How to ensure that a randomly-generated velocity vector moves the camera forward within a valid arc?

Question

I spawn a camera in a scene using random coordinates for x and y and setting z=0 and orient the camera so that it looks on the point (0, 0, 0). My goal is to move the camera forward using randomly-generated velocity vectors on the ground (so z stays 0). I want to ensure that the new position of the camera is within a valid range after moving it forward, defined in degrees with respect to the current focus point/direction. More specifically, the way I determine "valid range" is by ensuring that the new position is within 45 degrees of the old camera's focus point (-45 degrees to the left and +45 degrees to the right). Can someone write a pseudocode on how I can achieve this?

Here's my attempt to do this but this doesn't seem to be the correct way to help me achieve what I want:

camera_dir = (0, 0, 0) - current_cam_pos
while True:
    vel_vec=[uniform(-max_vel, max_vel), uniform(-max_vel, max_vel)] # generate a random velocity vector
    new_pos = camera_dir + vel_vec # compute a new position (and camera direction vector) for the camera
    if (compute_angle(new_pos, camera_dir) < 45 or compute_angle(new_pos, camera_dir) > 315):
        break 

Answer 1

Something like this should work:

forward_direction = normalize(-current_cam_pos)
right_direction = (forward_direction.y, -forward_direction.x, 0)
angle = uniform(-pi/4, pi/4)
new_direction = (cos(angle) * forward) + (sin(angle) * right)
new_pos = new_direction * how_far_you_want_to_move

Some notes:

• The normalize will blow up due to a division by 0 if your camera position is ever (0,0,0), so you may want to check for that case and use a default direction value instead.
• This also gets more complicated if your camera moves in Z: the (y, -x) trick for picking right_direction only works in a 2D plane.
• This doesn’t take care of aiming the camera back at the origin, since your sample code doesn’t mention how that works in your system, so after one move you’ll need to re-do that for the forward_direction (or camera_dir in your example) calculation to be correct.