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The Jacobi Mate of an Oval

Mircea Crasmareanu
Faculty of Mathematics, University "Al. I. Cuza", Iasi, 700506, Romania

mcrasm@uaic.ro

Abstract. We introduce and study the Jacobi mate Cj of an Euclidean oval C. We focus here on thecurvature of Cj and on some examples.
1. Introduction

The enormous influence of convexity in practically every area of mathematics is widely known.We highlight the idea of convex curve by limiting the discussion to geometry, namely Euclideanplane geometry. The recent book [2] dedicates an entire chapter, specifically chapter 6, to thistopic.This brief note aims to relate, via the first two Jacobi elliptic functions, a second curve, Cj , toa given specific convex curve C, called oval. Given that these elliptic functions are 1-parametricextensions of the standard cosinus and sinus functions, which determine C, this link makes sense.The support function defining C serves as the foundation for the full analysis of this pair of curves.More specifically, we concentrate on the curvature, which is the only differential invariant for aplane curve. As possible area of applications for our results we mention the very recent (computerbased) Shape Analysis or Topology Optimization.The following is a list of the contents. The differential (and integral) geometry of the ovals isreviewed in the second section. Our new idea of Jacobi mate of the given oval C is presented inthe next section. It is important to note that, apart from the pair (C,Cj), there exists another curve
P that is naturally connected to the support function p of C and hence we will call the support
curve. In fact, we study three curves. After the computation of P and Cj curvatures, we focus on afew cases. We point out that certain complicated calculations require software and we make useof WolframAlpha.

2. The differential geometry of Euclidean ovals
A brief overview of the differential geometry of ovals is given in this first part. Hence, ourframework is the Euclidean linear space E2 := (R2, 〈·, ·〉) with to the canonical inner product:
Received: 28 May 2024.
Key words and phrases. Jacobi elliptic functions; oval; support function; curvature.1

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https://doi.org/10.28924/ada/ma.4.18
https://orcid.org/0000-0002-5230-2751


Eur. J. Math. Anal. 10.28924/ada/ma.4.18 2

〈u, v〉 = x1y1 + x2y2, u = (x1, x2) ∈ R2, v = (y1, y2) ∈ R2, 0 ≤ ‖u‖2 = 〈u, u〉. (2.1)
Fix an open interval I ⊆ R and consider C ⊂ E2 a regular parametrized curve of equation:

C : r(t) = (x(t), y(t)), r ∈ C∞, ‖r ′(t)‖ > 0, t ∈ I. (2.2)
Suppose that C is closed, simple and strictly convex; then will be called oval. All its geometryis provided by a smooth support function p : I = [0, L > 0]→ (0,+∞) with:

p(0) = p(L), p(t) + p′′(t) > 0, t ∈ I (2.3)
through the relations:(

x(t)

y(t)

)
:= R(t) ·

(
p(t)

p′(t)

)
, R(t) :=

(
cos t − sin t
sin t cos t

)
∈ SO(2) = S1, ‖r(t)‖2 = (p(t))2 + (p′(t))2.

(2.4)We point out that the function p was firstly considered by Minkowski and the function t → ‖r(t)‖ >
0 is exactly the first Legendre transformation of the convex function p. Let F(C) = {T,N} be theFrenet frame of C and k : I = [0, L]→ R∗+ = (0,+∞) its curvature function. Then, it is well knownthat these main functions are given by:

p(t) := −〈r(t), N(t)〉 > 0, k(t) :=
1

p(t) + p′′(t)
=

1

‖r ′(t)‖ > 0 (2.5)
since:

T (t) = (− sin t, cos t) = ie it , N(t) = iT (t) = −e it = (− cos t,− sin t) (2.6)which means that the Frenet frame is universal for the set of ovals defined on the same interval I .The geometry of the ovals has two well-known integral relations:i) the Cauchy formula:
L =

∫ 2π
0

p(t)dt. (2.7)
ii) the Blaschke formula for the area A(C) enclosed by C:

A(C) =
1

2

∫ 2π
0

[(p(t))2 − (p′(t))2]dt ≤
1

2

∫ 2π
0

‖r(t)‖2dt, 4πA(C) ≤ L2 (2.8)
with equality in the isoperimetric inequality (2.8) provided by the circle; we will treat the circleas oval in the example 3.4.

Remarks 2.1 i) The decomposition of the position vector field r in the Frenet basis is:
r(t) = p′(t)T (t)− p(t)N(t). (2.9)

A plane curve satisfying k(t) = 1
‖r ′(t)‖ for all t is called flat-flow curve in [6]. Hence, any oval issuch a curve, a fact that explains the equality with 2π of its total curvature.ii) An important tool in one-dimensional dynamics is the Fermi-Walker derivative. Let X(C) be the

https://doi.org/10.28924/ada/ma.4.18


Eur. J. Math. Anal. 10.28924/ada/ma.4.18 3set of vector fields along the curve C. Then the Fermi-Walker derivative is the map ( [6, p. 420])
∇FW : X(C)→ X(C):

∇FW (X) :=
d

dt
X + ‖r ′(·)‖k [〈X,N〉T − 〈X,T 〉N]. (2.10)

The Frenet frame is Fermi-Walker conserved: ∇FW (T ) = ∇FW (N) = 0. For our oval C we derive:
∇FW (r)(t) = r ′(t)− ‖r ′(t)‖k(t)[p(t)T (t) + p′(t)N(t)] = p′′(t)T (t)− p′(t)N(t). (2.11)

Hence if we denote r = Rotation(p) then the curve t → ∇FW (r)(t) is exactly the curve
Rotation(p′).iii) Associated to the support function p there exists the width function W : [0, L/2] → (0,+∞),
W (t) := p(t) + p

(
t + L

2

). Hence, its period is L
2 .iv) Concerning the possible relationship between the periodicity and the curvature of a plane curvea very interesting problem is solved in the paper [1]: when is a periodic function the curvature ofa closed plane curve? 2

3. The Jacobi mate of an oval
Fix the real number ρ ∈ (−1, 1) as the modulus for the differential system ( [7, p. 130]):

du
dt = −wv, u(0) = 1,

dv
dt = wu, v(0) = 0,

dw
dt = −ρ

2uv, w(0) = 1.

(3.1)
Recall that its solutions are called Jacobi elliptic functions and there are usually denoted cn(·, ρ),
sn(·, ρ) respectively dn(·, ρ); we prefer the simple notation used above. As solutions of the ODEsystem (3.1) these functions satisfy two remarkable identities:

u2 + v2 = 1, ρ2v2 + w2 = 1. (3.2)
Also, both functions u(·) and v(·) are periodic with L = 4L̃ for ( [7, p. 131]):

L̃ = L̃(ρ) :=

∫ 1
0

ds√
(1− s2)(1− ρ2s2)

(3.3)
while w is periodic of period 2L̃. In particular, L̃(0) = arcsin s|10 = π

2 for the usual trigonometricalfunctions cn(·, 0) = cos(·) and sn(·, 0) = sin(·). The complementary modulus is ρ′ := √1− ρ2 ∈
(0, 1] and the third Jacobi function is bounded by:

0 < ρ′ ≤ w(t) ≤ 1. (3.4)
The self-complementary case ρ′ = ρ is provided by ρ = 1√

2
and being in the interval (0, 1) is theeccentricity of an ellipse, called self-complementary and studied in [5].

https://doi.org/10.28924/ada/ma.4.18


Eur. J. Math. Anal. 10.28924/ada/ma.4.18 4Due to the increasing interest in the geometry of ovals this short note defines the Jacobi matefor the given oval C. As basic tool we use the new rotation matrix:
Jacobi(t, ρ) :=

(
u(t) −v(t)
v(t) u(t)

)
∈ SO(2) = S1. (3.5)

Definition 3.1 The curve Cj is the ρ-Jacobi mate of C if its parametrization is:
rj(t) =

(
xj

yj

)
(t) := Jacobi(t, ρ)

(
p

p′

)
(t) =

(
p(t)u(t)− p′(t)v(t)
p′(t)u(t) + p(t)v(t)

)
, t ∈ I = [0, L].

(3.6)Since the derivative of rj is:
r ′j (t) = (p

′(t)u(t)(1− w(t))− v(t)(p(t)w(t) + p′′(t)), p′(t)v(t)(1− w(t)) + u(t)(p(t)w(t) + p′′(t)))(3.7)it results:
‖r ′j (t)‖2 = (p′(t))2[1−w(t)]2+[p(t)w(t)+p′′(t)]2 ∈ ((ρ′p(t)+p′′(t))2, (p′(t))2+[p(t)+p′′(t)]2)(3.8)and then Cj is a regular curve. It results also immediately:{

x ′′j = p
′′u(1− 2w) + p′v(ρ2u2 + w2 − 2w) + v(pρ2uv − p′′′)− pw2u

y ′′j = p
′′v(1− 2w) + p′u(ρ2v2 − w2 + 2w)− u(pρ2uv − p′′′)− pw2v

(3.9)
and then, considering the map (·, ρ)→ rj(·) as a flow of curves, we compute its first derivative witha possible application to a parabolic flow (for example, of curve shortening type, see the chapter 2in [3]):{

∂
∂ρ r
′′
j (t) = 2ρu(t)v(t)[p

′(t)(u(t), v(t)) + p(t)(v(t),−u(t))] = 2ρu(t)v(t)[−i rj(t)] = 2w ′(t)[i rj(t)],
‖ ∂∂ρ r

′′
j (t)‖ = 2|ρ||u(t)||v(t)|‖r(t)‖. (3.10)Therefore, ∂

∂ρ r
′′
j (t) is orthogonal to rj(t), for all t ∈ [0, L].

Remark 3.2 We point out that following the approach of [8] we can think C and Cj as theEuclidean and Jacobi deformations of the support curve t → P (t) := (p(t), p′(t)). We have
‖r(t)‖ = ‖P (t)‖ = ‖rj(t)‖, for all t . The expression of P recalls the well-known Weierstrassparametrization (℘(u), ℘′(u)) of the elliptic curve E(g2, g3) : y2 = 4x3 − g2x − g3; see [9, p. 77].
2 Our main theoretical result computes the curvature of the mate Cj through a long but straight-forward computation:

Theorem 3.3 i) If p is not a constant then the support curve P is a regular one having the
Euclidean curvature:

kP (t) =
p′(t)p′′′(t)− (p′′(t))2

[(p′(t))2 + (p′′(t))2]
3
2

. (3.11)

https://doi.org/10.28924/ada/ma.4.18


Eur. J. Math. Anal. 10.28924/ada/ma.4.18 5

Let r(t0) be a vertex of the oval C i.e. p′′′(t0) = −p′(t0). Then the curvature of P in t0 is:
kP (t0) =

−1
[(p′(t0))2 + (p′′(t0))2]

1
2

< 0.

ii) The curvature of the ρ-Jacobi mate Cj of the oval C is a quadratic function in ρ:

kj =
(p′)2(1− w)(2w − w2) + p′[(1− w)p′′′ − ρ2uv(p + p′′)] + [pw2 + p′′(2w − 1)](pw + p′′)

[(p′)2(1− w)2 + (pw + p′′)2]
3
2

.(3.12)
If the modulus ρ is zero then w ≡ 1 and kj reduces to the usual curvature k from (2.5). Moreover,
for the flow interpretation before the remark 3.2 we have:

∂2kj(t)

∂ρ2
|ρ=0 = −2u(t)v(t)p′(t)[k(t)]2. (3.13)

We focus now on some concrete examples.
Example 3.4 The circle C(O,R > 0) of the Euclidean plane geometry is the oval provided bythe constant support function p ≡ R and hence W ≡ 2R; the curve P consists in the unique point

(R, 0). Its ρ-Jacobi mate coincides cu C(O,R), hence kj = k ≡ 1
R , but now with the parametrization:

circlej(t) = R(u(t), v(t)), ci rcle
′
j (t) = Rw(t)(−v(t), u(t)), ‖circle ′j (t)‖ = Rw(t) ∈ [Rρ′, R].(3.14)Hence, the Frenet frame is:

T (t) = (−v(t), u(t)), N(t) = iT (t) = (−u(t),−v(t)) (3.15)
as natural generalization of (2.6). By defining a new function:

W (t) =

∫ t

0

w(λ)dλ (3.16)
we can write the parametrization by arc-length:

circlej(s) = R
(
u ◦W−1

( s
R

)
, v ◦W−1

( s
R

))
, s ∈ [0, 2πR]. (3.17)

The value of L̃ from (3.3) in the self-complementary case is:
L̃

(
1√
2

)
' 1.85 >

π

2
' 1.57 (3.18)

while the second identity from (3.2) provides, in the case ρ 6= 0, a second Jacobi parametrizationof the circle:
Scirclej(t) = R(ρv(t), w(t)), Scircle ′j (t) = Rρu(t)(w(t),−ρv(t)). (3.19)

Now, this second parametrization has singularities, namely the zeros L̃, 3L̃ of the function u. 2

Example 3.5 Fix the smooth real function p(t) := R − cos 3t; hence p(t) = p(t + 2π) andagain the width is constant W ≡ 2R. In [4, p. 23] it is proved that if R > 8 then p is the supportfunction of an oval C. If R > 8 is a positive integer then the oval C contains the integral point
(−(R + 1), 0) corresponding to t = π while the curve P contains the 4 integral points (R − 1, 0),

https://doi.org/10.28924/ada/ma.4.18


Eur. J. Math. Anal. 10.28924/ada/ma.4.18 6

(R, 3), (R + 1, 0), (R,−3) corresponding respectively to t = 0, t = π
2 , t = π and t = 3π

2 . Withthe derivatives:
p′(t) = 3 sin 3t, p′′(t) = 9 cos 3t, p′′′(t) = −27 sin 3t (3.20)

it results the curvatures:
kP (t) =

−3
[(sin 3t)2 + 9(cos 3t)2]

3
2

< 0, k(t) =
1

R + 8cos 3t
∈
[
1

R + 8
,
1

R − 8

]
. (3.21)

We note that kP does not depend on R while the curvature k solves the differential equation
∂k
∂R = −k

2. The Cauchy and the Blaschke formulae give:
L(C) = 2πR, A(C) = π(R2 − 4) > 60π. (3.22)

The length of the curve P is:
L(P ) = 3

∫ 2π
0

√
(sin 3t)2 + 9(cos 3t)2dt = 3

∫ 2π
0

√
5 + 4 cos 6tdt ' 40.09 (3.23)

and we point out that the argument 3t involved in its components recalls the Cayley sextic, whichis not an oval but a closed curve:
Cay ley(t) := cos3 t(cos 3t, sin 3t), t ∈ [0, 2π], L(Cay ley) = 3π. (3.24)

For the ρ-Jacobi mate we compute only the velocity since its curvature has a complicated expression:{
‖r ′j (t)‖2 = 9(sin 3t)2[1− w(t)]2 + [9 cos 3t + w(t)(R − cos 3t)]2,
0 < (ρ′R + (9− ρ′) cos 3t)2 < ‖r ′j (t)‖2 < 81(sin 3t)2 + (R + 8cos 3t)2.

(3.25)
2

Example 3.6 For α ∈ [1,+∞) the 2π-periodic function pα : [0, 2π]→ R∗+,
pα(t) :=

1
α

√
α4 cos2 t + sin2 t is the support function of an ellipse since:

pα(t) + p
′′
α(t) =

α3

(α4 cos2 t + sin2 t)
3
2

> 0. (3.26)
Now, the width function is non-constant being 2pα. As example, with WolframAlpha we obtain thelength L (Pα=2) ' 45.51. The RHS of the inequality (3.8) reads:

‖r ′j (t)‖2 <
(α4 − 1)2

2α2(α4 cos2 t + sin2 t)
+

α6

(α4 cos2 t + sin2 t)3
. (3.27)

With the same possible flow interpretation in mind we compute the first derivative of the supportfunction:
∂pα
∂α
(t) =

α4 cos2 t − sin2 t
α2
√
α4 cos2 t + sin2 t

. (3.28)
We note also that for α > 1 the given support function does not has an indicatrix i.e. the planecurve defined implicitly by {(α, t) ∈ R2; pα(t) = 1} is empty. The same fact holds for the supportfunction of the previous example when R > 8. 2

https://doi.org/10.28924/ada/ma.4.18


Eur. J. Math. Anal. 10.28924/ada/ma.4.18 7References
[1] J. Arroyo, O. J. Garay, J. J. Mencia, When is a periodic function the curvature of a closed plane curve, Am. Math.Mon. 115 (2005), 405-414. https://doi.org/10.1080/00029890.2008.11920543[2] H. Alencar, W. Santos, G. Silva Neto, Differential geometry of plane curves, American Mathematical Society,Providence, Rhode Island, 2022. https://doi.org/10.1090/stml/096.[3] B. Andrews, B. Chow, C. Guenther, M. Langford, Extrinsic geometric flows, American Mathematical Society, Provi-dence, Rhode Island, 2020. https://doi.org/10.1090/gsm/206.[4] W. Cieślak, W. Mozgawa, P. Wlaź, On the closest distance between a point and a convex body, Bull. Soc. Sci. Lettr.Łódź, Sér.: Rech. Déform. 67 (2017), 21-30. https://doi.org/10.26485/0459-6854/2017/67.2/2.[5] M. Crasmareanu, Magic conics, their integer points and complementary ellipses, An. Ştiinţ. Univ. Al. I. Cuza IaşiMat. 67 (2021), 129-148.[6] M. Crasmareanu, The flow-curvature of plane parametrized curves, Commun. Fac. Sci. Univ. Ankara Ser. A1 Math.Stat. 72 (2023), 417-428. https://doi.org/10.31801/cfsuasmas.1165123.[7] R.H. Cushman, L.M. Bates, Global aspects of classical integrable systems, Springer Basel, Basel, 2015. https:

//doi.org/10.1007/978-3-0348-0918-4.[8] B. Mazur, Perturbations, deformations, and variations (and “near-misses") in geometry, physics, and number theory,Bull. Amer. Math. Soc. 41 (2004), 307–336. https://doi.org/10.1090/S0273-0979-04-01024-9.[9] R. Takloo-Bighash, A pythagorean introduction to number theory: right triangles, sums of squares, and arithmetic,Springer, Cham, 2018. https://doi.org/10.1007/978-3-030-02604-2.

https://doi.org/10.28924/ada/ma.4.18
https://doi.org/10.1080/00029890.2008.11920543
https://doi.org/10.1090/stml/096
https://doi.org/10.1090/gsm/206
https://doi.org/10.26485/0459-6854/2017/67.2/2
https://doi.org/10.31801/cfsuasmas.1165123
https://doi.org/10.1007/978-3-0348-0918-4
https://doi.org/10.1007/978-3-0348-0918-4
https://doi.org/10.1090/S0273-0979-04-01024-9
https://doi.org/10.1007/978-3-030-02604-2

	1. Introduction
	2. The differential geometry of Euclidean ovals
	3. The Jacobi mate of an oval
	References

