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April  2018, 38(4): 2187-2206. doi: 10.3934/dcds.2018090

New periodic orbits in the planar equal-mass three-body problem

Rongchang Liu , Jiangyuan Li and  Duokui Yan ,

School of Mathematics and System Sciences, Beihang University, Beijing 100191, China

* Corresponding author: Duokui Yan

Received  June 2017 Revised  October 2017 Published  January 2018

Fund Project: The authors are supported by NSFC No.11432001 and the Fundamental Research Funds of the Central Universities.

It is known that there exist two sets of nontrivial periodic orbits in the planar equal-mass three-body problem: retrograde orbit and prograde orbit. By introducing topological constraints to a two-point free boundary value problem, we show that there exists a new set of periodic orbits for a small interval of rotation angle $ \mathit{\theta }$.

Keywords: Three-body problem, periodic orbit, variational method, topological constraint.
Mathematics Subject Classification: Primary: 70F07; Secondary: 70F10, 53D12.
Citation: Rongchang Liu, Jiangyuan Li, Duokui Yan. New periodic orbits in the planar equal-mass three-body problem. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 2187-2206. doi: 10.3934/dcds.2018090
References:
[1]

R. Broucke and D. Boggs, Periodic orbits in the planar general three-body problem, Celestial Mech., 11 (1975), 13-38.  doi: 10.1007/BF01228732.  Google Scholar

[2]

K. Chen, Existence and minimizing properties of retrograde orbits to the three-body problem with various choices of masses, Annals of Math., 167 (2008), 325-348.  doi: 10.4007/annals.2008.167.325.  Google Scholar

[3]

K. Chen and Y. Lin, On action-minimizing retrograde and prograde orbits of the three-body problem, Comm. Math. Phys., 291 (2009), 403-441.  doi: 10.1007/s00220-009-0769-5.  Google Scholar

[4]

A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses, Annals of Math., 152 (2000), 881-901.  doi: 10.2307/2661357.  Google Scholar

[5]

A. Chenciner, Action minimizing solutions in the Newtonian n-body problem: From homology to symmetry, Proceedings of the International Congress of Mathematicians (Beijing, 2002), Higher Ed. Press, Beijing, (2002), 279-294.  Google Scholar

[6]

W. B. Gordon, A minimizing property of Keplerian orbits, Amer. J. Math., 99 (1977), 961-971.  doi: 10.2307/2373993.  Google Scholar

[7]

M. Hénon, A family of periodic solutions of the planar three-body problem, and their stability, Celestial Mech., 13 (1976), 267-285.  doi: 10.1007/BF01228647.  Google Scholar

[8]

W. Kuang, T. Ouyang, Z. Xie and D. Yan, The Broucke-Hénon orbit and the Schubart orbit in the planar three-body problem with equal masses, preprint, arXiv: 1607.00580. Google Scholar

[9]

C. Marchal, How the method of minimization of action avoids singularities, Celest. Mech. Dyn. Astro., 83 (2002), 325-353.  doi: 10.1023/A:1020128408706.  Google Scholar

[10]

T. Ouyang and Z. Xie, Star pentagon and many stable choreographic solutions of the Newtonian 4-body problem, Physica D, 307 (2015), 61-76.  doi: 10.1016/j.physd.2015.05.015.  Google Scholar

[11]

B. Shi, R. Liu, D. Yan and T. Ouyang, Multiple periodic orbits connecting a collinear configuration and a double isosceles configuration in the planar equal-mass four-body problem, Adv. Nonlinear Stud., 17 (2017), 819-835.  doi: 10.1515/ans-2017-6028.  Google Scholar

[12]

G. Yu, Simple choreography solutions of the Newtonian N-body problem, Arch. Rational Mech. Anal., 225 (2017), 901-935.  doi: 10.1007/s00205-017-1116-1.  Google Scholar

[13]

G. Yu, Spatial double choreographies of the Newtonian $ 2n$-body problem, arXiv: 1608.07956. Google Scholar

show all references

References:
[1]

R. Broucke and D. Boggs, Periodic orbits in the planar general three-body problem, Celestial Mech., 11 (1975), 13-38.  doi: 10.1007/BF01228732.  Google Scholar

[2]

K. Chen, Existence and minimizing properties of retrograde orbits to the three-body problem with various choices of masses, Annals of Math., 167 (2008), 325-348.  doi: 10.4007/annals.2008.167.325.  Google Scholar

[3]

K. Chen and Y. Lin, On action-minimizing retrograde and prograde orbits of the three-body problem, Comm. Math. Phys., 291 (2009), 403-441.  doi: 10.1007/s00220-009-0769-5.  Google Scholar

[4]

A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses, Annals of Math., 152 (2000), 881-901.  doi: 10.2307/2661357.  Google Scholar

[5]

A. Chenciner, Action minimizing solutions in the Newtonian n-body problem: From homology to symmetry, Proceedings of the International Congress of Mathematicians (Beijing, 2002), Higher Ed. Press, Beijing, (2002), 279-294.  Google Scholar

[6]

W. B. Gordon, A minimizing property of Keplerian orbits, Amer. J. Math., 99 (1977), 961-971.  doi: 10.2307/2373993.  Google Scholar

[7]

M. Hénon, A family of periodic solutions of the planar three-body problem, and their stability, Celestial Mech., 13 (1976), 267-285.  doi: 10.1007/BF01228647.  Google Scholar

[8]

W. Kuang, T. Ouyang, Z. Xie and D. Yan, The Broucke-Hénon orbit and the Schubart orbit in the planar three-body problem with equal masses, preprint, arXiv: 1607.00580. Google Scholar

[9]

C. Marchal, How the method of minimization of action avoids singularities, Celest. Mech. Dyn. Astro., 83 (2002), 325-353.  doi: 10.1023/A:1020128408706.  Google Scholar

[10]

T. Ouyang and Z. Xie, Star pentagon and many stable choreographic solutions of the Newtonian 4-body problem, Physica D, 307 (2015), 61-76.  doi: 10.1016/j.physd.2015.05.015.  Google Scholar

[11]

B. Shi, R. Liu, D. Yan and T. Ouyang, Multiple periodic orbits connecting a collinear configuration and a double isosceles configuration in the planar equal-mass four-body problem, Adv. Nonlinear Stud., 17 (2017), 819-835.  doi: 10.1515/ans-2017-6028.  Google Scholar

[12]

G. Yu, Simple choreography solutions of the Newtonian N-body problem, Arch. Rational Mech. Anal., 225 (2017), 901-935.  doi: 10.1007/s00205-017-1116-1.  Google Scholar

[13]

G. Yu, Spatial double choreographies of the Newtonian $ 2n$-body problem, arXiv: 1608.07956. Google Scholar

Figure 2.  A picture of the periodic orbit extended by the action minimizer $\mathcal{P}_0$ with $\theta = \pi/10$. The three dots represent the starting configuration $Q_S$, and the three crosses represent the ending configuration $Q_E$ of $\mathcal{P}_0$. In the graph, the red curve is the trajectory of body 2, the blue curve is for body 1 and the black curve is for body 3.
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Figure 1.  The configurations $Q_S$ and $Q_E$ are shown, where blue dots represent $q_1$, red dots represent $q_2$ and black dots represent $q_3$. In $Q_S$, three masses are on the $x-$axis with an order $q_{2x} \leq q_{1x} \leq q_{3x}$. In $Q_E$, three masses form an isosceles triangle, whose symmetry axis is a counterclockwise $\theta$ rotation of the $x-$axis. $q_1$ is on the symmetry axis and $q_2$ is above the symmetry axis in $Q_E$.
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Figure 3.  In each figure, the horizontal axis is $ \theta/\pi$, and the vertical axis is the action value $ \mathcal{A}$. In each subfigure, the black curve is the graph of the test path's action $\mathcal{A}_{test}$; the purple curve is the graph of $\mathcal{A}_{Euler} = g(\theta)$ in (31) and the red curve is the graph of $f_2(\theta)$ in (31), which is the lower bound of $\mathcal{A}_{col}$.
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Table 1.  The positions of $\tilde{q}_{i, j} = \tilde{q}_{i}(\frac{j}{10}) \, (i = 1,2, \, j = 0,1, \dots, 10)$ in $\mathcal{P}_{test, \, \theta}$ corresponding to $\theta \in [0.181\pi, \, 0.183 \pi]$.
$\mathbf{\theta_0= 0.182 \pi, \, \, \, \, \, \theta \in [0.181 \pi, \, 0.183 \pi ]}$
$t$ $\tilde{q}_1$ $\tilde{q}_2$
$0$ $(0.1109, \, 0 )$ $( -1.1189,\, 0 )$
$0.1$ $( 0.11378134, \, 0.035120592)$ $ ( -1.1144897, \, 0.089194621)$
$0.2$ $(0.12212758, \, 0.068373015 )$ $( -1.1012708, \,0.17787543 )$
$0.3$ $(0.13510199, \, 0.098115021 )$ $(-1.0792782, \, 0.26552094 )$
$0.4$ $( 0.15147483, \, 0.12309328 )$ $( -1.0485685, \, 0.35159462 ) $
$0.5$ $( 0.16980434, \, 0.14251378 )$ $ ( -1.0092177, \, 0.43553762) $
$0.6$ $( 0.18860029, \, 0.15603011)$ $ (-0.96132044, \, 0.51676178)$
$0.7$ $(0.20644197, \, 0.16368275 )$ $ ( -0.90499096, \, 0.59464214) $
$0.8$ $( 0.22204784, \, 0.16582253 )$ $(-0.84036636, \, 0.66850889) $
$0.9$ $( 0.23430842, \, 0.16304077 )$ $( -0.76761388, \, 0.73763832) $
$1$ $(0.28822237, \, 0 ) R(\theta)$ $ (-0.144111185, \, 1.0455201)R(\theta)$
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$\mathbf{\theta_0= 0.182 \pi, \, \, \, \, \, \theta \in [0.181 \pi, \, 0.183 \pi ]}$
$t$ $\tilde{q}_1$ $\tilde{q}_2$
$0$ $(0.1109, \, 0 )$ $( -1.1189,\, 0 )$
$0.1$ $( 0.11378134, \, 0.035120592)$ $ ( -1.1144897, \, 0.089194621)$
$0.2$ $(0.12212758, \, 0.068373015 )$ $( -1.1012708, \,0.17787543 )$
$0.3$ $(0.13510199, \, 0.098115021 )$ $(-1.0792782, \, 0.26552094 )$
$0.4$ $( 0.15147483, \, 0.12309328 )$ $( -1.0485685, \, 0.35159462 ) $
$0.5$ $( 0.16980434, \, 0.14251378 )$ $ ( -1.0092177, \, 0.43553762) $
$0.6$ $( 0.18860029, \, 0.15603011)$ $ (-0.96132044, \, 0.51676178)$
$0.7$ $(0.20644197, \, 0.16368275 )$ $ ( -0.90499096, \, 0.59464214) $
$0.8$ $( 0.22204784, \, 0.16582253 )$ $(-0.84036636, \, 0.66850889) $
$0.9$ $( 0.23430842, \, 0.16304077 )$ $( -0.76761388, \, 0.73763832) $
$1$ $(0.28822237, \, 0 ) R(\theta)$ $ (-0.144111185, \, 1.0455201)R(\theta)$
Table 2.  The positions of $\tilde{q}_{i, j} = \tilde{q}_{i}(\frac{j}{10}) \, (i = 1,2, \, j = 0,1, \dots, 10)$ in $\mathcal{P}_{test, \, \theta}$ corresponding to $\theta \in [0.176\pi, \, 0.181 \pi]$.
$\mathbf{\theta_0= 0.18 \pi, \, \, \, \, \, \theta \in [0.176 \pi, \, 0.181 \pi ]}$
$t$ $\tilde{q}_1$ $\tilde{q}_2$
$0$ $( 0.1198, \, 0 )$ $(-1.1166, \, 0 )$
$0.1$ $( 0.12300003, \, 0.038069081 )$ $ (-1.1122108, \, 0.087838457 )$
$0.2$ $(0.13225041, \, 0.074032720 )$ $( -1.0990555, \, 0.17518597 )$
$0.3$ $(0.14657647, \, 0.10606411 )$ $( -1.0771702, \, 0.26154328 )$
$0.4$ $(0.16456742, \, 0.13280675 )$ $( -1.0466117, \, 0.34639491 ) $
$0.5$ $(0.18460176, \, 0.15344287 )$ $ (-1.0074550, \, 0.42920169 ) $
$0.6$ $( 0.20503978, \, 0.16765957 )$ $ ( -0.95979063, \, 0.50939364 )$
$0.7$ $(0.22435124, \, 0.17556129 )$ $ ( -0.90372458, \, 0.58636242 ) $
$0.8$ $( 0.24118113, \, 0.17757208 )$ $(-0.83938055, \, 0.65945294 ) $
$0.9$ $( 0.25437250, \, 0.17435316 )$ $(-0.76690612, \, 0.72795349 ) $
$1$ $( 0.31146657, \, 0 ) R(\theta)$ $ (-0.155733285, \, 1.0357709)R(\theta)$
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$\mathbf{\theta_0= 0.18 \pi, \, \, \, \, \, \theta \in [0.176 \pi, \, 0.181 \pi ]}$
$t$ $\tilde{q}_1$ $\tilde{q}_2$
$0$ $( 0.1198, \, 0 )$ $(-1.1166, \, 0 )$
$0.1$ $( 0.12300003, \, 0.038069081 )$ $ (-1.1122108, \, 0.087838457 )$
$0.2$ $(0.13225041, \, 0.074032720 )$ $( -1.0990555, \, 0.17518597 )$
$0.3$ $(0.14657647, \, 0.10606411 )$ $( -1.0771702, \, 0.26154328 )$
$0.4$ $(0.16456742, \, 0.13280675 )$ $( -1.0466117, \, 0.34639491 ) $
$0.5$ $(0.18460176, \, 0.15344287 )$ $ (-1.0074550, \, 0.42920169 ) $
$0.6$ $( 0.20503978, \, 0.16765957 )$ $ ( -0.95979063, \, 0.50939364 )$
$0.7$ $(0.22435124, \, 0.17556129 )$ $ ( -0.90372458, \, 0.58636242 ) $
$0.8$ $( 0.24118113, \, 0.17757208 )$ $(-0.83938055, \, 0.65945294 ) $
$0.9$ $( 0.25437250, \, 0.17435316 )$ $(-0.76690612, \, 0.72795349 ) $
$1$ $( 0.31146657, \, 0 ) R(\theta)$ $ (-0.155733285, \, 1.0357709)R(\theta)$
Table 3.  The positions of $\tilde{q}_{i, j} = \tilde{q}_{i}(\frac{j}{10}) \, (i = 1,2, \, j = 0,1, \dots, 10)$ in $\mathcal{P}_{test, \, \theta}$ corresponding to $\theta \in [0.165\pi, \, 0.176 \pi]$.
$\mathbf{\theta_0= 0.173 \pi, \, \, \, \, \, \theta \in [0.165 \pi, \, 0.176 \pi ]}$
$t$ $\tilde{q}_1$ $\tilde{q}_2$
$0$ $( 0.1454,\, 0 )$ $( -1.1064,\, 0 )$
$0.1$ $( 0.14967657,\, 0.046886901 )$ $ (-1.1020407,\, 0.083918190 )$
$0.2$ $(0.16194195,\, 0.090810510 )$ $(-1.0889788,\, 0.16740323 )$
$0.3$ $(0.18067059,\, 0.12932570 )$ $( -1.0672593,\, 0.25001086 )$
$0.4$ $(0.20377305,\, 0.16080303 )$ $( -1.0369499,\, 0.33127610 ) $
$0.5$ $( 0.22901187,\, 0.18445175 )$ $ ( -0.99813266,\, 0.41070550 ) $
$0.6$ $(0.25429011,\, 0.20016641 )$ $ ( -0.95089814,\, 0.48777071 )$
$0.7$ $(0.27778726,\, 0.20832504 )$ $ ( -0.89534096,\, 0.56190183 ) $
$0.8$ $(0.29798679,\, 0.20961764 )$ $( -0.83155952,\, 0.63247939 ) $
$0.9$ $(0.31364961,\, 0.20493038 )$ $(-0.75965878,\, 0.69882326 ) $
$1$ $( 0.37739476,\, 0 ) R(\theta)$ $ ( -0.18869738,\, 1.0021646)R(\theta)$
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$\mathbf{\theta_0= 0.173 \pi, \, \, \, \, \, \theta \in [0.165 \pi, \, 0.176 \pi ]}$
$t$ $\tilde{q}_1$ $\tilde{q}_2$
$0$ $( 0.1454,\, 0 )$ $( -1.1064,\, 0 )$
$0.1$ $( 0.14967657,\, 0.046886901 )$ $ (-1.1020407,\, 0.083918190 )$
$0.2$ $(0.16194195,\, 0.090810510 )$ $(-1.0889788,\, 0.16740323 )$
$0.3$ $(0.18067059,\, 0.12932570 )$ $( -1.0672593,\, 0.25001086 )$
$0.4$ $(0.20377305,\, 0.16080303 )$ $( -1.0369499,\, 0.33127610 ) $
$0.5$ $( 0.22901187,\, 0.18445175 )$ $ ( -0.99813266,\, 0.41070550 ) $
$0.6$ $(0.25429011,\, 0.20016641 )$ $ ( -0.95089814,\, 0.48777071 )$
$0.7$ $(0.27778726,\, 0.20832504 )$ $ ( -0.89534096,\, 0.56190183 ) $
$0.8$ $(0.29798679,\, 0.20961764 )$ $( -0.83155952,\, 0.63247939 ) $
$0.9$ $(0.31364961,\, 0.20493038 )$ $(-0.75965878,\, 0.69882326 ) $
$1$ $( 0.37739476,\, 0 ) R(\theta)$ $ ( -0.18869738,\, 1.0021646)R(\theta)$
Table 4.  The positions of $\tilde{q}_{i, j} = \tilde{q}_{i}(\frac{j}{10}) \, (i = 1,2, \, j = 0,1, \dots, 10)$ in $ \mathcal{P}_{test, \, \theta}$ corresponding to $\theta \in [0.146\pi, \, 0.165 \pi]$.
$\mathbf{\theta_0= 0.16 \pi, \, \, \, \, \, \theta \in [0.146 \pi, \, 0.165 \pi ]}$
$t$ $\tilde{q}_1$ $\tilde{q}_2$
$0$ $(0.1803, \, 0 )$ $( -1.0860,\, 0 )$
$0.1$ $( 0.18658246,\, 0.059576674 )$ $ (-1.0816238,\, 0.078768710)$
$0.2$ $( 0.20429715,\, 0.11438956 )$ $(-1.0685224,\, 0.15716562)$
$0.3$ $( 0.23056861,\, 0.16092393 )$ $( -1.0467705,\, 0.23480016)$
$0.4$ $( 0.26186968,\, 0.19737903 )$ $( -1.0164725,\, 0.31125003) $
$0.5$ $( 0.29489886,\, 0.22338233 )$ $ ( -0.97774453,\, 0.38605495) $
$0.6$ $(0.32695826,\, 0.23945199 )$ $ ( -0.93070180,\, 0.45871449)$
$0.7$ $(0.35597786,\, 0.24655415 )$ $ ( -0.87545128,\, 0.52868620) $
$0.8$ $(0.38039590,\, 0.24583791 )$ $( -0.81208876,\, 0.59538118) $
$0.9$ $(0.39901814,\, 0.23851003 )$ $( -0.74069823,\, 0.65815496 ) $
$1$ $(0.46894463,\, 0) R(\theta)$ $ ( -0.234472315,\, 0.94630060)R(\theta)$
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$\mathbf{\theta_0= 0.16 \pi, \, \, \, \, \, \theta \in [0.146 \pi, \, 0.165 \pi ]}$
$t$ $\tilde{q}_1$ $\tilde{q}_2$
$0$ $(0.1803, \, 0 )$ $( -1.0860,\, 0 )$
$0.1$ $( 0.18658246,\, 0.059576674 )$ $ (-1.0816238,\, 0.078768710)$
$0.2$ $( 0.20429715,\, 0.11438956 )$ $(-1.0685224,\, 0.15716562)$
$0.3$ $( 0.23056861,\, 0.16092393 )$ $( -1.0467705,\, 0.23480016)$
$0.4$ $( 0.26186968,\, 0.19737903 )$ $( -1.0164725,\, 0.31125003) $
$0.5$ $( 0.29489886,\, 0.22338233 )$ $ ( -0.97774453,\, 0.38605495) $
$0.6$ $(0.32695826,\, 0.23945199 )$ $ ( -0.93070180,\, 0.45871449)$
$0.7$ $(0.35597786,\, 0.24655415 )$ $ ( -0.87545128,\, 0.52868620) $
$0.8$ $(0.38039590,\, 0.24583791 )$ $( -0.81208876,\, 0.59538118) $
$0.9$ $(0.39901814,\, 0.23851003 )$ $( -0.74069823,\, 0.65815496 ) $
$1$ $(0.46894463,\, 0) R(\theta)$ $ ( -0.234472315,\, 0.94630060)R(\theta)$
Table 5.  The positions of $\tilde{q}_{i, j} = \tilde{q}_{i}(\frac{j}{10}) \, (i = 1,2, \, j = 0,1, \dots, 10)$ in $ \mathcal{P}_{test, \, \theta}$ corresponding to $\theta \in [0.12\pi, \, 0.146 \pi]$.
$\mathbf{\theta_0= 0.132 \pi, \, \, \, \, \, \theta \in [0.12 \pi, \, 0.146\pi ]}$
$t$ $\tilde{q}_1$ $\tilde{q}_2$
$0$ $(0.2306, \, 0)$ $( -1.0421, \, 0)$
$0.1$ $( 0.24197875, \, 0.079876150 )$ $ (-1.0375636, \, 0.072168012 )$
$0.2$ $( 0.27249601, \, 0.14948693 )$ $( -1.0240177, \, 0.14401235 )$
$0.3$ $( 0.31440456, \, 0.20354167 )$ $( -1.0016223, \, 0.21516044 )$
$0.4$ $(0.36051601, \, 0.24151617 )$ $( -0.97057606, \, 0.28517262 ) $
$0.5$ $(0.40588817, \, 0.26512985 )$ $ ( -0.93107630, \, 0.35354922 ) $
$0.6$ $(0.44748116, \, 0.27663491 )$ $ ( -0.88330444, \, 0.41974435 )$
$0.7$ $(0.48345138, \, 0.27815124 )$ $ ( -0.82742450, \, 0.48317696 ) $
$0.8$ $(0.51262863, \, 0.27151295 )$ $( -0.76358450, \, 0.54323643 ) $
$0.9$ $(0.53419879, \, 0.25829537 )$ $( -0.69191492, \, 0.59928148 ) $
$1$ $(0.59692629, \, 0) R(\theta)$ $ ( -0.298463145, \, 0.84227284 )R(\theta)$
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$\mathbf{\theta_0= 0.132 \pi, \, \, \, \, \, \theta \in [0.12 \pi, \, 0.146\pi ]}$
$t$ $\tilde{q}_1$ $\tilde{q}_2$
$0$ $(0.2306, \, 0)$ $( -1.0421, \, 0)$
$0.1$ $( 0.24197875, \, 0.079876150 )$ $ (-1.0375636, \, 0.072168012 )$
$0.2$ $( 0.27249601, \, 0.14948693 )$ $( -1.0240177, \, 0.14401235 )$
$0.3$ $( 0.31440456, \, 0.20354167 )$ $( -1.0016223, \, 0.21516044 )$
$0.4$ $(0.36051601, \, 0.24151617 )$ $( -0.97057606, \, 0.28517262 ) $
$0.5$ $(0.40588817, \, 0.26512985 )$ $ ( -0.93107630, \, 0.35354922 ) $
$0.6$ $(0.44748116, \, 0.27663491 )$ $ ( -0.88330444, \, 0.41974435 )$
$0.7$ $(0.48345138, \, 0.27815124 )$ $ ( -0.82742450, \, 0.48317696 ) $
$0.8$ $(0.51262863, \, 0.27151295 )$ $( -0.76358450, \, 0.54323643 ) $
$0.9$ $(0.53419879, \, 0.25829537 )$ $( -0.69191492, \, 0.59928148 ) $
$1$ $(0.59692629, \, 0) R(\theta)$ $ ( -0.298463145, \, 0.84227284 )R(\theta)$
Table 6.  The positions of $\tilde{q}_{i, j} = \tilde{q}_{i}(\frac{j}{10}) \, (i = 1,2, \, j = 0,1, \dots, 10)$ in $\mathcal{P}_{test, \, \theta}$ corresponding to $\theta \in [0.089\pi, \, 0.12 \pi]$.
$\mathbf{\theta_0= 0.1 \pi, \, \, \, \, \, \theta \in [0.089 \pi, \, 0.12\pi ]}$
$t$ $\tilde{q}_1$ $\tilde{q}_2$
$0$ $(0.2680, \, 0)$ $( -0.9999, \, 0 )$
$0.1$ $(0.28699887, \, 0.097087533 )$ $ ( -0.99513360, \, 0.068868347)$
$0.2$ $(0.33390510, \, 0.17420817 )$ $( -0.98094971, \, 0.13739837 )$
$0.3$ $(0.39193700, \, 0.22664276 )$ $( -0.95760880, \, 0.20514818)$
$0.4$ $(0.45044849, \, 0.25815778 )$ $( -0.92539244, \, 0.27156414 ) $
$0.5$ $(0.50439276, \, 0.27354220 )$ $ (-0.88455797, \, 0.33601715 ) $
$0.6$ $(0.55150942, \, 0.27662332 )$ $ ( -0.83534366, \, 0.39783468 )$
$0.7$ $(0.59074871, \, 0.27023456 )$ $ ( -0.77798406, \, 0.45632275 ) $
$0.8$ $(0.62156413, \, 0.25649856 )$ $( -0.71271941, \, 0.51078103 ) $
$0.9$ $(0.64359767, \, 0.23707560 )$ $( -0.63979436, \, 0.56051180 ) $
$1$ $(0.69032914, \, 0) R(\theta)$ $ ( -0.34516457, \, 0.74809598 )R(\theta)$
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$\mathbf{\theta_0= 0.1 \pi, \, \, \, \, \, \theta \in [0.089 \pi, \, 0.12\pi ]}$
$t$ $\tilde{q}_1$ $\tilde{q}_2$
$0$ $(0.2680, \, 0)$ $( -0.9999, \, 0 )$
$0.1$ $(0.28699887, \, 0.097087533 )$ $ ( -0.99513360, \, 0.068868347)$
$0.2$ $(0.33390510, \, 0.17420817 )$ $( -0.98094971, \, 0.13739837 )$
$0.3$ $(0.39193700, \, 0.22664276 )$ $( -0.95760880, \, 0.20514818)$
$0.4$ $(0.45044849, \, 0.25815778 )$ $( -0.92539244, \, 0.27156414 ) $
$0.5$ $(0.50439276, \, 0.27354220 )$ $ (-0.88455797, \, 0.33601715 ) $
$0.6$ $(0.55150942, \, 0.27662332 )$ $ ( -0.83534366, \, 0.39783468 )$
$0.7$ $(0.59074871, \, 0.27023456 )$ $ ( -0.77798406, \, 0.45632275 ) $
$0.8$ $(0.62156413, \, 0.25649856 )$ $( -0.71271941, \, 0.51078103 ) $
$0.9$ $(0.64359767, \, 0.23707560 )$ $( -0.63979436, \, 0.56051180 ) $
$1$ $(0.69032914, \, 0) R(\theta)$ $ ( -0.34516457, \, 0.74809598 )R(\theta)$
Table 7.  The positions of $\tilde{q}_{i, j} = \tilde{q}_{i}(\frac{j}{10}) \, (i = 1,2, \, j = 0,1, \dots, 10)$ in $\mathcal{P}_{test, \, \theta}$ corresponding to $\theta \in [0.084\pi, \, 0.089 \pi]$.
$\mathbf{\theta_0= 0.085 \pi, \, \, \, \, \, \theta \in [0.084 \pi, \, 0.089\pi ]}$
$t$ $\tilde{q}_1$ $\tilde{q}_2$
$0$ $(0.2818, \, 0)$ $( -0.9827, \, 0 )$
$0.1$ $( 0.30528479, \, 0.10386817 )$ $ ( -0.97782312, \, 0.068296014 )$
$0.2$ $( 0.36044290, \, 0.18168802 )$ $( -0.96333464, \, 0.13622832 )$
$0.3$ $(0.42517243, \, 0.23070675 )$ $( -0.93953669, \, 0.20329795 )$
$0.4$ $( 0.48803512, \, 0.25746053 )$ $( -0.90673504, \, 0.26887596 ) $
$0.5$ $(0.54454099, \, 0.26800959 )$ $ (-0.86520333, \, 0.33225532 ) $
$0.6$ $(0.59300170, \, 0.26663595 )$ $ (-0.81520260, \, 0.39268889 )$
$0.7$ $(0.63277000, \, 0.25631044 )$ $ ( -0.75700467, \, 0.44941510 ) $
$0.8$ $(0.66356766, \, 0.23917705 )$ $( -0.69090435, \, 0.50167763 ) $
$0.9$ $(0.68521757, \, 0.21687569 )$ $( -0.61721738, \, 0.54873962 ) $
$1$ $(0.72320338, \, 0) R(\theta)$ $ ( -0.36160169, \, 0.71048767 )R(\theta)$
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$\mathbf{\theta_0= 0.085 \pi, \, \, \, \, \, \theta \in [0.084 \pi, \, 0.089\pi ]}$
$t$ $\tilde{q}_1$ $\tilde{q}_2$
$0$ $(0.2818, \, 0)$ $( -0.9827, \, 0 )$
$0.1$ $( 0.30528479, \, 0.10386817 )$ $ ( -0.97782312, \, 0.068296014 )$
$0.2$ $( 0.36044290, \, 0.18168802 )$ $( -0.96333464, \, 0.13622832 )$
$0.3$ $(0.42517243, \, 0.23070675 )$ $( -0.93953669, \, 0.20329795 )$
$0.4$ $( 0.48803512, \, 0.25746053 )$ $( -0.90673504, \, 0.26887596 ) $
$0.5$ $(0.54454099, \, 0.26800959 )$ $ (-0.86520333, \, 0.33225532 ) $
$0.6$ $(0.59300170, \, 0.26663595 )$ $ (-0.81520260, \, 0.39268889 )$
$0.7$ $(0.63277000, \, 0.25631044 )$ $ ( -0.75700467, \, 0.44941510 ) $
$0.8$ $(0.66356766, \, 0.23917705 )$ $( -0.69090435, \, 0.50167763 ) $
$0.9$ $(0.68521757, \, 0.21687569 )$ $( -0.61721738, \, 0.54873962 ) $
$1$ $(0.72320338, \, 0) R(\theta)$ $ ( -0.36160169, \, 0.71048767 )R(\theta)$
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