How to draw an ellipsoid using tilted or rotated lines using Plotly?

Question

I am drawing an ellipsoid using the following function from empet plotly community form:

def ellipse(x_center=0, y_center=0, ax1 = [1, 0],  ax2 = [0,1], a=1, b =1,  N=100):
    # x_center, y_center the coordinates of ellipse center
    # ax1 ax2 two orthonormal vectors representing the ellipse axis directions
    # a, b the ellipse parameters
    if np.linalg.norm(ax1) != 1 or np.linalg.norm(ax2) != 1:
        raise ValueError('ax1, ax2 must be unit vectors')
    if  abs(np.dot(ax1, ax2)) > 1e-06:
        raise ValueError('ax1, ax2 must be orthogonal vectors')
    t = np.linspace(0, 2*pi, N)
    #ellipse parameterization with respect to a system of axes of directions a1, a2
    xs = a * cos(t)
    ys = b * sin(t)
    #rotation matrix
    R = np.array([ax1, ax2]).T
    # coordinate of the  ellipse points with respect to the system of axes [1, 0], [0,1] with origin (0,0)
    xp, yp = np.dot(R, [xs, ys])
    x = xp + x_center 
    y = yp + y_center
    return x, y

However, I am getting unperfect ellipsoids around the lines. Refer to the image attached for more clarification. How we can update the function to consider the rotational axis of the lines and draw perfect ellipsoids around the lines?

The code to call the function and plot the ellipsoid as follow:

x22 = tw2_df['x']
z22 = tw2_df['z']

x5 = tw5_df['x']
z5 = tw5_df['z']

# Create figure
fig = go.Figure()

# Add scatter traces
fig.add_trace(go.Scatter(x=x22, y=z22, name="line-2", mode="markers", marker_color='gray'))
fig.add_trace(go.Scatter(x=x5, y=z5, name="line-3", mode="markers", marker_color='gray'))

# well-2
x_center=h2
y_center=k2
x22, y22 = ellipse(x_center=x_center, y_center=y_center, 
                   ax1 =[cos(pi/2), sin(pi/2)],  ax2=[-sin(pi/2),cos(pi/2)],
                   a=a2, b =b2)

fig.add_scatter(
            x=x22,
            y=y22,
            mode = 'lines')

# well-5
x_center=h5
y_center=k5
x5, y5 = ellipse(x_center=x_center, y_center=y_center, 
                   ax1 =[cos(pi/2), sin(pi/2)],  ax2=[-sin(pi/2),cos(pi/2)],
                   a=a5, b =b5)

fig.add_scatter(
            x=x5,
            y=y5,
            mode = 'lines')

fig.update_layout(showlegend=True)
fig.show()

Plotly points represent lines and ellipsoids

Data: h2=540.05, k2=-54.75, a2=36.67, b2=577.62 h5=550.28, k5=-246.58, a5=43.17, b5=590.69

This is the link that has the x and y data as CSV files. https://drive.google.com/drive/folders/1JGKdnfqu9aZy8YFtL_2kL-NnJ-fJHcMM?usp=sharing

Any help on this will be highly appreciated!

Answer 1

One of the solutions I found is to include the rotational angel when we creating the ellipsoids. The following is an updated version of the angle:

def ellipse(x_center=0, y_center=0, ax1 = [1, 0],  ax2 = [0,1], a=1, b =1,  N=100):
   # x_center, y_center the coordinates of ellipse center
   # ax1 ax2 two orthonormal vectors representing the ellipse axis directions
   # a, b the ellipse parameters
   if np.linalg.norm(ax1) != 1 or np.linalg.norm(ax2) != 1:
       raise ValueError('ax1, ax2 must be unit vectors')
   if  abs(np.dot(ax1, ax2)) > 1e-06:
       raise ValueError('ax1, ax2 must be orthogonal vectors')
   #rotation matrix   
   R = np.array([ax1, ax2]).T
   if np.linalg.det(R) <0: 
       raise ValueError("the det(R) must be positive to get a  positively oriented ellipse reference frame")
   t = np.linspace(0, 2*pi, N)
   #ellipse parameterization with respect to a system of axes of directions a1, a2
   xs = a * cos(t)
   ys = b * sin(t)
   
   # coordinate of the  ellipse points with respect to the system of axes [1, 0], [0,1] with origin (0,0)
   xp, yp = np.dot(R, [xs, ys])
   x = xp + x_center 
   y = yp + y_center
   return x, y

Moreover, when we call the function, and we define the rotational angle as follow: Example-1: Notice that we define pi/1.93, which will result in the rotational angel in my model. This will be user define and verified visually by the user.

# well-4 
x_center = h4
y_center = k4
angel_4 = pi/1.93
x, y = ellipse(x_center=x_center, y_center=y_center, ax1 =[cos(angel_4), sin(angel_4)],  ax2=[-sin(angel_4),cos(angel_4)], a=a4, b =b4)
fig.add_scatter(x=x, y=y, mode = 'lines', name='Zone-1 - well-4', fill='toself', opacity=0.5)

Example-2: Notice that we define pi/1.935 to rotate the ellipse.

# well-5
x_center = h5
y_center = k5
angel_5 = pi/1.935
x, y = ellipse(x_center=x_center, y_center=y_center, ax1 =[cos(angel_5), sin(angel_5)],  ax2=[-sin(angel_5),cos(angel_5)], a=a5-20, b =b5-140)
fig.add_scatter(x=x, y=y, mode = 'lines', name='Zone-1 - well-5', fill='toself', opacity=0.5)

Example-3 Notice that we define pi/1.921 to rotate the ellipse.

# well-6
x_center=h6
y_center=k6
angel_6 = pi/1.921
x, y = ellipse(x_center=x_center, y_center=y_center, ax1 =[cos(angel_6), sin(angel_6)],  ax2=[-sin(angel_6),cos(angel_6)], a=a6-25, b =b6-140)
fig.add_scatter(x=x, y=y, mode = 'lines', name='Zone-1 - well-6', fill='toself', opacity=0.8)

Please see the attached figure as an example.

enter image description here