move along given path relative to the inital position

Question

how can I move an object along a path "relative" to its inital position? I want it so if the object is initially positioned "around" the path instead of in the center, it will move along the path keeping that "displacement", as the below images

The blue line is the path and the red line is what the object, in this case the zombie, would follow if it had been initially positioned in the left and right sides of the "blue path", respectively:

Hope you understood. So far I've this

func move_zombie(delta):
    if target == null:
        return

    var target_pos = target.get_position()
    var pos = get_position()
    var dir = (target_pos - pos).normalized()
    var zombie_speed = 200.0
    position += delta * zombie_speed * dir

Answer 1

You'll track two separate position variables here: your current on-path position, and your current actual position, shifted from the path.

Store your offset from the start of the path when you first begin traversing:

waypoint_index = 0;
path_position = path.get_waypoint(waypoint_index);
offset = actual_position - path_position;

When you update your path position, something like this...

var next_waypoint = path.get_waypoint(waypoint_index);
var difference = next_waypoint - path
var step_size = speed * delta_time

if(dot(difference, difference) <= step_size * step_size) {
    // Snap to the waypoint instead of overshooting.
    // (You can also track how much "loose change" movement you have left
    //  and spend it advancing toward the next waypoint after this one)
    path_position = next_waypoint
    waypoint_index++
} else {
    path_position += difference.normalized * step_size
}

Then you also update your actual position by adding your offset back in:

actual_position = path_position + offset