Casio fx-CG 100 Python: Clothoid Curve Analysis
Introduction
The clothoid is a mathematical curve where its curvature is in proportion to the distance traveled from the origin. This property allows the curve to serve many applications including connecting railways, designing roller coasters, and traffic distribution.
Let:
L: arc length of the curve traveled
R: radius from the center of the clothoid to the point on the curve
(x, y): point on the clothoid curve with curve length L and radius R
A: parameter, where A = √(R * L)
Θ: angle between the radius and line of the center point and a point on the x-axis, where Θ = L^2 ÷ (2 * A^2)
The point on the curve is determined by a variation of the Fresnel Integrals:
x = A * √2 * ∫( cos(u^2) du, u = 0 to u = t)
y = A * √2 * ∫( sin(u^2) du, u = 0 to u = t)
The Python program uses infinite series to calculate the point (x, y).
The Clothoid curve is also known as the Cornu spiral or the Euler spiral.
Casio fx-CG 100 Program: clothoid.py
# Clothoid Curve Analysis
# template of infinite series
# Eddie W. Shore, 11/23/2025
from math import *
# factorial function
def fact(n):
f=1
if n<=1:
return 1
else:
for i in range(2,n+1):
f*=i
return f
# main program
print("Clothiod Analysis\nCornu Spiral")
r=eval(input("radius: "))
l=eval(input("arc length: "))
a=sqrt(r*l)
t=l**2/(2*a**2)
# c: cosine, s=sine
c=0
s=0
# set term artificially high
w=100
# set counter at beginning
n=0
# series loop
while abs(w)>=1e-20:
cc=(-1)**n*t**(4*n+1)/(fact(2*n)*(4*n+1))
ss=(-1)**n*t**(4*n+3)/(fact(2*n+1)*(4*n+3))
w=max(cc,ss)
c+=cc
s+=ss
n+=1
# answer
x=a*sqrt(2)*c
y=a*sqrt(2)*s
print("constant: {0:.12f}".format(a))
print("angle: {0:.12f}".format(t))
print("x: {0:.12f}".format(x))
print("y: {0:.12f}".format(y))
Example
Input:
Radius: r = 1.75
Arc Length: l = 4.00
Results:
constant (a): 2.645751311065
angle (degrees): 1.142857142857°
x: 3.602081584381
y: 1.646831998544
Sources
Autodesk, Inc. “About Spiral Definitions” Autodesk Civil 3D Help. https://help.autodesk.com/view/CIV3D/2025/ENU/?guid=GUID-DD7C0EA1-8465-45BA-9A39-FC05106FD822. 2025. Retrieved November 19, 2025.
Constantin. “The Clothoid” A railway track blog. https://railwaytrackblog.com/2016/07/03/the-clothoid/comment-page-1/ March 7, 2016. Retrieved November 19, 2025.
Gombáu, Alberto. “The clothoid: geometry that unites mathematics, engineer and design”. https://medium.com/@gombau/the-clothoid-geometry-that-unites-mathematics-engineering-and-design-6323de37e979. April 11, 2025. Retrieved November 19, 2025.
Eddie
All original content copyright, © 2011-2026. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.
