How to obtain the extreme edges from an identified contour?

I need to know how can I capture the extreme edges of a line(starting and ending coordinates) after identifying the contours. Currently, I am identifying the contours for the shapes(different types of lines) in the following image and drawing them back to a new image. I have already tried to obtain the topmost, the bottom-most, the leftmost, and the right-most coordinates from the contours array but they will not accurate to a line that has curves like below. So is there any way to capture those starting and ending points from the contours array?

Source Code

import cv2
import numpy as np

# Let's load a simple image with 3 black squares
image = cv2.imread("C:/Users/Hasindu/3D Objects/edge-test-188.jpg")
cv2.waitKey(0)

# Grayscale
gray = cv2.cvtColor(image, cv2.COLOR_BGR2GRAY)

# Find Canny edges
edged = cv2.Canny(gray, 30, 200)
cv2.waitKey(0)

# Finding Contours
# Use a copy of the image e.g. edged.copy()
# since findContours alters the image
contours, hierarchy = cv2.findContours(edged,
    cv2.RETR_EXTERNAL, cv2.CHAIN_APPROX_NONE)

cv2.imshow('Canny Edges After Contouring', edged)
cv2.waitKey(0)

print("Number of Contours found = " + str(len(contours)))
print(contours)
topmost = tuple(contours[0][contours[0][:,:,1].argmin()][0]);
bottommost = tuple(contours[0][contours[0][:,:,1].argmax()][0]);

print(topmost);
print(bottommost);



# Draw all contours
# -1 signifies drawing all contours
cv2.drawContours(image, contours, -1, (0, 255, 0), 3)


cv2.imshow('Contours', image)
cv2.waitKey(0)
cv2.destroyAllWindows()

Input 1

Output 1

EDIT:

I have followed the solution suggested by stateMachine but it was not 100 percent accurate on my all the inputs.You can see clearly some of the endpoints on the 2nd input image are not detected by the solution.

Input 2

Output 2

from How to obtain the extreme edges from an identified contour?