DISCRETE EULERʼS ELASTICA ‒ CHARACTERIZATION AND APPLICATION
Sebastián Elías Graiﬀ Zurita¹, Kenji Kajiwara²¹ Graduate School of Mathematics, Kyushu University,e-mail: ma217063@math.kyushu-u.ac.jp,² Institute of Mathematics for Industry, Kyushu University

Overview

Recently, Brander et al. [1] have constructed
an algorithm to approximate a given plane
curve segment by an elastica, in a continuous
framework, motivated mainly by thedevelopment of robotic hot-blade cutting
technology.

An Euler’s elastica curve is a plane curve
whose shape is that of an ideal infinitesimallythin elastic rod bent on a plane, without
stretching.
One way of characterizing this family of
curves is via a variational problem, i.e. theminimization of the bending energy yields a
diﬀerential equation that defines the elasticafamily.

arg min L(p) :=p

In this work, in view of higher quality
numerical implementations, we’ve applied adiscrete analogue of that algorithm toapproximate a given discrete plane curve by
an integrable discrete elastica [2].

Euler’s elastica
✓
◆

N
X
1

2
n=0

k⇣n (p)nk

2)

Initialguess

Gradient-driven tool
IPOPT

k = 0.2x(s)y(s)

2 R2 : arc length parametrized planar curve(s : arc length , k 0 (s)k = 1)
T (s) = 0 (s): unit tangent vector(s) =<latexit sha1_base64="pzx/MA0CnRfRW+8rr5KEbc4eKt0=">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</latexit>
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General solution<latexit sha1_base64="ORhdpbN5QVC+r0lZPkO9BrU7sS0=">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k = 0.82<latexit sha1_base64="6mxrjvcXP/5oJgRhFZXpXiJ5JY0=">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</latexit>
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✓(s)
sin
= k sn (q + zs, k)
2: angle function✓(s)<latexit sha1_base64="mL92Cbk5f8JZyIS9N1O7bBJ1DL8=">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</latexit>
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(s) = ✓0 (s): curvature<latexit sha1_base64="FBQHA0eemJYwh9eYJisRPXReODQ=">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</latexit>
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k = 0.992

sn(u, k) : a Jacobi elliptic function

Definition
00✓ (s) + µ sin ✓(s) = 0

q, z

2
µ
=
z
: real parameters,k: positive modulus<latexit sha1_base64="xo1sgYtUCCK9hYkAAMqXEkrJTFg=">AAAXkHicrVhLb9tGEN6kr9jpw0kPPvTCxgicVw0pTdGgRVonVoqkSGzXj8St5RgURVmM+VBJyokh6N5/0luv7bn/ov+mM98OKZKiJNqICIq7szPfzszOzC7Z6rlOFNdq/124+N77H3z40aW5+csff/LpZwtXrr6Ign5o2btW4AbhXsuMbNfx7d3YiV17rxfaptdy7Zet4zUef3lih5ET+Dvxac8+8Mwj3+k4lhkT6XDhWjPu2rG5vHwjumncNppevxk5viYy6YFRO1xYqq3U8DPGG3VpLCn5bQZXLt9WTdVWgbJUX3nKVr6Kqe0qU0V07au6qqke0Q7UgGghtRyM22qo5km2T1w2cZhEPab/I+rtC9WnPmNGkLZoFpfukCQNdV142tTugKqfPL+R4Z00xwD/AfBj4jsF6vRrjq7r9NwmGZueRYQVmqNNWhjUD+ifrWSbfVC6kGHPnNJoH3oG8I0DvSOSN9QO+ELwmvJkS2KMszXFWdmv92BdD5Kz7chbs4LLEB/msUbUb0qoieQ89OrC2jaeDmnZhYUP6K6r+7CTuTziOQZnDP/EdNmyTiHW3FZvsH4eeH1CvEU2NglhFetopR4bgr5N9x2JpioIfdV6hzhV0LI2BxIJIbzG/p8Unxbx+dS2ENPV1zV7Mf5j0mVHPURsbdHzV6Ksq4baIMomUTbUI/WMaM9pbJtot+heHpt9hXSMKdqWadSHFTrf+vALSyXWWELV6/qGvMIWO0QZkARHRh/+ZT8VPVcFw1a/05iZ83cZzqTa0kNesSYuXVxZNCUiHofaIz8nGfJQsE8wq53md4vaR9DKEAyeI0SedAnPSDU3ED8G8XvIhFNktYMYMyCRRIWuFJpb/zvwsK6s06pZMi/X17dkxZxcbMEuZAypT3fQ4tXl1mtq65bOasaO1L/qz5z+WtIVD7xV3xOtL6hawhBPNNQL9ZSeHuGx3vOpJok2O0TdE3ReA7a0D596sq4WPXnkVGq5T763sXuMZvuKrh9SzcY15Nkm+4t3lkjm68J6TeGel+lHpINHKz15tyrz/WTeLuHpKso50AEvc+v1D0D1JNdcSPRgkyNx3ZS6lOcfFPqcEzsFyn4hSw6kPoTYYW34No/Cmd6H9ayvg4gZ7bY2LOPYyONO1tFCO8DKmrLn7he4OB/XSvhmeakNb7KOTq4OTNKlnL9Mm0Ypp/ZdGUrea1FmH28jHuxMPhsiMc3v0RQrOM5N1BR9oirT/3GBR2seydkqu55Zvuo69LBWPbqjmb7cnMD7bnXiXVq32PPHyIatXF/vvWu5vT1S3wFR23NCurUR11meQTpjIFms9/QQ9n4NGxOO8ZPDEp2E9OkgqTuMwvS7RF8i+aHE7CwdRueTaXpMO8VU1WV+osUsq6M4jxNKZOqRoXgkwH7nYdWjsVNWGfaWnBGKyJp+ftynUh+LuJo+GXfW6XXamulzY5vGOqj53FtFz0eMx6ikHJ2vMNYgbTrqJ+FMejw2TGNE157sKK/p+IoW7Rz1I5zs9JtIG7jjlr9KY4XfPALKvRYsj2RnswvalPPwrB3MpM9+2ncjj7DlR9QPsLeXI+fHGDFBaQjKdjraVF/SzdS3Y7PcmCGp9TMhya0I1HKs/XeI1ZxpUxaNd538/7hEErVVIlPv8Cdp/diXGtJB9li5+B3FhK4W2TEDtWPEUaWiJW9V1bTopd8R+IxYpkueo1yjqjo1JuqUn+UQ+Fw1ixpVncum3BimVTWSTElytazCVrfFgPxvsKEuNhg5TM7ZATiynqqKvDUTeeucyOszkdfPiexjR/BS/CT7Osg/l95i9BsH4zYRA/qbBreTseqz8Vzt3FyJFfpdZ0Dv4UOqTTzbzTPj2rLz1zPxqUe4dSIn0B5OXS7OXOMZqussv31H2BFYsgWrmaK/6HXTeGtAI1eQn4stG4X9Vke2rm8eKtdQvgMe4ftIQjsvoiPv/1nE0ZtSNcQB7Eu+I5xd1hJZ6xyybZFtn0P2dXryOpFK1MXeYZ4ZaQvfc/QpOUij+izyT2jX0fK6dTb5Z6l80po/XFiqF787jzde3F2p11bqv9xbWn0k36QvqS/UNcqjuvqWTlhP6I1jl7T4Q/2l/lb/LF5dvL/44+JDzXrxgsh8rnK/xZ//B6g1cl4=</latexit>

k=1

T (s)

Variational principle
Z L
1 02
E=
✓ (s) + µ cos ✓(s) ds
20<latexit sha1_base64="qP8Uq0r94bKBGbryFqBLsv8DMPM=">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</latexit>

(s)

✓(s)

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k = 2.5<latexit sha1_base64="rVs6L7pPudwngEcx2U8Dc5M4kxs=">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</latexit>

Discrete Euler’s elastica
n 2R

k n+1

2

nk =

Tn =

Characterization of the discrete Euler’s elastica: discrete planar curve= const.

n+1

n

k n+1

: unit tangent vector

nk

sin

⇣
⌘1
0
cos ✓n +✓2 n+1
Tn = @ ⇣ ✓ +✓ ⌘ A
sin n 2 n+1: potential angle function✓nDefinition
sin

✓

✓n+1

2✓n + ✓n 1
4◆✓

✓n+1 + 2✓n + ✓n 1
+ " sin
4

◆

=0

Tn

Variational principle
S=

N
X1 
n=0

sin

2

✓

✓n+1 ✓n
4

◆

" sin

2

✓

✓n+1 + ✓n4

Approximation process(✓n + ✓n+1 )/2

◆

✓n+1

2✓n + ✓n 1
4◆✓

✓n+1 + 2✓n + ✓n 1
+ " sin
4◆=0General solution(
sin ✓n2 = k sn(q + zn, k),
1 "1+" = cn(z, k)
Reconstructionof the curve!
8>< = x0 ,
0
y0
>
:
Tn 1
n = n 1+(x0 , y0 )Parametersnp = (x0 , y0 ,Input
Initial guess
Output

Input

✓

&

, , z, q, k)More examples
x0 = 0.076
y0 = 0.115= 0.202= 15.5z = 0.101
q = 2.081(

k = 0.556

Minimization problem
NX1
arg min L(p) :=k⇣n (p)
2
p
n=0

nk

Gradient-driven tool
IPOPT

2

)

Initialguess

x0 = 0.014
y0 =Output0.411= 0.200= 12.3z = 0.083
q = 2.086
k = 0.533

References[1] D. Brander, J. Gravesen and T. B. Nørbjerg, Approximation by planar elastic curves, Adv. Comput. Math. 43 (2017) 25–43.
[2] K. Sogo, Variational discretization of Euler’s elastica problem, J. Phys. Soc. Jpn. 75 (2006) 064007.
[3] J. Inoguchi, K. Kajiwara, N. Matsuura and Y. Ohta, Motion and Bäcklund transformations of discrete planar curves, Kyushu J. Math. 66 (2012) 303–324.
[4] A. Wächter and L.T. Biegler, Math. Program., On the implementation of an interior-point filter line-search algorithm for large-scale non-linear programming, Ser. A 106 (2006) 25–57.
[5] S. E. Graiﬀ Zurita, K. Kajiwara, Fairing of discrete planar curves by Euler’s elasticae, In progress.