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Mikhail Vladimirovich Taldykin // A remarkable property of cycloidal curves, 9-19

DOI: http://dx.doi.org/10.46925//rdluz.33.02

A remarkable property of cycloidal curves

Mikhail Vladimirovich Taldykin *

ABSTRACT

The purpose of this theoretical work is to establish a connection between the most important

properties of plane curves: cycloids and sinusoids. For this, a drawing mechanism is

considered, which simultaneously draws a sinusoid and two cycloids. Based on the results

obtained using this mechanical method of obtaining curves, the following important,

previously unknown, theoretical facts are established. Firstly, new in theoretical terms is

that the sinusoid is not represented as a graph of a trigonometric function, but as a locus of

points equidistant from the current points of two cycloids: an ordinary and another cycloid

congruent to the original one, inverted and shifted along the axis by half a period. Secondly,

the line passing through the current points of these cycloids is nothing like a normal to the

resulting sinusoid. This property greatly simplifies the graphical construction of such a

normal. And, finally, a simple trigonometric relationship was established between the angle

of rotation of the generating circle and the angle of deviation of the normal from the vertical.

KEY WORDS: angle of rotation of the generating circle; cycloid; cycloidal curves; generating

circle; mechanisms for drawing curves; normal to sinusoid; shortened cycloid; sinusoid;

tangent to a sinusoid.

*

Head of technical support unit of Institute of Mechanics and Engineering - Subdivision of

the Federal State Budgetary Institution of Science “Kazan Scientific Center of the Russian

Academy of Sciences”, Kazan, Russia. ORCID: https://orcid.org/0000-0001-9977-1224 . E-

mail: mvtkazan@gmail.com

Recibido: 10/02/2021

Aceptado: 6/04/2021

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REVISTA DE LA UNIVERSIDAD DEL ZULIA. 3ª época. Año 12 N° 33, 2021

Mikhail Vladimirovich Taldykin // A remarkable property of cycloidal curves, 9-19

DOI: http://dx.doi.org/10.46925//rdluz.33.02

Una propiedad destacable de las curvas cicloidales

RESUMEN

El propósito de este trabajo teórico es establecer una conexión entre las propiedades más

importantes de las curvas planas: cicloides y sinusoides. Para esto, se considera un mecanismo

de dibujo, que presenta simultáneamente una sinusoide y dos cicloides. Con base en los

resultados obtenidos mediante este método mecánico de obtención de curvas, se establecen

los siguientes hechos teóricos importantes, previamente desconocidos. En primer lugar, lo

nuevo en términos teóricos es que la sinusoide no se representa como un gráfico de una

función trigonométrica, sino como un lugar geométrico de puntos equidistantes de los puntos

actuales de dos cicloides: una ordinaria y otra cicloide congruente con la original, invertida y

desplazada a lo largo del eje por medio punto. En segundo lugar, la línea que pasa por los

puntos actuales de estas cicloides no se parece en nada a una normal a la sinusoide resultante.

Esta propiedad simplifica enormemente la construcción gráfica de una normal de este tipo.

Y, finalmente, se estableció una relación trigonométrica simple entre el ángulo de rotación

del círculo generador y el ángulo de desviación de la normal respecto a la vertical.

PALABRAS CLAVE: ángulo de rotación del círculo generador; cicloide; curvas cicloidales;

círculo generador; mecanismos para dibujar curvas; normal a sinusoide; cicloide acortado;

sinusoide; tangente a una sinusoide.

Introduction

The purpose of this article is to develop new theoretical knowledge regarding

cycloidal and sinusoidal curves. On the basis of this theoretical knowledge, it is possible to

construct various devices that associate rotational motion with harmonic motion, such as

sinus mechanisms, propellers in water with a fish-like working organ, peristaltic pumps with

a delicate effect on the pumped liquid, for example, blood, and other similar devices.

The cycloid was first considered by the French mathematician Roberval in 1634.

When calculating the area under the cycloid graph, he considered an auxiliary curve formed

by the projection of a point on a circle rolling in a straight line onto the vertical diameter of

this circle. He carefully and vaguely called this curve the companion of the cycloid. This curve

turned out to be an ordinary sinusoid (Gindikin, 2001).

Roberval, as it were, “encrypted” another approach to defining a sinusoid, not as a

graph of a harmonic function, but as a kind of cycloidal curves with a general kinematic

approach, in the same mathematical terms as for an ordinary cycloid. “The true spirit of

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REVISTA DE LA UNIVERSIDAD DEL ZULIA. 3ª época. Año 12 N° 33, 2021

Mikhail Vladimirovich Taldykin // A remarkable property of cycloidal curves, 9-19

DOI: http://dx.doi.org/10.46925//rdluz.33.02

geometry means something more: it requires an approach to the study of geometric images

not from one, but from different points of view, because only this path leads to complete

knowledge” (Litzman, 1960). In the case of a sinusoid, the first approach is a sinusoid as a

graph of a sine, and an alternative approach, a sinusoid, is a quasi-cycloidal curve (the main

parameters are the radius of the generating circle, its angle of rotation, the distance from a

point to the center of the circle, in the case of considering a shortened cycloid). The term

“

wavelength” (cycloid period) is not used, it is used for reference only. The provisions of

Roberval's treatise concerning the cycloid and the auxiliary satellite line (sinusoid) almost

unchanged “migrated” into the modern Handbook of Higher Mathematics: “Cycloid and

M

Sinusoid. The locus of the bases of the perpendiculars dropped from the point of the

cycloid to the diameter of the generating circle passing through the fulcrum is a sinusoid with

a wavelength and amplitude d. The axis of this sinusoid coincides with the line of the

centers of the cycloid” (Vygodsky, 2006).

2 R

“

By the end of the seventeenth century, mathematicians had discovered all the

secrets of the cycloid and paid attention to other curves. It has often happened in

the history of mathematics that a certain idea or problem will appear at exactly

the right time. This was the case with the cycloid. The discoveries of its beautiful

geometric and mechanical properties are closely related to the history of analytic

geometry and differential calculus. The missions and battles that were fought over

them led to significant achievements. No other curve could serve the same

purpose” (Martin, 2010).

“

The heroic history of the cycloid ended at the end of the 17th century. It arose so

mysteriously in solving a variety of problems that no one doubted that it played a

completely exclusive role. The piety before the cycloid held out for a long time,

but time passed, and it became clear that it was not connected with the

fundamental laws of nature, like, say, conical sections. The problems that led to

the cycloid played a huge role in the formation of mechanics and mathematical

analysis, but when the magnificent buildings of these sciences were built, it

turned out that these problems are private, far from the most important. An

instructive historical illusion took place. However, getting acquainted with the

instructive history of the cycloid, it is possible to see many fundamental facts from

the history of science” (Gindikin, 2001).

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Mikhail Vladimirovich Taldykin // A remarkable property of cycloidal curves, 9-19

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These two quotes kind of summarize the study of the properties of cycloidal curves at

the end of the 17th century. However, even in our time, cycloids often become the solution to

scientific problems, for example, associated with tsunamis. “The optimal trajectory for two

arbitrary points in the ocean, as in the case of one point on the coast, will also be a cycloid

passing through these two points” (Shokin et al., 1989).

It should also be noted that all of the above refers to one isolated cycloid, while this

article discusses the properties of the mechanism of two cycloids.

Cycloid properties are currently actively used for educational purposes. “Cycloids are

a great example of not only the need for parametric equations, but an example of how to

integrate and differentiate them; they also require many of the necessary skills and abilities

to use these skills to solve problems” (Roidt, 2011). “The fascinatingly presented biographies

of great scientists will interest the widest circles of readers, from high school students to

adults; those interested in mathematics will enjoy and benefit from getting to know the

scientific achievements of the heroes of the book” (Gindikin, 2001).

1

. Methodology

Let's set the task: to invent a mechanism for plotting curves that could draw

simultaneously such “mechanical curves” as cycloids and sinusoids. Is it possible to build

such a mechanism? Mathematics answers in the affirmative. “You can build other hinge

mechanisms, at least theoretically, which will draw ellipses, hyperbolas and even any

predetermined curve, whatever its degree” (Courant and Robbins, 2001). Such a mechanism

for plotting cycloidal curves was found, and it turned out to be extremely simple and

informative.

There is no up-to-date information on this area, since the topic has long been

considered well-established and quite classical, and the expectation of new works is

considered unlikely. A publication similar to this one could have appeared three hundred

years ago, and it is surprising that this did not happen earlier. This article provides new

knowledge for the uderstanding of cycloidal and sinusoidal curves; the provisions outlined

in this article can be extended to other, more complex cycloidal curves (for example, epi- and

hypocycloids). Existing newpublications on cycloids are often teaching material for students

and even for high school students. It should be noted that articles on cycloids are descriptive

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or educational in nature. This work sheds light on the unknown properties of cycloidal

curves.

2

. Results

Let us refer to Figure 1. A point

P

located on the circumference of the upper generating

1

disc at a distance

R

from the center of the disc describes, when rolling without sliding along

), a well-known ordinary cycloid, which is a series of arches with

points” downward. The curve is periodic, located in the upper positive half-plane; Period

a straight line (axis

X

“

(

basis of the cycloid) 2 R. Similarly, a point P₂located on the circumference of the lower

from the center of the disk describes, when rolling

without sliding along a straight line (axis ), a congruent, mirrored, inverted cycloid

“points” up), shifted along the axis by half a period relative to the upper cycloid. This

shift is set initially. This cycloid is located already in the lower, negative half-plane. The angle

in radians is a generalized coordinate for both cycloids, since the producing discs roll

synchronously without sliding.

generating disk at the same distance

R

X

(

X

t

Figure 1. Scheme for plotting cycloidal curves and an ordinary sinusoid

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Let's connect the points

P and P₂, find the midpoint P₃, equidistant from the points

1

P

and P₂; that is, P₃is the midpoint of a segment P P₂of variable length. It will be shown

1

below that the point P₃draws nothing more than the old familiar Roberval's auxiliary line,

the companion of the cycloid – a sinusoid of the same period as the cycloid and with an

amplitude equal to the radius of the generating disk.

However, what is most surprising and striking is the fact that the normal to the

sinusoid at an arbitrary angle

t

passes through the same current points

P

and P₂the cycloid

1

corresponding to the same value of the generalized coordinate (angle

t

). A complete and

generalized proof for the case of an arbitrary value of the ratio r R is given below.

Thus, a cycloid, as a cycloidal curve and a sinusoid, as a kind of quasi - cycloidal curve

(

the point P₃is not located on the extension of the radius of the generating circle, but is

constructed in some way) are “equalized in rights” so that they can be considered not only

fellow travelers” but also “relatives”, but rather even “Siamese twins” of geometry. The

“

properties of these curves can be considered from a general point of view and are determined

by the radius of the generating circle, its angle of rotation and the distance from a point to its

center (for shortened cycloids), without involving the concepts of a cycloid basis and

wavelength for a sinusoid.

Let two generating circles of radius

R

(Figure 1) roll synchronously along a direct

. The

straight line y  0 (axis ) without sliding in the positive direction of the axis

X

condition of synchronicity means the presence of a common point of contact of the circles

with the guide at any moment of rolling, i.e. the angles of rotation of the circles are always

equal. One circle rolls over the “positive” side of the base (top) ( y  0), the other under the

“bottom” ( y  0).

The starting points of the upper and lower cycloid are chosen in such a way that the

lower inverted arch of the cycloid is offset along the guide by half the base. Then the following

theorem holds, which consists of two points.

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Mikhail Vladimirovich Taldykin // A remarkable property of cycloidal curves, 9-19

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Figure 2. Scheme for obtaining harmonic oscillations (sinusoids) using two shortened

cycloids

Taldykin's theorem:

1

. The locus of the points of the midpoints of the segments connecting the current

points of two congruent cycloids, mirrored relative to the directing line and shifted

displaced) along the half-base (half-period) guide is a sinusoid of the same period as the

(

cycloids, with an amplitude equal to the distance from the point describing the cycloid to the

center of the generating circle.

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2. The normal to such a sinusoid passes through the current points of the upper and

lower cycloid.

Evidence. Let us consider a more general case when the points

P

and P₂during

1

synchronous rolling of the generating circles along the guide (x-axis) without sliding

describe shortened cycloids. Parametric equations of these cycloids

for

P

:

x₁ Rt 



Rsint

,

y₁ R Rcost ; (1)

y₂ R Rcost

1

for P₂

:

x₂ Rt 



Rsint

,

;

where x , y , x , y are current coordinates of points P1 and P₂

;

t

is the angle of rotation of

is the radius of each producing circle; is

to the center of the generating circle; Truncated cycloids are

r R ; At

1 we get an ordinary cycloid, at = 0 is a straight

line along which the center of the generating circle moves.

1

2

each circle in radians (generalized coordinate);

the distance from the point

characterized by the ratio

R

r

P



=



The following properties are valid for such a mechanism of two synchronously rolling

producing circles with a common point of contact with the directing line.

1

. The point

circles describes a sinusoid with an amplitude

cycloid

The coordinates of the midpoint P₃of the segment P₁P₂are equal to the half-sum of

the coordinates of the points

and P₂

P



t



is the middle of the segment P₁P₂when rolling the generating

3



R

and period equal to the periods of the

2 R

.

P

1

2

1

2

,

(2)

x_s



x₁ x₂







Rt 



Rsint  Rt 



Rsint



 Rt

1

푦_푠= (푦 + 푦 ) = (푅 − 휀푅 푐표ꢂ 푡 − 푅 − 휀푅 cos 푡) = −휀푅 cos 푡

ꢀ

ꢁ

2

Equations for a point P₃are parametric equations of a sinusoid:

y_s Rcost (3)

. Tangent and normal to a sinusoid. It is known from the course in differential

x_s Rt

,

2

   

, the

geometry that for a plane smooth curve given in parametric form x  x t y  y t

equation of the normal to this curve is as follows:

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y'



Y  y



 x'



X  x



 0, (4)

where X,Y are current coordinates of normal points; x, y are coordinates of the

curve point . Substituting here the parametric equations of the sinusoid (3) and the

equations of the first derivative of the sinusoid:

M

′

푥 = 푅; 푦 = 휀푅 sin 푡; (5)

we get

2

.

(6)

Y sint  Rcost sint  X  Rt  0

Thus, on the one hand, the equations of the normal to a sinusoid, given in a parametric

form x  Rt

,

y  



Rcost are as follows

1

t

;

(7)

Y  X

 R

cost  R



sint



sint

On the other hand, the equation of a straight line passing through two given points

   

P x₁; y₁P x₂; y₂

, is:

1 2

y  y₁

y₂ y₁x₂ x₁

x  x₁



; (8)

Substituting the current values for the points here

P



x₁; y₁



,

P



x₂; y₂



(1)

1

2

get

y 



R Rcost



x 



Rt Rsint





.

(9)



R Rcost







R Rcost Rt Rsint







Rt Rsint



After a number of transformations, equation (9) can be written in the form

ꢀ

ꢇ

푦 = −푥

− 푅휀 cos 푡 + 푅

(10)

ꢃ ꢄꢅꢆ ꢇ

Comparing the equation of a straight line passing through two given points

 

P x₁; y₁

1

,

P



x₂; y₂



(10) with the equation of the normal to the sinusoid (1), we see that they are

identical. Therefore, we can conclude that the normal to the sinusoid at an arbitrary point

and the line passing through the points and that

2

M



x  at; y  Rcos



P



t



 

P t

2

1

draw the cycloids coincide, that is, the normal to the sinusoid is a line passing through the

current points of the upper and lower truncated cycloids.

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Basic relations between the angle of rotation of the generating circles (generalized

coordinate) and the angle between the normal to the sinusoid and the vertical. It follows

directly from Figure 2 that

P A  P B 



Rsin

; (11)

Rcos

1

2

AO  O P  O B 





; (12)

1

3

2

then for a sinusoid:

P₁A

AP₃



Rsint

tg





sint ; (13)

R

It is well known that the normal to the cycloid passes through the fulcrum of the

generating circle (Gindikin, 2001). Let's draw normals to cycloids, connecting points and

P

1

P₂with a common point O₃of support of two generating circles. Tangent of the angle

between the normal to the cycloid and the vertical for the upper cycloid





sint

tg₁

tg₂



; (14)

1



cost

sint

cost



1

for the lower cycloid. (15)

Conclusion

The theorem on the construction of a flat curve was first formulated in the work:

sinusoid using two truncated cycloids, one of which is of the usual form with its points

downward, the other is inverted and shifted by half a period, relative to the original one. In

this case, a point equidistant from the current points of the cycloid during the rolling of two

identical generating circles draws a sinusoid of the same period as the cycloid, and the line

passing through the current points of the cycloid is the normal to the constructed sinusoid.

An elementary relationship has been established between the angle of rotation of the

generating circle and the angle of deviation of the normal to the sinusoid from the vertical.

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Acknowledgment

The author is sincerely grateful to Professor Rinat Gerfanovich Zaripov for constant

attention to the work, senior researcher Irina Veniaminovna Morenko for help in preparing

the article and Galina Ivanovna Budnikova for a number of valuable advice and support.

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Center of Continuous Mathematical Education, 568.

Gindikin, S. G. (2001). Stories about physicists and mathematicians. Secrets of the cycloid.

3

rd ed., Extended. Moscow: Moscow Center of Continuous Mathematical Education, 448.

Litzman, V. (1960). Old and new about the circle. Moscow: Publishing house of physical and

mathematical literature, 60.

Martin, J. (2010). The Helen of Geometry. The College Mathematics Journal, 41, 17-28.

Roidt, T. (2011). Cycloids and Paths (PDF) (MS). Portland State University, 4.

Shokin, Yu. I., Chubarov, L. B., Marchuk, An. G. and Simonov, K. V. (1989). Computational

experiment in the tsunami problem. Novosibirsk: Science. Siberian branch, 168.

Vygodsky, M. Ya. (2006). Handbook of Higher Mathematics. Moscow: AST, 991.

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