[Python] Generate trajectory along surface of a sphere with velocity

I am trying to generate trajectory between 2 points A and B along surface of a hemisphere.

Constraints:

• The end point of the trajectory(Point B) should have velocity in a certain direction given by (vx, vy, vz).

I have implemented simple SLERP(spherical linear interpolation) for this but how do I give velocity at point B?

Current Code:

# Simple SLERP

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D


def slerp(A, B, t):
# Interpolates between points A and B on a sphere using Slerp.
A = A / np.linalg.norm(A)
B = B / np.linalg.norm(B)

dot_product = np.clip(np.dot(A, B), -1.0, 1.0) # Clamped to avoid precision errors
omega = np.arccos(dot_product) # Angular distance

if np.isclose(omega, 0):
return A

return (np.sin((1 - t) * omega) * A + np.sin(t * omega) * B) / np.sin(omega)


A_cartesian = np.array([1, 0, 0]) # Point A on unit sphere
B_cartesian = np.array([0, 1, 0]) # Point B on unit sphere


num_points = 10 # Number of trajectory points
trajectory = np.array([slerp(A_cartesian, B_cartesian, t) for t in np.linspace(0, 1, num_points)])

# Visualization
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot(trajectory[:, 0], trajectory[:, 1], trajectory[:, 2], 'bo-', label="Trajectory")
ax.scatter(*A_cartesian, color='red', label='Start (A)')
ax.scatter(*B_cartesian, color='green', label='End (B)')

# Sphere surface
u, v = np.mgrid[0:2 * np.pi:20j, 0:np.pi:10j]
x = np.cos(u) * np.sin(v)
y = np.sin(u) * np.sin(v)
z = np.cos(v)
ax.plot_wireframe(x, y, z, color='gray', alpha=0.3)

ax.set_xlabel("X")
ax.set_ylabel("Y")
ax.set_zlabel("Z")
ax.legend()
plt.show()



Let me know if there is any confusion regarding "What does velocity at point means".

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