Curve Intersection or Overlapping detection algorithm

Question

I am attempting to build a Python GUI that creates a canvas where I can draw using a mouse or touch pen. The canvas will mainly consist of a cursive font, such as "Happy Birthday," for my DIY chocolate 3D printer.

Thus far, I have successfully created a canvas where I can draw, which yields coordinates. However, I am having a hard time figuring out an algorithm to detect an intersection. After encountering an intersection, the curve overlaps the existing drawn curve by changing print height so that I can have continuous extrusion. Later, I will add a G-code converter to move my printer.

Although I have tried all existing slicers that I'm aware of, the consistency is not what a continuous printing would yield.

Below, I have included the code for the canvas creation and display of coordinates:

import tkinter as tk

class TrajectoryApp:
   def __init__(self, master):
        self.master = master
        self.canvas_width = 320
        self.canvas_height = 280
        self.canvas = tk.Canvas(self.master, width=self.canvas_width, height=self.canvas_height, bg='white')
        self.canvas.pack(side=tk.LEFT)

        self.canvas.bind('<Button-1>', self.start_trajectory)
        self.canvas.bind('<B1-Motion>', self.draw_trajectory)
        self.canvas.bind('<ButtonRelease-1>', self.end_trajectory)

        self.coordinates = []

        self.text_area = tk.Text(self.master, height=self.canvas_height // 10, width=30)
        self.text_area.pack(side=tk.RIGHT)

   def start_trajectory(self, event):
        self.coordinates.append([])
        self.coordinates[-1].append((event.x, event.y))
        
   def draw_trajectory(self, event):
        self.canvas.create_line(self.coordinates[-1][-1][0], self.coordinates[-1][-1][1], event.x, event.y)
        self.coordinates[-1].append((event.x, event.y))
        
   def end_trajectory(self, event):
        self.display_coordinates()

   def display_coordinates(self):
        self.text_area.delete(1.0, tk.END)
        for trajectory in self.coordinates:
            self.text_area.insert(tk.END, "Trajectory:\n")
            for point in trajectory:
                x = point[0] * (self.canvas_width / self.canvas.winfo_width())
                y = point[1] * (self.canvas_height / self.canvas.winfo_height())
                self.text_area.insert(tk.END, f"({x:.2f}, {y:.2f})\n")
            self.text_area.insert(tk.END, "\n")

if __name__ == '__main__':
    root = tk.Tk()
    app = TrajectoryApp(root)
    root.mainloop()