In this code I create the cone using given parameters. However the cone base center shifts from its expected position due to inclination. Correct the cone axis length to ensure cone base center and cone apex are fixed, but ensure the cone base itself is still correctly projected. Your changes should affect cone base center and its size, but not its shape.

# Define cone parameters
cone_top = SkyCoord('7h28m43.5s 69d11m24s', frame='icrs')  # Cone top (apex) as SkyCoord
cone_base_center = SkyCoord('7h28m38s 69d12m28s', frame='icrs')  # Cone base center as SkyCoord
cone_opening_angle = 20 * u.deg  # Opening angle of the cone
cone_orientation = 10 * u.deg  # In-plane orientation angle
cone_inclination = 50 * u.deg  # Tilt relative to the plane of the sky

# Load WCS for the image (assumed to be already defined)
wcs = WCS(HD(h_ha))

def radec_to_cartesian(ra_deg, dec_deg):
    """Convert RA and Dec in degrees to Cartesian coordinates."""
    ra_rad = np.deg2rad(ra_deg)
    dec_rad = np.deg2rad(dec_deg)
    x = np.cos(dec_rad) * np.cos(ra_rad)
    y = np.cos(dec_rad) * np.sin(ra_rad)
    z = np.sin(dec_rad)
    return np.array([x, y, z])

def cartesian_to_radec(x, y, z):
    """Convert Cartesian coordinates to RA and Dec in degrees."""
    hyp = np.hypot(x, y)
    ra_rad = np.arctan2(y, x)
    dec_rad = np.arctan2(z, hyp)
    ra_deg = np.rad2deg(ra_rad) % 360.0
    dec_deg = np.rad2deg(dec_rad)
    return ra_deg, dec_deg

def cone_base_vertices(cone_top, cone_base_center, cone_opening_angle, cone_inclination, num_points=100):
    """
    Generate vertices for the cone's base, projected onto the sky,
    taking into account the cone_inclination.
    """
    # Convert apex and base center to Cartesian coordinates
    apex_vector = radec_to_cartesian(cone_top.ra.deg, cone_top.dec.deg)
    base_center_vector = radec_to_cartesian(cone_base_center.ra.deg, cone_base_center.dec.deg)
    
    # Compute the axis vector from apex to base center
    axis_vector = base_center_vector - apex_vector
    distance = np.linalg.norm(axis_vector)
    axis_vector = axis_vector / distance
    
    # Compute current inclination angle between apex_vector and axis_vector
    current_inclination = np.arccos(np.dot(apex_vector, axis_vector))
    
    # Convert cone_inclination to radians
    cone_inclination_rad = cone_inclination.to(u.radian).value
    
    # Compute the angle difference
    delta_inclination = cone_inclination_rad - current_inclination
    
    # If there is a difference, rotate axis_vector around rotation_axis by delta_inclination
    if not np.isclose(delta_inclination, 0):
        # Compute rotation axis (perpendicular to both apex_vector and axis_vector)
        rotation_axis = np.cross(apex_vector, axis_vector)
        if np.linalg.norm(rotation_axis) == 0:
            # apex_vector and axis_vector are parallel, choose an arbitrary perpendicular axis
            rotation_axis = np.cross(apex_vector, np.array([1, 0, 0]))
        rotation_axis /= np.linalg.norm(rotation_axis)
        
        # Rotate axis_vector by delta_inclination using Rodrigues' rotation formula
        axis_vector_rotated = (axis_vector * np.cos(delta_inclination) +
                               np.sin(delta_inclination) * np.cross(rotation_axis, axis_vector) +
                               (1 - np.cos(delta_inclination)) * np.dot(rotation_axis, axis_vector) * rotation_axis)
        axis_vector = axis_vector_rotated / np.linalg.norm(axis_vector_rotated)
    
    # Now axis_vector has the correct inclination
    # Recompute base_center_vector using adjusted axis_vector
    base_center_vector = apex_vector + axis_vector * distance
    base_center_vector /= np.linalg.norm(base_center_vector)
    
    # Compute base radius
    cone_opening_angle_rad = cone_opening_angle.to(u.radian).value
    base_radius = np.tan(cone_opening_angle_rad) * distance
    
    # At base center, compute orthonormal vectors perpendicular to axis_vector
    # Find a vector not parallel to axis_vector
    if np.allclose(axis_vector, [0, 0, 1]) or np.allclose(axis_vector, [0, 0, -1]):
        v1 = np.array([1, 0, 0])
    else:
        v1 = np.array([0, 0, 1])
    # Compute perpendicular vectors
    v_east = np.cross(axis_vector, v1)
    v_east /= np.linalg.norm(v_east)
    v_north = np.cross(axis_vector, v_east)
    v_north /= np.linalg.norm(v_north)
    
    # Generate circle points around the base
    theta = np.linspace(0, 2 * np.pi, num_points)
    circle_points = base_center_vector[:, np.newaxis] + base_radius * (
        v_east[:, np.newaxis] * np.cos(theta) + v_north[:, np.newaxis] * np.sin(theta)
    )
    circle_points /= np.linalg.norm(circle_points, axis=0)  # Normalize to unit sphere
    
    # Convert circle points to RA and Dec
    ra_base_vertices, dec_base_vertices = cartesian_to_radec(
        circle_points[0], circle_points[1], circle_points[2]
    )
    
    # Return base vertices as SkyCoord
    base_vertices = SkyCoord(ra=ra_base_vertices * u.deg, dec=dec_base_vertices * u.deg, frame='icrs')
    return base_vertices

# Generate the base vertices
cone_base = cone_base_vertices(cone_top, cone_base_center, cone_opening_angle, cone_inclination)