diff --git a/ktch/outline/_elliptic_Fourier_analysis.py b/ktch/outline/_elliptic_Fourier_analysis.py
index e4bd099..143b356 100644
--- a/ktch/outline/_elliptic_Fourier_analysis.py
+++ b/ktch/outline/_elliptic_Fourier_analysis.py
@@ -330,7 +330,7 @@ def _transform_single(
 
         return X_transformed
 
-    def _normalize(self, an, bn, cn, dn):
+    def _normalize(self, an, bn, cn, dn, keep_start_point=False):
         """Normalize Fourier coefficients.
 
         Todo:
@@ -343,18 +343,29 @@ def _normalize(self, an, bn, cn, dn):
         c1 = cn[1]
         d1 = dn[1]
 
-        theta = (1 / 2) * np.arctan2(
-            2 * (a1 * b1 + c1 * d1), (a1**2 + c1**2 - b1**2 - d1**2)
+        theta = (1 / 2) * np.arctan(
+            2 * (a1 * b1 + c1 * d1) / (a1**2 + c1**2 - b1**2 - d1**2)
         )
-        if theta < 0:
-            theta = theta + 2 * np.pi
+
+        [[a_s, b_s], [c_s, d_s]] = np.array([[a1, b1], [c1, d1]]).dot(
+            rotation_matrix_2d(theta)
+        )
+        s1 = a_s**2 + c_s**2
+        s2 = b_s**2 + d_s**2
+
+        if s1 < s2:
+            if theta < 0:
+                theta = theta + np.pi / 2
+            else:
+                theta = theta - np.pi / 2
 
         a_s = a1 * np.cos(theta) + b1 * np.sin(theta)
         c_s = c1 * np.cos(theta) + d1 * np.sin(theta)
         scale = np.sqrt(a_s**2 + c_s**2)
         psi = np.arctan2(c_s, a_s)
-        if psi < 0:
-            psi = psi + 2 * np.pi
+
+        if keep_start_point:
+            theta = 0
 
         coef_norm_list = []
         r_psi = rotation_matrix_2d(-psi)
diff --git a/pyproject.toml b/pyproject.toml
index 8f4a951..0249461 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -25,7 +25,7 @@ version = "0.3.1"
 [tool.poetry.dependencies]
 numpy = ">=1.22"
 pandas = ">=1.5"
-python = "^3.8"
+python = "^3.9"
 scikit-learn = ">=1.2"
 scipy = ">=1.8"
 toml = ">=0.10"
@@ -48,20 +48,20 @@ jupyter = "^1.0.0"
 jupyterlab = "^3.4.4"
 jupytext = "^1.14.5"
 liccheck = "^0.7.2"
-llvmlite = "^0.39.1"
-matplotlib = "^3.5.2"
+llvmlite = "^0.41"
 module-name = "^0.6.0"
-numba = "^0.56.4"
+numba = "^0.58"
 opencv-python = "^4.7.0.68"
 plotly = "^5.9.0"
 poetry2conda = "^0.3.0"
 pytest = "^7.1"
 pytest-cov = "^3.0"
-seaborn = "^0.11.0"
 setuptools = "<60.0"
 tqdm = "^4.64.1"
 tslearn = "^0.5.3.2"
 xarray = "^2022.6.0"
+matplotlib = "^3.8.2"
+seaborn = "^0.13.0"
 
 [tool.poetry.group.doc.dependencies]
 myst-nb = "^0.17.2"