is given. The trajectory of focus in
such rolling is known curves. The formation of flat
curves by given kinematic parameters is considered
in [4]. The basics of designing non-circular wheels
for gears are given in [5]. Geometric modelling of a
centroid of non-circular wheels was further developed
in the works [6–8]. The use of non-circular wheels in

gears has been considered in [9–17], in chain drives
in a monograph [18]. The purpose of the article is
to develop an analytical description of a curvilinear
closed profile, in which an equilateral polygon will be
rolled without sliding and its centre will move in a
circle of a given radius.

2. Material and method
Consider the rolling of a polygon by the example of
a square. We need to find a form of a flat profile in
which the square will be rolled without sliding, and
its centre will move in a circle of radius r (Fig. 1,a).
Rolling of a square can be considered as an example
of rolling of a triangle, the basis of which is the side
of the square, and the altitude - the distance AC=a
from its centre to the side. In the initial position, the
altitude of the AC is located on the Ox axis (Fig. 1,a).
When rolling a square, its centre moves in a circle of
a radius r, and the side rolls along the curve, that
will be found. The moment comes when the point
of contact of the square with the curve becomes its
vertex. The diagonal A’C’ in this position passes
through the origin of coordinates - the point O. It is
obvious that the length s of the arc AA’ is equal to
the length of half of the side of the square (Fig. 1,a).
When rolling, the side of the square is tangent to the
curve, and in the position where the point of contact
is its vertex, both sides of the square touch the curves.
With further rolling, the process is repeated. In such
a way that a rectilinear plane profile will consist of
equal arcs that intersect on a circle of radius ro at
right angle (Fig. 1,a).

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T. Kresan, S. Pylypaka, Z. Ruzhylo et al. Acta Polytechnica

Figure 1. Graphic illustrations for rolling a square along a curvilinear contour: a) schematic representation of the
two positions of the square when it is rolling; b) the current point of contact A’ in the polar coordinate system.

In Fig. 1,a, it is shown that in the position when
the point of contact with the curve is the vertex of
the square, its centre is on the radius-vector OC’.
This applies to any current point of contact of the
side of the square with the curve. Let the triangle,
which is the fourth part of the square, touch the
curve at the current point A’ (Fig. 1,b). As it is
rolling without sliding, the current point of contact
A’ can be considered as the instantaneous centre of
the rotation of the segment A’C’ around it. In this
case, the direction of the velocity of the point C’
must be perpendicular to the segment A’C’. However,
on condition that the point C’ moves in a circle of
a radius r, that is, it must be perpendicular to the
radius-vector OC’. Thus, the centre of the square C’
(or the vertex of the corresponding triangle, whose
base is the side of the square) and the point of contact
of the side of square A’ are located on a common
radius-vector that starts at the origin. Due to this,
the equation of the curve of the profile is conveniently
considered in the polar coordinate system.

Denote the distance from the origin of coordinates
to the point of contact OA’=ρ (Fig. 1,b), where ρ
is a function of the angle α: ρ=ρ(α). The constant
distance r is the sum of two segments of variable
lengths: r=ρ+A’C’. The length of the hypotenuse
A’C’ can be found from the right triangle A’C’B. We
find the expressions of the lengths of the legs A’B
and BC’ for the current tangency point A’. For a
better understanding of the rolling process, consider
the moving coordinate system - the accompanying
trihedral of the curve of the profile, in which the
unit vector τ is tangent to it, the unit vector n is
perpendicular, and the unit vector of the binormal b
is projected to a point. The vertex of the trihedral is
point À in its initial position, with the altitude AC
lying on the Ox axis (Fig. 1, a, b), which coincides with
the unit normal vector n (Fig. 1, b). When rolling the
triangle, the point of contact A’, which is the vertex of
the trihedral, moves along the curve and unit vector
τ remains tangent to the curve. In the initial position

τn of the trihedral, points A and B coincided, and
the altitude a of the triangle coincided with the unit
vectorn. When rolling a triangle along the curve, the
trihedral occupies a new position τ′n′ with the vertex
at point A’ (Fig. 1,b). The coordinates of the point
C’ in its system are as follows. The segment A’B
is equal to the length of the arc AA’ of the curve:
AA’=A’B=s. The altitude a of the triangle when it
is rolling remains parallel to the principal normaln′ .
So the length of the segment A’C’ is determined by
the Pythagorean theorem: A′C′ =

√
s2 + a2. The

expression r=ρ+A’C’ can be written as follows:

r = ρ +
√
s2 + a2. (1)

Let‘s solve equation (1) for s:

s =
√

(r −ρ)2 −a2. (2)

Let‘s write the parametric equations of the curve
of the profile in the polar coordinate system:

x = ρ cos α ;
y = ρ sin α. (3)

Let‘s find the expression of the arc length s of the
curve (3). To do this, we define its first derivatives:

x′ = ρ′ cos α− ρ sin α;
y′ = ρ′ sin α + ρ cos α. (4)

By the known formula we write:
ds

dα
=
√
x′2 + y′2 =

√
ρ2 + ρ′2. (5)

The derivative of arc s can be found by a differenti-
ation of expression (2):

ds

dα
= −

ρ′ (r −ρ)√
(r −ρ)2 −a2

. (6)

We equate expressions (5) and (6) and solve for ρ’:

dρ

dα
=
ρ

a

√
(r −ρ)2 −a2. (7)

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The differential equation of first order (7) is ob-
tained on the basis of the equality of the arcs of the
profile curve and the side of the square that rolls on
it without sliding. The altitude a of the triangle can
be found through the angle ε : a = R · cos ε, where R
is the length of the side of the triangle that is equal
to the radius of the circle circumscribed about the
square. Let’s extend this expression to a polygon
with an arbitrary number of sides n. The angle ε, in
this case, will depend on the number of sides of the
polygon: ε=π/n. The differential equation (7) for a
polygon with an arbitrary number of sides n, being
inscribed in a circle of radius R, will be written:

dρ

dα
=

ρ

R cos (π/n)

√
(r −ρ)2 −R2 cos2 (π/n). (8)

Differential equation (8) has an analytical solution.
On a condition that α=0 ρ=r-a=r-R·cos(π/n), we
find the corresponding value of the constant of integra-
tion. With consideration of this constant, the solution
of equation (8) takes the final form:

ρ =
r2 −R2 cos2 (π/n)

r + R cos (π/n) cosh
(
α
√

r2

R2 cos2(π/n) − 1
). (9)

Substitution (9) in (3) will give parametric equa-
tions of the curve. We need a limited arc to construct
a profile. The magnitude of this arc is due to the
minimum value of the radius-vector ρ=ro (Fig. 1a).
Inversely, ro=r-A’C’=r-R. Let‘s substitute in (9) ρ=r-
R and solve for α:

α0 = ±
R cos (π/n)√

r2 −R2 cos2 (π/n)
Arc cosh(

r −R cos2 (π/n)
(r −R) cos (π/n)

)
. (10)

The wanted arc of the curve at given values of r, R
and n is constructed according to equations (3) taking
into account (9) when the angle α changes within α=
–αî...αî. However, in this case, we will not be able
to place the required number of arcs so that a closed
profile is obtained. If we want to construct a profile
of four arcs, then the angle αî=±π/4 (this case is
shown in Fig. 1,a). The measure of the angle αo is
determined by dividing the number π by the number
of arcs. Hence, for a given number of arcs of a profile
and the number of sides of a polygon of two radii r
and R, we can only specify one of them, since the
angle αî will also be given. Equation (10) cannot
be solved with respect to one of the radii r or R, so
numerical techniques must be used.

3. Result
It should be noted that the curvilinear profile can
consist of one arc. If the polygon is a square, then
at αi = ±π, n = 4, r = 100, we find: R = 95.28. In
Fig. 2, we can see a curvilinear profile and a square in

Figure 2. Curvilinear profile, with complete rolling
during which the square makes quarter of a turn.

two positions on opposite sides. When the profile is
completely rolled, only one side of the square contacts
it, that is, the square makes quarter of a turn.
If a curvilinear profile consists of four elements,

then the square during one complete rolling of the
profile makes one turn. Repeating the calculation at
αi = ±π/4, n = 4, r = 100, we find: R = 62.27. In
Fig. 3, we can see a curvilinear profile of four elements
and a set of positions of a square when it is rolled
along one of the elements. Sequential movement of
the centre of the square when it is rolled is shown by
circles. In the extreme positions of a square, when
its vertices are points of contact with the profile, the
sides of the square are depicted thickened.
It should be noted that for the physical rolling of

a polygon along a curvilinear profile, there is a limit
on the number of its sides. The number of sides of a
polygon cannot be less than four. This is explained as
follows. When rolling a polygon, its vertex describes
a known curve - evolvent. Its property is that at
the moment of detachment from the curve, the point
of the straight that rolls on it (in our case the end
of the side of the square) moves perpendicular to
it. This can be seen from the enlarged fragment in
Fig. 3, b. If the polygon was a triangle, then the angle
between neighbouring elements would be 60◦ and
physical rolling would be impossible. The considered
approach allows constructing a curvilinear profile that
would provide the required number of revolutions of
the polygon at complete rolling along the profile. It
is determined by the ratio of the number of profile
elements to the number of sides of the polygon.

In Fig. 4,a, a profile consisting of 8 elements is con-
structed. At r = 100, the radius of the circumscribed
circle is R = 41.76. When completely rolling along
the profile, the square makes 2 turns. To ensure 4
turns, the profile must be 16 elements, with a radius
of R = 25.13 (Fig. 4,b).

With an unlimited increase in the number n of the
sides of a polygon, it transforms into a circle, and the
radius-vector ρ into a constant value r–R, that is, rî
(Fig. 1,a). The number of turns of a circle of radius
R, after a completed rolling along a circle of radius rî,
is determined by the ratio of these radii.

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T. Kresan, S. Pylypaka, Z. Ruzhylo et al. Acta Polytechnica

(a). (b).

Figure 3. Curvilinear profile of four elements, when completely rolling along it, the square makes one turn: a)
the set of positions of the square when it is rolling along the arc of the profile; b) enlarged fragment of the profile
element

(a). (b).

Figure 4. Curvilinear profiles with different number of turns of a square: a) a square, when completely rolling along
the profile, makes two turns; b) a square, when completely rolling along the profile, makes four turns.

Figure 5. Polygons and curvilinear profiles with different numbers of sides and curvilinear elements at r = 100: a) a
hexagon with a radius of the circumscribed circle R = 54.93 and a corresponding curvilinear profile with six elements;
b) a pentagon with a radius of the circumscribed circle R = 71.47 and a corresponding curvilinear profile with three
elements.

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The correspondence of the number of sides of a
polygon and elements of a curvilinear profile can be
different (Fig. 5). Such figures can roll one by one
with simultaneous rotation around fixed centres O
and O1 with angular velocities ω and ω1, that is, they
can serve as centroids for the design of non-circular
gears [6–8]. The predetermined value r is the centre
to centre distance.

When rotating one non-circular wheel at a constant
angular velocity, the second will rotate at a variable
angular velocity. This is due to the variable radius
from the centres of the wheels to the point of the con-
tact during the rotation. The radius-vector ρ changes
from the maximum value at α = 0 to the minimum
ρ = ri = r − R (Fig. 1,a, points A and A’). This
distance difference is the difference between the maxi-
mum and minimum values of the distance from the
axis of rotation of the polygon to the point of contact
with the curvilinear profile. It decreases as the number
of sides of a polygon increases as well as increase the
number of curvilinear elements of the profile that can
be traced by the example of the square in Fig. 2 - 4.

4. Conclusions and prospects for
further research

An analytical description of the rolling of a polygon
along a curvilinear closed profile can be used to design
non-circular gear wheels. The number of sides of a
polygon must be at least four in order to be physically
rolling. To design a curvilinear centroid, it is necessary
to specify the distance from centre to centre, the
number of sides of a polygon, that is another centroid,
and the correspondence of the number of turns of
these centroids. The length of the centroid element
is equal to the length of the side of the polygon. The
number of elements of a centroid can be any integer,
starting with one. Increasing the number of curvilinear
centroid elements and the number of sides of a polygon
increases the rotation uniformity of one centroid with
respect to another.

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http://dx.doi.org/10.1016/j.mechmachtheory.2005.06.003
http://dx.doi.org/10.1016/j.mechmachtheory.2007.07.004
http://dx.doi.org/10.1016/S0007-8506(07)60811-7

	Acta Polytechnica 60(4):1–5, 2020
	1 Introduction
	2 Material and method
	3 Result
	4 Conclusions and prospects for further research
	References