diff --git a/.gitignore b/.gitignore
index df99b7a..ad43a04 100644
--- a/.gitignore
+++ b/.gitignore
@@ -5,4 +5,5 @@
 *.log
 *.gif
 __pycache__/
-.idea/
\ No newline at end of file
+.idea/
+/setup.cfg
\ No newline at end of file
diff --git a/A/4/simulator.py b/A/4/simulator.py
index 508a89d..12ff5d0 100644
--- a/A/4/simulator.py
+++ b/A/4/simulator.py
@@ -1,40 +1,133 @@
 from loong import *
 import json
-class BestOrbit(Orbit):
+import numpy as np
+import sys
+import matplotlib.pyplot as plt
+
+
+class GoodOrbit(Orbit):
+
   def __init__(self):
-    self.kAlpha = mp.mpf('1.7') / (2 * mp.pi)
+    self.kAlpha = mp.mpf("1.7") / (2 * mp.pi)
+    self.kCriticalTheta = 2.86 / ((2 / 3) * self.kAlpha)
+    self.r = (1 / 3) * self.kAlpha * mp.sqrt(1 + self.kCriticalTheta**2)
+    self.point_A_cartesian = (
+        self.kAlpha * self.kCriticalTheta * mp.cos(self.kCriticalTheta),
+        self.kAlpha * self.kCriticalTheta * mp.sin(self.kCriticalTheta),
+    )
+    self.point_B_cartesian = (-self.kAlpha * self.kCriticalTheta * mp.cos(self.kCriticalTheta),
+                              -self.kAlpha * self.kCriticalTheta * mp.sin(self.kCriticalTheta))
+    self.kPhi = mp.atan(self.kCriticalTheta)
+    dx, dy = self.point_A_cartesian[0] - self.point_B_cartesian[0], self.point_A_cartesian[1] - self.point_B_cartesian[1]
+    self.angle = mp.atan2(dy, dx)
+    self.point_C1_cartesian = (self.point_A_cartesian[0] - 2 * self.r * dx, self.point_A_cartesian[1] - 2 * self.r * dy)
+    self.point_C2_cartesian = (self.point_B_cartesian[0] + 1 * self.r * dx, self.point_B_cartesian[1] + 1 * self.r * dy)
+    self.radius_of_C1 = 2 * self.r
+    self.radius_of_C2 = 1 * self.r
+    self.arclength = 6 * self.r * self.kPhi
+    self.edge_k = self.kAlpha * mp.sqrt(1 + self.kCriticalTheta * self.kCriticalTheta)
+    self.n = -1
+    for i in range(3, 20, 2):
+      self.a = (self.arclength - 2 * self.edge_k * self.kCriticalTheta) / (2 * (1 - i) * self.kCriticalTheta**i)
+      self.b = (i * self.arclength - 2 * self.edge_k * self.kCriticalTheta) / (2 * (i - 1) * self.kCriticalTheta)
+      if self.a > 0 and self.b > 0:
+        self.n = i
+        break
+    print(f"arclength={self.arclength}", file=sys.stderr)
+    print(f"edge_k={self.edge_k}", file=sys.stderr)
+    print(f"a={self.a}", file=sys.stderr)
+    print(f"b={self.b}", file=sys.stderr)
+    print(f"n={self.n}", file=sys.stderr)
+    print(f"now k={self.n*self.a*self.kCriticalTheta**(self.n-1)+self.b}", file=sys.stderr)
+    if self.n == -1:
+      raise Exception("n must be set")
+    self.edge_raw_C = self.kAlpha * 0.5 * (
+        self.kCriticalTheta * mp.sqrt(1 + self.kCriticalTheta * self.kCriticalTheta) -
+        mp.log(-self.kCriticalTheta + mp.sqrt(1 + self.kCriticalTheta * self.kCriticalTheta)))
+
   def InitIdx(self):
-    return mp.mpf('0.0')
+    return mp.mpf("0.0")
+
   def InitC(self):
-    return mp.mpf('0.0')
-  def Idx2C(self, idx):
-    return idx / self.kAlpha
+    return mp.mpf("0.0")
+
+  def Idx2C(self, idx):  # this function must be monotonically increasing
+    if idx >= 0:
+      theta = idx + self.kCriticalTheta
+      tmp = mp.sqrt(1 + theta * theta)
+      return self.kAlpha * 0.5 * (theta * tmp - mp.log(-theta + tmp)) - self.edge_raw_C
+    elif idx >= -2 * self.kCriticalTheta:
+      x = idx + self.kCriticalTheta
+      y = (self.a * (x**self.n) + self.b * x) - 0.5 * self.arclength
+      return y
+    else:
+      theta = -idx - self.kCriticalTheta
+      tmp = mp.sqrt(1 + theta * theta)
+      return -self.kAlpha * 0.5 * (theta * tmp - mp.log(-theta + tmp)) + self.edge_raw_C - self.arclength
+
   def Idx2Cartesian(self, idx):
     return mp.matrix([mp.cos(idx), mp.sin(idx)])
+
   def C2Idx(self, C):
-    return C * self.kAlpha
+
+    def f(idx):
+      return self.Idx2C(idx) - C
+
+    return mp.findroot(f, 0, solver='secant')
+
   def GenerateNextPointIdx(self, cur_point_idx, expected_distance):
     return cur_point_idx + expected_distance
 
+
+orbit = GoodOrbit()
+
+
+def f(x):
+  return float(orbit.Idx2C(x))
+
+
+# 获取 kCriticalTheta
+kCriticalTheta = float(orbit.kCriticalTheta)
+print(f"kCriticalTheta={kCriticalTheta}")
+
+# 定义范围 [-kCriticalTheta-1, kCriticalTheta +1]
+start = -2.5 * kCriticalTheta - 1
+end = 0.5 * kCriticalTheta + 1
+
+# 生成范围内的点
+x_vals = np.linspace(start, end, 1000)
+
+# 计算 f(x) 的值
+y_vals = [f(mp.mpf(x)) for x in x_vals]
+
+# 绘制图像
+plt.figure(figsize=(10, 6))
+plt.plot(x_vals, y_vals, label='Idx2C(x)')
+plt.title('Idx2C function plot')
+plt.xlabel('Index (x)')
+plt.ylabel('C Value')
+plt.grid(True)
+plt.axhline(0, color='black', linewidth=0.5)
+plt.axvline(0, color='black', linewidth=0.5)
+plt.legend()
+plt.show()
+sys.exit()
 if __name__ == "__main__":
-  orbit=BestOrbit()
-  loong=Loong(orbit, 224, mp.mpf('2.0'), mp.mpf('1e-8'))
-  res_list=[]
-  for ti in range(-100,101):
+  orbit = GoodOrbit()
+  loong = Loong(orbit, 224, mp.mpf("2.0"), mp.mpf("1e-8"))
+  res_list = []
+  for ti in range(-100, 101):
     print(f"calculating time_point={ti}")
     res_list.append(loong.CalcStatusListByTime(mp.mpf(ti)))
     # 转换成内置浮点数并保留6位
-  float_res_list = [
-    [
-      {
-        "idx": round(float(node["idx"]),6),
-        "node": [round(float(node["node"][0]),6), round(float(node["node"][1]),6)],
-        "C": round(float(node["C"]),6),
-        "v": round(float(node["v"]),6)
-      }
-      for node in res
-    ]
-    for res in res_list
-  ]
+  float_res_list = [[{
+      "idx": round(float(node["idx"]), 6),
+      "node": [
+          round(float(node["node"][0]), 6),
+          round(float(node["node"][1]), 6),
+      ],
+      "C": round(float(node["C"]), 6),
+      "v": round(float(node["v"]), 6),
+  } for node in res] for res in res_list]
   with open("A4_res.json", "w") as file:
-    json.dump(float_res_list, file, indent=4)
\ No newline at end of file
+    json.dump(float_res_list, file, indent=4)
diff --git a/A/4/testplot.py b/A/4/testplot.py
index cd297d4..7d10286 100644
--- a/A/4/testplot.py
+++ b/A/4/testplot.py
@@ -5,10 +5,10 @@ kPitch = 1.7
 kAlpha = kPitch / (2 * np.pi)
 kCriticalRadius = 4.5
 
-theta_max = (kCriticalRadius) / kAlpha + 2*2*np.pi
-kPlotingRadius = theta_max  * kAlpha
+theta_max = (kCriticalRadius) / kAlpha + 2 * 2 * np.pi
+kPlotingRadius = theta_max * kAlpha
 
-kCriticalTheta =  2.86 / ((2/3)*kAlpha)
+kCriticalTheta = 2.86 / ((2 / 3) * kAlpha)
 # 生成角度数组
 theta = np.linspace(kCriticalTheta, theta_max, 1000)
 
@@ -34,36 +34,40 @@ circle_theta = np.linspace(0, 2 * np.pi, 1000)
 circle_r = np.full_like(circle_theta, kCriticalRadius)
 ax.plot(circle_theta, circle_r, linestyle='--')
 
-point_A_cartesian = (kAlpha*kCriticalTheta*np.cos(kCriticalTheta),kAlpha*kCriticalTheta*np.sin(kCriticalTheta))
-point_B_cartesian = (-kAlpha*kCriticalTheta*np.cos(kCriticalTheta),-kAlpha*kCriticalTheta*np.sin(kCriticalTheta))
+point_A_cartesian = (kAlpha * kCriticalTheta * np.cos(kCriticalTheta), kAlpha * kCriticalTheta * np.sin(kCriticalTheta))
+point_B_cartesian = (-kAlpha * kCriticalTheta * np.cos(kCriticalTheta),
+                     -kAlpha * kCriticalTheta * np.sin(kCriticalTheta))
 kPhi = np.arctan(kCriticalTheta)
-r = (1/3) * kAlpha * np.sqrt(1 + kCriticalTheta**2)
+r = (1 / 3) * kAlpha * np.sqrt(1 + kCriticalTheta**2)
 dx, dy = point_A_cartesian[0] - point_B_cartesian[0], point_A_cartesian[1] - point_B_cartesian[1]
 angle = np.arctan2(dy, dx)
-dx, dy = np.cos(angle - (0.5*np.pi-kPhi)), np.sin(angle - (0.5*np.pi-kPhi))
-point_C1_cartesian = (point_A_cartesian[0] - 2*r*dx, point_A_cartesian[1] - 2*r*dy)
-point_C2_cartesian = (point_B_cartesian[0] + 1*r*dx, point_B_cartesian[1] + 1*r*dy)
-radius_of_C1 = 2*r
-radius_of_C2 = 1*r
+dx, dy = np.cos(angle - (0.5 * np.pi - kPhi)), np.sin(angle - (0.5 * np.pi - kPhi))
+point_C1_cartesian = (point_A_cartesian[0] - 2 * r * dx, point_A_cartesian[1] - 2 * r * dy)
+point_C2_cartesian = (point_B_cartesian[0] + 1 * r * dx, point_B_cartesian[1] + 1 * r * dy)
+radius_of_C1 = 2 * r
+radius_of_C2 = 1 * r
+
 
 # 定义用于绘制圆的函数
 def draw_circle(ax, center, radius, num_points, beg_angle, span_angle):
-    t = np.linspace(beg_angle, beg_angle+span_angle, num_points)
-    x = center[0] + radius * np.cos(t)
-    y = center[1] + radius * np.sin(t)
-    r, theta = np.sqrt(x**2 + y**2), np.arctan2(y, x)
-    ax.plot(theta, r)
+  t = np.linspace(beg_angle, beg_angle + span_angle, num_points)
+  x = center[0] + radius * np.cos(t)
+  y = center[1] + radius * np.sin(t)
+  r, theta = np.sqrt(x**2 + y**2), np.arctan2(y, x)
+  ax.plot(theta, r)
+
 
 # 绘制圆C1
-draw_circle(ax, point_C1_cartesian, radius_of_C1, 100, angle+0.5*np.pi-kPhi-np.pi, 2*kPhi)
+draw_circle(ax, point_C1_cartesian, radius_of_C1, 100, angle + 0.5 * np.pi - kPhi - np.pi, 2 * kPhi)
 
 # 绘制圆C2
-draw_circle(ax, point_C2_cartesian, radius_of_C2, 100, angle+0.5*np.pi-kPhi, 2*kPhi)
+draw_circle(ax, point_C2_cartesian, radius_of_C2, 100, angle + 0.5 * np.pi - kPhi, 2 * kPhi)
 
 print(f"Total length={6*r*kPhi}")
+print(f"kCriticalTheta={kCriticalTheta}")
 
-x_ticks = np.arange(-int(kPlotingRadius)-1, int(kPlotingRadius)+1, 1)
-y_ticks = np.arange(-int(kPlotingRadius)-1, int(kPlotingRadius)+1, 1)
+x_ticks = np.arange(-int(kPlotingRadius) - 1, int(kPlotingRadius) + 1, 1)
+y_ticks = np.arange(-int(kPlotingRadius) - 1, int(kPlotingRadius) + 1, 1)
 X, Y = np.meshgrid(x_ticks, y_ticks)
 X = X.flatten()
 Y = Y.flatten()
@@ -79,4 +83,4 @@ ax.scatter(theta_grid[valid_points], r_grid[valid_points], color='grey', s=10)
 plt.title("The Moving Path")
 
 # 显示图像
-plt.show()
\ No newline at end of file
+plt.show()