cubic bezier curve arc length approximation using gaussian quadrature in python

cubic_bezier_length.py

import math
"""
based on 
https://pomax.github.io/bezierinfo/#arclength
we can determine the approximate length of a Bézier curve by computing the Legendre-Gauss sum.
"""
# Gaussian quadrature coefficients
T = [-0.9061798459, -0.5384693101, 0, 0.5384693101, 0.9061798459]
C = [0.2369268850, 0.4786286705, 0.5688888889, 0.4786286705, 0.2369268850]

def derivative(t, p0, p1, p2, p3):
    mt = 1 - t
    dx = 3 * mt * mt * (p1[0] - p0[0]) + 6 * mt * t * (p2[0] - p1[0]) + 3 * t * t * (p3[0] - p2[0])
    dy = 3 * mt * mt * (p1[1] - p0[1]) + 6 * mt * t * (p2[1] - p1[1]) + 3 * t * t * (p3[1] - p2[1])
    return dx, dy

def arc_length_element(t, derivative):
    return math.sqrt(derivative[0] * derivative[0] + derivative[1] * derivative[1])

def approximateCubicArcLength(p0, p1, p2, p3):
    z = 0.5
    total_length = 0
    
    for t_val, c_val in zip(T, C):
        t = z * t_val + z
        deriv = derivative(t, p0, p1, p2, p3)
        total_length += c_val * arc_length_element(t, deriv)
            
    return z * total_length


p1 = (0, 0)
p2 = (50, 50)
p3 = (90, 90)
p4 = (230, 230)
l = approximateCubicArcLength(p1, p2, p3, p4)
print(l)