a necessary condition to determine the symmetry center of

Adv. Mat. Sci. & Technol.
Recibido 27 10 2013
A NECESSARY
CONDITION TO DETERMINE THE SYMMETRY CENTER OF A REAL-ANALYTIC PLANE CURVE
V 7 Nº3 Art 1 pp 1-3,
2013
Aceptado 23 11 2013
ISSN 1316-2012
Publicado 01 12 2013
Depósito Legal pp 96-0071
© 2013 CIRES
A NECESSARY CONDITION TO DETERMINE
THE SYMMETRY CENTER OF A REAL-ANALYTIC
PLANE CURVE
GUILLERMO F. MIRANDA & EDUARDO J. SANCHEZ*
<[email protected]>
Adjunct Faculty at San Diego State University, San Diego, California, USA.
*
Teaching Assistant and Ph.D. Student at San Diego State University,
San Diego, California, USA.
ABSTRACT
A necessary condition to be satisfied by the Cartesian coordinates of the symmetry center of real
analytic plane curves is derived. Real analytic means that the implicit function of two real
variables used to describe the curve is real analytic in a neighborhood of the symmetry center.
This necessary condition amounts to the vanishing of the implicit function´s gradient at the
symmetry center, assumed to exist, and its use is exemplified by means of a
Bernoullianlemniscate.
Key Words: Plane curves. Symmetry. Critical points. Bernoullianlemniscate.
UNA CONDICIÓN NECESARIA PARA DETERMINAR
EL CENTRO DE SIMMETRIA DE CURVAS PLANAS
REAL-ANALITICAS
RESUMEN
Se deduce una condición necesaria a ser satisfecha por las coordenadas Cartesianas del centro de
simetría de curvas planas analíticas reales. Que una curva sea analítica real significa que la función
implícita de dos variables reales empleada para describir la curva, admite un desarrollo de Taylor
convergente en una vecindad del centro de simetría. Esta condición necesaria equivale a la
anulación del gradiente de la función implícita en el centro de simetría, supuesto existente, y se
ejemplifica su uso mediante una lemniscata de Bernoulli.
Palabras Clave: Curvas planas. Simetría. Puntos críticos. Lemniscata de Bernoulli.
FICHA
MIRANDA GUILLERMO F. & EDUARDO J. SANCHEZ, 2013.- A
NECESSARY CONDITION TO DETERMINE THE SYMMETRY CENTER
OF A REAL-ANALYTIC PLANE CURVE Adv. Mat. Sci. & Technol. 7(3): 1-3.
ISSN 1316-2012
1
MIRANDA GUILLERMO F. & EDUARDO J. SANCHEZ
symmetry center of the aforesaid curve. There is no
loss of generality here, since a translation of
coordinates preserves analyticity of F.
Problem Statement
Consider a real analytic plane curve possessing a
symmetry center, that is, a plane curve which can be
implicitly described in the Cartesian Plane by means
of the relationF(x,y) = 0, where F is a real function
of two real variables x and y, expandable in a Taylor
Series about the origin (0,0), assumed to be the
The symmetry assumption implies that F(-x,-y) =
F(x,y) identically.
By the analyticity assumption, we must have:
Hence, it holds identically in x and y:
These two scalar equations can be rewritten as a
single vector equation, namely, grad F ( 0,0 ) = 0.
Thus, the symmetry center coordinates satisfy the
same pair of equations which are satisfied by a
critical point of F. This also has the obvious
implication that while considering the one parameter
family of plane curves defined by F(x,y) = c, (c is
constant), which characterize the level curves of the
surface defined in 3-D space by z = F(x,y), it follows
that if (0,0) is a common symmetry center for them,
then (0,0) might be a point of maximum or
minimum for F.
Let y tend to zero first, so that:
identically in x, so that, after canceling a factor x,
we must have
An example will exhibit the usefulness of our
necessary condition to be satisfied by the Cartesian
coordinates of the symmetry center of a given real
analytic plane curve. Suppose you are given the
following quartic implicit equation in order to detect
a possible symmetry center for the associated plane
curve in the Cartesian x-y Plane:
Now let x tend to zero, so that:
identically in y. This implies, after canceling a factor
y, that
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A NECESSARY CONDITION TO DETERMINE THE SYMMETRY CENTER OF A REAL-ANALYTIC PLANE CURVE
Our necessary condition to be satisfied by the coordinates x and y of a possible symmetry center has the form
of the 2 by 2 system:
and
An obvious solution will occur when y = x, so that
we have to solve the following cubic equation:
that is a standard Bernoulli Lemniscate, something
that would have been difficult to ascertain as easily
as it has been done with our necessary condition.
or
CONCLUSIONS
It is readily found that x = - 1 is the only real root of
this cubic, so that our necessary condition says that
if the given quartic curve possesses a symmetry
center, it must be x = - 1, y = - 1. It is readily found
that this is the case, since after a Coordinate
translation defined by x + 1 = X, y + 1 = Y, the
original implicit equation for our quartic becomes:
Our result admits an immediate generalization for
finding symmetry centers of surfaces defined by
F( x,y,z ) = 0, with F analytic, since repeating the
previous argument almost verbatim, we see that
F( -x, -y, -z ) = F( x,y,z ) identically implies grad
F( 0,0,0 ) = 0.
This curious connection between symmetry and
eventual optimization might turn out to have
applications in physics and other areas. There are no
references included since we have not been able to
find anything similar in the literature.
In this form, it is immediately seen that the new
Coordinate Origin is a symmetry center for our
curve, since f( -X. -Y ) = f(X,Y). Besides, by an axis
rotation defined by x* = X - Y, y* = X + Y, which
eliminates the XY term, the implicit equation
becomes:
ADRESS
GUILLERMO F. MIRANDA
<[email protected]>
Adjunct Faculty at San Diego State University, San Diego, California,
USA.
EDUARDO J. SANCHEZ
<[email protected]>
Teaching Assistant and Ph.D. Student at San Diego State University,
San Diego, California, USA.
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