A Curvature Invariant Inspired by Leonhard Euler's Inequality
A Curvature Invariant Inspired by Leonhard Euler's Inequality
A Curvature Invariant Inspired by Leonhard Euler's Inequality
Abstract. It is of major interest to point out natural connections between the ge-
ometry of triangles and various other areas of mathematics. In the present work
we show how Euler’s classical inequality between circumradius and inradius in-
spires, by using a duality between triangle geometry and three-dimensional hy-
persurfaces lying in R4 , the definition of a curvature invariant. We investigate
this invariant by relating it to other known curvature invariants.
1. A historical motivation
Leonhard Euler believed that one could not define a good measure of curvature
for surfaces. He wrote in 1760 that [5]: “la question sur la courbure des surfaces
n’est pas susceptible d’une réponse simple, mais elle exige à la fois une infinité de
déterminations.” Euler’s main objection to the very idea of the curvature of sur-
faces was that if one can approximate a planar curve with a circle (as Isaac Newton
described ), then for a surface it would be impossible to perform a similar con-
struction. For example, Euler wrote in his investigation, what would we do with a
saddle surface if we intend to find the sphere that best approximates it at a saddle
point? On which side of the saddle should an approximating sphere lie? There are
two choices and both appear legitimate. Today we see that Euler’s counterexample
corresponds to the case of a surface with negative Gaussian curvature at a given
point, but in Euler’s period no definition of curvature for a surface was available.
Since Leonhard Euler contributed so much to the knowledge of mathematics in his
time, one might expect that he was responsible for introducing and investigating
a measure for the curvature of surfaces. However, this is not what happened. In-
stead, Leonhard Euler wrote a profound paper in which he obtained a well-known
theorem about normal sections and stopped right there.
Moreover, Euler’s paper [5] was extremely influential in the mathematical world
of his times. When Jean-Baptiste Meusnier came to see his mentor, Gaspard
Monge, for the first time, Monge instructed him to think of a problem related to
the curvature of curves lying on a surface. Monge asked Meusnier to study Euler’s
paper [5], yet before reading it, in a stroke of genius the following night, Meusiner
proved a theorem that today bears his name, and he did so by using a different geo-
metric idea than Euler’s. Decades afterwards, C.F. Gauss [7] introduced a measure
for the curvature of surfaces that was complemented by the very inspired Sophie
Germain [8], who introduced the mean curvature.
Perhaps the most direct way to explain the two curvature invariants for surfaces
is the following. Consider a smooth surface S lying in R3 , and an arbitrary point
P ∈ S. Consider NP , the normal to the surface at P, and the family of all planes
passing through P that contain the line through P with the same direction as NP .
These planes yield a family of curves on S called normal sections. We now de-
termine the curvature κ(P ) of the normal sections, viewed as planar curves. Then
κ(P ) has a maximum, denoted κ1 , and a minimum, denoted κ2 . The curvatures
κ1 and κ2 are called the principal curvatures. Using these principal curvatures,
one may define the Gaussian curvature K(P ) as, K(P ) = κ1 (P ) · κ2 (P ), and
the mean curvature H(P ) as, H(P ) = 12 [κ1 (P ) + κ2 (P )] . Note that in the 18th
century these constructions were not possible. They are related in a very profound
way to the birth of non-Euclidean geometry and the revolution that took place in
geometry at the beginning of the 19th century.
Taking into account this important historical context, we would like to investi-
gate the following question. Was it possible for Leonhard Euler to have determined
himself a curvature invariant, perhaps a concept for which all the algebra that he
needed was in place, but in which it was just a matter of interpretation to define it?
We are here in the territory of speculations, of alternative history, and only our sur-
prise while we read and re-read Euler’s original works leads us to such a question.
However, this question makes a lot of sense as Euler had many contributions in ge-
ometry - from plane Euclidean geometry to the geometry of surfaces - and one of
them in particular could have been tied to his investigations on curvature. Hence,
was there any geometric property that was actually obtained by Euler which leads
us to a property related to curvature?
The converse of this claim is also true. Namely, if a, b, c are the sides of a
triangle in the Euclidean plane, then the system given through the equations a =
y+z, b = z+x, and c = x+y, has a unique solution x = s−a, y = s−b, z = s−c,
where we denote by s the semiperimeter of the triangle, namely s = 12 (a + b + c).
Some classical facts in advanced Euclidean geometry can be proved by substi-
tuting a, b, c, instead of the variables x, y, z,. This technique is called Ravi substi-
tution.
We can view these three positive real numbers x, y, z in a different way. We
could view them as principal curvatures in the geometry of a three-dimensional
smooth hypersurface lying in the four dimensional Euclidean space. To every tri-
angle with sides a, b, c there corresponds a triple x, y, z > 0, and these numbers
can be interpreted as principal curvatures, given pointwise, at some point p lying
on the hypersurface. We call this correspondence a Ravi transformation and we
would like to see whether this duality leads us to some interesting geometric inter-
pretations of facts from triangle geometry into the geometry of hypersurfaces into
R4 endowed with the canonical Euclidean product.
x=s−a
x=s−a
E
F
I z =s−c
y =s−b
B D
y =s−b z =s−c C
Figure 1
To better see how we use Ravi’s substitutions, we recall here a few useful for-
mulae for a triangle ΔABC lying in the Euclidean plane. Denote the area of the
triangle by A, its circumradius by R , its inradius by r,its semiperimeter by s, and
perimeter by P. Then Heron’s formula is A = s(s − a)(s − b)(s − c) =
its
xyz(x + y + z). The inradius can be obtained as
A xyz
r= = ,
s x+y+z
A curvature invariant inspired by Leonhard Euler’s inequality R ≥ 2r 123
E
F
I
r
O R
B D C
Figure 2
Corollary 2.
A · E ≥ H̄ · |K| ≥ A · |K|.
This relation naturally relates the amalgamatic curvature, the Gaussian curva-
ture, and the curvature invariant E, which we introduced inspired by Euler’s in-
equality for triangles in Euclidean geometry, R ≥ 2r. In particular, note that we
have the following.
Corollary 3.
E(p) ≥ |K(p)| (1)
at any point p of M3⊂ R4 . The equality holds at a point p if and only if the point
is absolutely umbilical.
It may be of interest to note that this inequality may be viewed as an extension
of the inequality H 2 (p) ≥ K(p) from the geometry of surfaces.
We conclude this section by pointing out that all the algebra was definitely in
place in Euler’s times, and the only new part was the geometric interpretation in
terms of curvature invariants. This vision was greatly enhanced after the major
revisitation of curvature invariants described in Bang-Yen Chen’s comprehensive
monograph [2], which best represents a research direction with many developments
(see e.g. [10, 12], among many other works), which inspires also the context of our
present investigation.
References
[1] B. Brzycki, M. D. Giesler, K. Gomez, L. H. Odom, and B. D. Suceavă, A ladder of curva-
tures for hypersurfaces in the Euclidean ambient space, Houston Journal of Mathematics,
40 (2014) 1347–1356.
[2] B.-Y. Chen, Pseudo-Riemannian geometry, δ−invariants and applications, World Scien-
tific, 2011.
[3] C. T. R. Conley, R. Etnyre, B. Gardener, L. H. Odom and B. D. Suceavă, New Curva-
ture Inequalities for Hypersurfaces in the Euclidean Ambient Space, Taiwanese Journal of
Mathematics, 17 (2013) 885–895.
[4] M. P. do Carmo, Riemannian Geometry, Birkhäuser, 1992.
[5] L. Euler, Recherches sur la courbure des surfaces, Memoires de l’academie des sciences de
Berlin (written in 1760, published in 1767), 16: 119–143.
[6] L. Euler, Solutio facilis problematum quorundam geometricorum difficillimorum, Novi
Commentarii academiae scientiarum Petropolitanae 11 (1767) 103–123.
A curvature invariant inspired by Leonhard Euler’s inequality R ≥ 2r 127
Nicholas D. Brubaker: California State University Fullerton, 800 N. State College Blvd, 154
McCarthy Hall, Fullerton, California 92831-6850, U.S.A.
E-mail address: nbdubaker@fullerton.edu
Jasmine Camero: California State University Fullerton, 800 N. State College Blvd, 154 McCarthy
Hall, Fullerton, California 92831-6850, U.S.A.
E-mail address: jasminecamero@csu.fullerton.edu
Oscar Rocha Rocha: California State University Fullerton, 800 N. State College Blvd, 154 Mc-
Carthy Hall, Fullerton, California 92831-6850, U.S.A.
E-mail address: oscrocha167@csu.fullerton.edu
Roberto Soto: California State University Fullerton, 800 N. State College Blvd, 154 McCarthy
Hall, Fullerton, California 92831-6850, U.S.A.
E-mail address: rcsoto@fullerton.edu
Bogdan D. Suceavă: California State University Fullerton, 800 N. State College Blvd, 154 Mc-
Carthy Hall, Fullerton, California 92831-6850, U.S.A.
E-mail address: bsuceava@fullerton.edu