[Python] Trying to create a great arc using 3 points

I am trying to create a program that creates a top-down polar plot of the paths of satellites using 3 points: start azimuth and elevation, maximum azimuth and elevation, and end azimuth and elevation as it passes over the horizon.



Similar to this but from a top-down view.

Currently, I have been using spherical linear interpolation (SLERP) to create two great arcs from start -> max then max -> end, then connecting them together to create a single arc. However the issue that I'm having is that the arc doesn't look like a normal arc, there is a bump as it approaches the maximum point.



The real graph should be smooth, I have a feeling that using the two SLERP lines is causing this bump. Is there a better equation to create a single, smooth great arc using 3 points that I am just missing?

import numpy as np
import matplotlib.pyplot as plt

def spherical_to_cartesian(az_deg, el_deg, radius=1.0):
"""Convert spherical (azimuth, elevation, radius) to Cartesian coordinates."""
az = np.radians(az_deg)
el = np.radians(el_deg)

x = radius * np.cos(el) * np.sin(az)
y = radius * np.cos(el) * np.cos(az)
z = radius * np.sin(el)
return np.array([x, y, z])

def slerp(v0, v1, t):
"""Perform spherical linear interpolation between two vectors."""
v0 = v0 / np.linalg.norm(v0)
v1 = v1 / np.linalg.norm(v1)
dot = np.clip(np.dot(v0, v1), -1.0, 1.0)
omega = np.arccos(dot)

if np.isclose(omega, 0):
return v0
return (np.sin((1 - t) * omega) * v0 + np.sin(t * omega) * v1) / np.sin(omega)

def interpolate_hemisphere_arc(start_az, start_el, mid_az, mid_el, end_az, end_el, samples=50, radius=1.0):
"""Generate an interpolated arc on the upper hemisphere between three az/el points."""
start = spherical_to_cartesian(start_az, start_el, radius)
mid = spherical_to_cartesian(mid_az, mid_el, radius)
end = spherical_to_cartesian(end_az, end_el, radius)

path_xy = []

# Interpolate start → mid
for t in np.linspace(0, 1, samples // 2 + 1):
p = slerp(start, mid, t)
path_xy.append((p[0], p[1]))

# Interpolate mid → end (skip the duplicate mid point)
for t in np.linspace(0, 1, samples // 2 + 1)[1:]:
p = slerp(mid, end, t)
path_xy.append((p[0], p[1]))

return path_xy

if __name__ == "__main__":
# Test case: plot a single interpolated satellite arc
path = interpolate_hemisphere_arc(
start_az=169.67,
start_el=0,
mid_az=256.75,
mid_el=75.92,
end_az=345.34,
end_el=0
)

xs, ys = zip(*path)
xs = np.array(xs)
ys = np.array(ys)

r = 90 - 90 * np.sqrt(xs**2 + ys**2)
theta = np.arctan2(xs, ys)

plt.figure(figsize=(6, 6))
ax = plt.subplot(111, polar=True)
ax.plot(theta, r)

ax.set_theta_zero_location('N')
ax.set_theta_direction(-1)
ax.set_title("Top-Down Polar View of Interpolated Satellite Pass")
ax.set_rlim(90.0, 0.0)

plt.tight_layout()
plt.show()

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