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楼主: mathe

[原创] 反演会保持田埂最短的特性吗?

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 楼主| 发表于 2019-4-3 15:20:32 | 显示全部楼层
改用Pari/gp可以计算获得:
t=1.145869754950095508247099850069722495324714873543556586284342371924883601241352833592462102584559613
d=0.2209696780891027977378513676362321110743346941807911312188610942298283317783824735498763241214062325
然后
I=[0.9192025346407358292306227753980566994513716914556202813450633827660510970474677965171545232091257802, 0.3937850940678771399585922872986842020798388222487138733021052945666379501076960565677860404215727946]
F=[0.4991574547946068226787752384923353770964861094094286582449413056625206272419988763890220633181731312, -0.8665113013244374565777339575364399012246819294942637875876538464556307906906338763229881758119514035]
B=[-0.05049072689722888952170518173930602603539630333402870218696573616533218839983143172228601348066893592, 0.9987245298366258947940528940386819558149148013150071278173336181831481433060257309566067346523348876]
C=[-0.005516997341733283071050005947320227288672836943175290939750257551052317406672856306266778809573949534, 0.1091281689694765362720605584799146840444020451405389291319828382217985837518922661795789409501447797]
D=[0.2209696780891027977378513676362321110743346941807911312188610942298283317783824735498763241214062325,0]
C1=[2.373238735071780516122518462665177432016514277997764994381435898296641075515169341887236268252632947, 4.756451555592603278431214350095370950446435502368938363924652157659346777317362240225141792759893316]
C2=[2.373238735071780516122518462665177432016514277997764994381435898296641075515169341887236268252632947,
-3.000334645429469396166805981966068437547593604613223408672154189470610668722859382440929592887206127]
C3=[2.373238735071780516122518462665177432016514277997764994381435898296641075515169341887236268252632947, 0.2130610487551819888546261440068205314114757021316985645862782258850129129915499490104359515005921626]
最终弧长4.833846643527396783657715928557596371715336638885537942540338590143011601591030504537006038698250498
  1. rotate60(x)=
  2. {
  3.     local(y);
  4.     y=vector(2);
  5.    y[1]=x[1]*0.5+x[2]*sqrt(3)/2;
  6.    y[2]=-sqrt(3)/2*x[1]+0.5*x[2];
  7.    y
  8. }

  9. get_circle_line1(cc, l)=
  10. {
  11.     local(c1,c2,c3,delt);
  12.     local(u,v,w,a,b,c,y2);
  13.     u=cc[1];v=cc[2];w=cc[3];
  14.     a=l[1];b=l[2];c=l[3];
  15.     c1=(b*b)/(a*a)+1.0;
  16.     c2=-b*u/a+v+2*b*c/(a*a);
  17.     c3=-c*u/a+w+c*c/(a*a);
  18.     delt = sqrt(c2*c2-4*c1*c3);
  19.     y2=(-c2+delt)/(2*c1);
  20.    [-b/a*y2-c/a,y2]
  21. }

  22. get_circle_line2(cc, l)=
  23. {
  24.     local(c1,c2,c3,delt);
  25.     local(u,v,w,a,b,c,y2);
  26.     u=cc[1];v=cc[2];w=cc[3];
  27.     a=l[1];b=l[2];c=l[3];
  28.     c1=(b*b)/(a*a)+1.0;
  29.     c2=-b*u/a+v+2*b*c/(a*a);
  30.     c3=-c*u/a+w+c*c/(a*a);
  31.     delt = sqrt(c2*c2-4*c1*c3);
  32.     y2=(-c2-delt)/(2*c1);
  33.    [-b/a*y2-c/a,y2]
  34. }

  35. get_circle_circle2(c1, c2)=
  36. {
  37.     get_circle_line2(c1,c2-c1)
  38. }

  39. get_circle_circle1(c1, c2)=
  40. {
  41.     get_circle_line1(c1,c2-c1)
  42. }

  43. area(a,b)=
  44. {
  45.     (a[1]*b[2]-a[2]*b[1])/2
  46. }

  47. atan2(s,h)=
  48. {
  49.     local(tv);
  50.     tv=atan(s/h);
  51.     if(h>0, tv,
  52.          if(tv<0, tv+Pi, tv-Pi)
  53.     )
  54. }

  55. get_arc_area(c,f,t)=
  56. {
  57.      local(rs,da);
  58.      rs=(f[1]-c[1])*(f[1]-c[1])+(f[2]-c[2])*(f[2]-c[2]);
  59.      da=atan2(t[2]-c[2],t[1]-c[1])-atan2(f[2]-c[2],f[1]-c[1]);
  60.      rs*(da-sin(da))/2.0
  61. }

  62. get_arc_length(c, f, t)=
  63. {
  64.      local(rs,da);
  65.       rs=(f[1]-c[1])*(f[1]-c[1])+(f[2]-c[2])*(f[2]-c[2]);
  66.       da=atan2(t[2]-c[2],t[1]-c[1])-atan2(f[2]-c[2],f[1]-c[1]);
  67.       sqrt(rs)*da
  68. }

  69. start_with_c(d,t,c)=
  70. {
  71.      local(u,tv,tv1,tv2,II,J);
  72.      local(rc,b,s1,s2);
  73.      u=(d+1/d)/2.0;tv=(1/d-d)/2.0*tan(t);
  74.      tv2=(1/d-d)/2.0*tan(t-Pi/3);
  75.      tv1=(1/d-d)/2.0*tan(t+Pi/3);
  76.      J=get_circle_circle2([0,0,-1],[-2*u,-2*tv2,-d*d+2*d*u]);
  77.      II=get_circle_circle1([0,0,-1],[-2*u,-2*tv1,-d*d+2*d*u]);
  78.      rc=sqrt(c[1]*c[1]+c[2]*c[2]);b=c/rc;
  79.      s1=area(b,c)+area(c,[d,0])+area([d,0],II)+area(II,b);
  80.      s1+=get_arc_area([u,tv],c,[d,0]);
  81.      s1+=get_arc_area([u,tv1],[d,0],II);
  82.      s1+=get_arc_area([0,0],II,b);
  83.      s2=area([d,0],J)+area(J,II)+area(II,[d,0]);
  84.      s2+=get_arc_area([u,tv2],[d,0],J);
  85.      s2+=get_arc_area([0,0],J,II);
  86.      s2+=get_arc_area([u,tv1],II,[d,0]);
  87.      [s1,s2]
  88. }

  89. start_out_with_c(d,t,c)=
  90. {
  91.      local(u,tv,tv1,tv2,II,J);
  92.      local(rc,b,s1,s2,alen);
  93.      u=(d+1/d)/2.0;tv=(1/d-d)/2.0*tan(t);
  94.      tv2=(1/d-d)/2.0*tan(t-Pi/3);
  95.      tv1=(1/d-d)/2.0*tan(t+Pi/3);
  96.      J=get_circle_circle2([0,0,-1],[-2*u,-2*tv2,-d*d+2*d*u]);
  97.      II=get_circle_circle1([0,0,-1],[-2*u,-2*tv1,-d*d+2*d*u]);
  98.      print("I=" II);
  99.      print("F=" J);
  100.      rc=sqrt(c[1]*c[1]+c[2]*c[2]);b=c/rc;
  101.      print("B=" b);
  102.      print("C="c);
  103.      print("d="d);
  104.      printf("u="u);
  105.      print("tv0=" tv);
  106.      print("tv1=" tv1);
  107.      print("tv2=" tv2);
  108.      alen=1.0-rc;
  109.      alen+=2.0*abs(get_arc_length([u,tv],c,[d,0]));
  110.      alen+=2.0*abs(get_arc_length([u,tv1],[d,0],II));
  111.      alen+=2.0*abs(get_arc_length([u,tv2],[d,0],J));
  112.      printf("len="alen);
  113.      s1=area(b,c)+area(c,[d,0])+area([d,0],II)+area(II,b);
  114.      s1+=get_arc_area([u,tv],c,[d,0]);
  115.      s1+=get_arc_area([u,tv1],[d,0],II);
  116.      s1+=get_arc_area([0,0],II,b);
  117.      s2=area([d,0],J)+area(J,II)+area(II,[d,0]);
  118.      s2+=get_arc_area([u,tv2],[d,0],J);
  119.      s2+=get_arc_area([0,0],J,II);
  120.      s2+=get_arc_area([u,tv1],II,[d,0]);
  121.      [s1,s2]
  122. }

  123. start_with_f2(d,t, f2)=
  124. {
  125.     local(u,tv,c);
  126.     u=(d+1/d)/2.0;tv=(1/d-d)/2.0*tan(t);
  127.     c=get_circle_circle2([-2*u,-2*tv,-d*d+2*d*u], [-2*f2[1],-2*f2[2],0]);
  128.     start_with_c(d,t,c)
  129. }

  130. start_out_with_f2(d,t, f2)=
  131. {
  132.     local(u,tv,c);
  133.     u=(d+1/d)/2.0;tv=(1/d-d)/2.0*tan(t);
  134.     c=get_circle_circle2([-2*u,-2*tv,-d*d+2*d*u], [-2*f2[1],-2*f2[2],0]);
  135.     start_out_with_c(d,t,c)
  136. }

  137. scan_one_config(d,t)=
  138. {
  139.     local(u,v,tt,f2);
  140.     u=(d+1.0/d)/2.0;v=(1/d-d)/2.0;
  141.     tt=tan(t);
  142.     f2=rotate60([u,tt*v]);
  143.     start_with_f2(d,t,f2)
  144. }

  145. output_data(d,t)=
  146. {
  147.     local(u,v,tt,f2);
  148.     u=(d+1.0/d)/2.0;v=(1/d-d)/2.0;
  149.     tt=tan(t);
  150.     f2=rotate60([u,tt*v]);
  151.     start_out_with_f2(d,t,f2)
  152. }
复制代码

然后比如调用
find_d(t)=solve(d=0.20,0.24,scan_one_config(d,t)[1]-Pi/5)
solve(t=1.14,1.15,scan_one_config(find_d(t),t)[2]-Pi/5)
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2019-4-7 21:52:41 | 显示全部楼层
mathe 发表于 2019-3-31 12:39
如果不考虑面积限制,那么仅仅给出所有内点的坐标和边的连接关系,我们就应该可以确定每条边的曲率。
假设 ...



由于代数计算的表达式比较复杂,我做了下列的数值计算:

\(A[x_1,y_1],B[x_2,y_2],C[x_3,y_3],O[x_0,y_0],B'[x_{12},y_{12}],C'[x_{13},y_{13}],F[x_{01},y_{01}],F'[x_{00},y_{00}]\)

过\(AOF',BOF',COF'\)的圆分别为

\((x-x_{21})^2+(y-y_{21})^2=r_1^2\)

\((x-x_{22})^2+(y-y_{22})^2=r_2^2\)

\((x-x_{23})^2+(y-y_{23})^2=r_3^2\)


取坐标\(x_0 = 0, x_1 = 0, y_0 = 0, y_1 = 1, x_2 = -2, y_2 = -\frac{1}{2}, x_3 = \frac{1}{3}, y_3 = -\frac{2}{3}\)


可以求得:

\(r_1 = 1.563789149, r_2 = 22.01863033, r_3 = 1.460091708, x_0 = 0, x_{00} = -2.525004700, x_{01}= -.3703591928, x_1 = 0, x_{11} = 0, x_{12}= -\frac{8}{17}, x_{13}= \frac{3}{5}, x_2 = -2, x_{21} = -1.481700544, x_{22} = 4.334447332, x_{23} = -1.096022181, x_3 =\frac{1}{3}, y_0 = 0, y_{00}= -0.6648832305, y_{01}= -00.9752283492, y_1 = 1, y_{11} = 1, y_{12} = -\frac{2}{17}, y_{13} = -\frac{6}{5}, y_2 = -\frac{1}{2}, y_{21}= 0.5000000000, y_{22}= -21.58778933, y_{23}= -0.9646777573, y_3 = -\frac{2}{3}\)


即:经过AOF',BOF',COF'的三个圆:

\((x+1.481700544)^2+(y-0.5000000000)^2=2.445436503\) ,\(\theta_1=0.6509021186=37.29394426^{\circ}, \widehat{AO}= 1.017873670,AO=1,r_1=1.563789149\)

\((x-4.334447332)^2+(y+21.58778933)^2=484.8200816\),\(\theta_2 =0.09366189084= 5.366431045^{\circ},\widehat{BO}= 2.062306550,BO=2.061552813,r_2=22.01863033\)

\((x+1.096022181)^2+(y+0.9646777573)^2=2.131867796\),\(\theta_3 =0.5161977674=29.57595346^{\circ}, \widehat{CO}= 2.288803009,CO=0.7453559923,r_3=1.460091708\)


注:\(\frac{1}{r_1}+\frac{1}{r_2}-\frac{1}{r_3}=\frac{1}{1.563789149}+\frac{1}{22.01863033}-\frac{1}{1.460091708}=0\)


画图:





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毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2019-4-8 18:45:11 | 显示全部楼层
对于一般三角形,若存在内点P及三条分割线圆弧\(\widehat{AP},\widehat{BP},\widehat{CP}\)将三角形分成面积相等的三等分,且使其分割线总长最小,根据mathe得到的条件:

设\(A[x_1,y_1],B[x_2,y_2],C[x_3,y_3],P[x_0,y_0]\),取\(x_2=0,y_2=0,x_3=a,y_3=0\),三角形三边长及面积为\(a,b,c,s\),三条分割线半径及圆弧角\(r_1,r_2,r_3,2t_1,2t_2,2t_3\),

线段长\(AP=x,BP=y,CP=z\)

则有下列方程:

\(\sin(t_1)yz+\sin(t_2)xz+\sin(t_3)xy=0\)

\(12ay_0\sin(t_2)^2\sin(t_3)^2-3y^2\sin(2t_2)\sin(t_3)^2+3z^2\sin(2t_3)\sin(t_2)^2-8s\sin(t_2)^2\sin(t_3)^2-6z^2t_3\sin(t_2)^2+6y^2t_2\sin(t_3)^2=0\)

\(-6\sin(t_1)^2\sin(t_2)^2a^2y_0+6\sin(t_1)^2\sin(t_2)^2b^2y_0-6\sin(t_1)^2\sin(t_2)^2c^2y_0-3x^2\sin(2t_1)\sin(t_2)^2a+3y^2\sin(2t_2)\sin(t_1)^2a-8s\sin(t_1)^2\sin(t_2)^2a+24x_0s\sin(t_1)^2\sin(t_2)^2-6y^2t_2\sin(t_1)^2a+6x^2t_1\sin(t_2)^2a=0\)

\(\sin(t_1)a^2yz-2\sin(t_1)yzx_0a-\sin(t_1)b^2yz+\sin(t_1)c^2yz-2\sin(t_2)xzx_0a+2\sin(t_3)xya^2-2\sin(t_3)xyx_0a+2\cos(t_1)yzy_0a+2\cos(t_2)xzy_0a+2\cos(t_3)xyy_0a-4\cos(t_1)yzs=0\)

\(2\sin(t_1)xyy_0a+2\sin(t_1)yzy_0a+2\sin(t_2)xzy_0a-\cos(t_1)a^2yz+2\cos(t_1)yzx_0a+\cos(t_1)b^2yz-\cos(t_1)c^2yz+2\cos(t_2)xzx_0a-2\cos(t_3)xya^2+2\cos(t_3)xyx_0a-4\sin(t_1)xys-4\sin(t_1)yzs=0\)

\(-x_0^2+y^2-y_0^2=0\)

\(-a^2+2ax_0-x_0^2-y_0^2+z^2=0\)

\(a^2x_0-ac^2+ax^2-ax_0^2-ay_0^2-b^2x_0+c^2x_0+4sy_0=0\)


取\(a=5,b=4,c=3,s=6\)得到


  1. sin(t1)*y*z+sin(t2)*x*z+sin(t3)*x*y, 60*y0*sin(t2)^2*sin(t3)^2-3*y^2*sin(2*t2)*sin(t3)^2+3*z^2*sin(2*t3)*sin(t2)^2-48*sin(t2)^2*sin(t3)^2-6*z^2*t3*sin(t2)^2+6*y^2*t2*sin(t3)^2, -108*sin(t1)^2*sin(t2)^2*y0-15*x^2*sin(2*t1)*sin(t2)^2+15*y^2*sin(2*t2)*sin(t1)^2-240*sin(t1)^2*sin(t2)^2+144*x0*sin(t1)^2*sin(t2)^2-30*y^2*t2*sin(t1)^2+30*x^2*t1*sin(t2)^2, 18*sin(t1)*y*z-10*sin(t1)*y*z*x0-10*sin(t2)*x*z*x0+50*sin(t3)*x*y-10*sin(t3)*x*y*x0+10*cos(t1)*y*z*y0+10*cos(t2)*x*z*y0+10*cos(t3)*x*y*y0-24*cos(t1)*y*z, 10*sin(t1)*x*y*y0+10*sin(t1)*y*z*y0+10*sin(t2)*x*z*y0-18*cos(t1)*y*z+10*cos(t1)*y*z*x0+10*cos(t2)*x*z*x0-50*cos(t3)*x*y+10*cos(t3)*x*y*x0-24*sin(t1)*x*y-24*sin(t1)*y*z, -x0^2+y^2-y0^2, -x0^2-y0^2+z^2+10*x0-25, 5*x^2-5*x0^2-5*y0^2+18*x0+24*y0-45
复制代码




得不到我们想要的解,谁有兴趣解一下上面的方程组看能否得到非平凡解?

我用maple只得到了下列平凡解:

  1. {t1 = 0, t2 = t2, t3 = 0, x = 4, x0 = 5, y = 5, y0 = 0, z = 0}, {t1 = 0, t2 = t2, t3 = Pi, x = 4, x0 = 5, y = 5, y0 = 0, z = 0}, {t1 = 0, t2 = t2, t3 = 0, x = 4, x0 = 5, y = -5, y0 = 0, z = 0}, {t1 = 0, t2 = t2, t3 = Pi, x = 4, x0 = 5, y = -5, y0 = 0, z = 0}, {t1 = 0, t2 = t2, t3 = 0, x = -4, x0 = 5, y = 5, y0 = 0, z = 0}, {t1 = 0, t2 = t2, t3 = Pi, x = -4, x0 = 5, y = 5, y0 = 0, z = 0}, {t1 = 0, t2 = t2, t3 = 0, x = -4, x0 = 5, y = -5, y0 = 0, z = 0}, {t1 = 0, t2 = t2, t3 = Pi, x = -4, x0 = 5, y = -5, y0 = 0, z = 0}, {t1 = Pi, t2 = 0, t3 = 0, x = 4, x0 = 5, y = 5, y0 = 0, z = 0}, {t1 = Pi, t2 = 0, t3 = Pi, x = 4, x0 = 5, y = 5, y0 = 0, z = 0}, {t1 = Pi, t2 = 0, t3 = 0, x = 4, x0 = 5, y = -5, y0 = 0, z = 0}, {t1 = Pi, t2 = 0, t3 = Pi, x = 4, x0 = 5, y = -5, y0 = 0, z = 0}, {t1 = Pi, t2 = 0, t3 = 0, x = -4, x0 = 5, y = 5, y0 = 0, z = 0}, {t1 = Pi, t2 = 0, t3 = Pi, x = -4, x0 = 5, y = 5, y0 = 0, z = 0}, {t1 = Pi, t2 = 0, t3 = 0, x = -4, x0 = 5, y = -5, y0 = 0, z = 0}, {t1 = Pi, t2 = 0, t3 = Pi, x = -4, x0 = 5, y = -5, y0 = 0, z = 0}, {t1 = Pi, t2 = Pi, t3 = 0, x = 4, x0 = 5, y = 5, y0 = 0, z = 0}, {t1 = Pi, t2 = Pi, t3 = Pi, x = 4, x0 = 5, y = 5, y0 = 0, z = 0}, {t1 = Pi, t2 = Pi, t3 = 0, x = 4, x0 = 5, y = -5, y0 = 0, z = 0}, {t1 = Pi, t2 = Pi, t3 = Pi, x = 4, x0 = 5, y = -5, y0 = 0, z = 0}, {t1 = Pi, t2 = Pi, t3 = 0, x = -4, x0 = 5, y = 5, y0 = 0, z = 0}, {t1 = Pi, t2 = Pi, t3 = Pi, x = -4, x0 = 5, y = 5, y0 = 0, z = 0}, {t1 = Pi, t2 = Pi, t3 = 0, x = -4, x0 = 5, y = -5, y0 = 0, z = 0}, {t1 = Pi, t2 = Pi, t3 = Pi, x = -4, x0 = 5, y = -5, y0 = 0, z = 0}, {t1 = 0, t2 = Pi, t3 = Pi, x = 0, x0 = 9/5, y = 3, y0 = 12/5, z = 4}, {t1 = 0, t2 = Pi, t3 = Pi, x = 0, x0 = 9/5, y = 3, y0 = 12/5, z = -4}, {t1 = 0, t2 = Pi, t3 = Pi, x = 0, x0 = 9/5, y = -3, y0 = 12/5, z = 4}, {t1 = 0, t2 = Pi, t3 = Pi, x = 0, x0 = 9/5, y = -3, y0 = 12/5, z = -4}, {t1 = Pi, t2 = Pi, t3 = Pi, x = 0, x0 = 9/5, y = 3, y0 = 12/5, z = 4}, {t1 = Pi, t2 = Pi, t3 = Pi, x = 0, x0 = 9/5, y = 3, y0 = 12/5, z = -4}, {t1 = Pi, t2 = Pi, t3 = Pi, x = 0, x0 = 9/5, y = -3, y0 = 12/5, z = 4}, {t1 = Pi, t2 = Pi, t3 = Pi, x = 0, x0 = 9/5, y = -3, y0 = 12/5, z = -4}, {t1 = t1, t2 = 0, t3 = t3, x = 3, x0 = 0, y = 0, y0 = 0, z = 5}, {t1 = t1, t2 = 0, t3 = t3, x = 3, x0 = 0, y = 0, y0 = 0, z = -5}, {t1 = t1, t2 = 0, t3 = t3, x = -3, x0 = 0, y = 0, y0 = 0, z = 5}, {t1 = t1, t2 = 0, t3 = t3, x = -3, x0 = 0, y = 0, y0 = 0, z = -5}, {t1 = t1, t2 = Pi, t3 = t3, x = 3, x0 = 0, y = 0, y0 = 0, z = 5}, {t1 = t1, t2 = Pi, t3 = t3, x = 3, x0 = 0, y = 0, y0 = 0, z = -5}, {t1 = t1, t2 = Pi, t3 = t3, x = -3, x0 = 0, y = 0, y0 = 0, z = 5}, {t1 = t1, t2 = Pi, t3 = t3, x = -3, x0 = 0, y = 0, y0 = 0, z = -5}, {t1 = 0, t2 = 0, t3 = t3, x = 0, x0 = 9/5, y = 3, y0 = 12/5, z = 4}, {t1 = 0, t2 = 0, t3 = t3, x = 0, x0 = 9/5, y = 3, y0 = 12/5, z = -4}, {t1 = 0, t2 = 0, t3 = t3, x = 0, x0 = 9/5, y = -3, y0 = 12/5, z = 4}, {t1 = 0, t2 = 0, t3 = t3, x = 0, x0 = 9/5, y = -3, y0 = 12/5, z = -4}, {t1 = Pi, t2 = 0, t3 = t3, x = 0, x0 = 9/5, y = 3, y0 = 12/5, z = 4}, {t1 = Pi, t2 = 0, t3 = t3, x = 0, x0 = 9/5, y = 3, y0 = 12/5, z = -4}, {t1 = Pi, t2 = 0, t3 = t3, x = 0, x0 = 9/5, y = -3, y0 = 12/5, z = 4}, {t1 = Pi, t2 = 0, t3 = t3, x = 0, x0 = 9/5, y = -3, y0 = 12/5, z = -4}, {t1 = 0, t2 = t2, t3 = 0, x = 0, x0 = 9/5, y = 3, y0 = 12/5, z = -4}, {t1 = 0, t2 = t2, t3 = 0, x = 0, x0 = 9/5, y = 3, y0 = 12/5, z = 4}, {t1 = 0, t2 = t2, t3 = 0, x = 0, x0 = 9/5, y = -3, y0 = 12/5, z = 4}, {t1 = 0, t2 = t2, t3 = 0, x = 0, x0 = 9/5, y = -3, y0 = 12/5, z = -4}, {t1 = Pi, t2 = t2, t3 = 0, x = 0, x0 = 9/5, y = 3, y0 = 12/5, z = 4}, {t1 = Pi, t2 = t2, t3 = 0, x = 0, x0 = 9/5, y = -3, y0 = 12/5, z = -4}, {t1 = Pi, t2 = t2, t3 = 0, x = 0, x0 = 9/5, y = -3, y0 = 12/5, z = 4}, {t1 = Pi, t2 = t2, t3 = 0, x = 0, x0 = 9/5, y = 3, y0 = 12/5, z = -4}
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