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We remind here the expression of the curvature for a given set of parametric equations $(x(t), y(t))$ : $$ c(t) = \frac{x'(t) y''(t) - y'(t) x''(t) }{\left(x'(t)^2 + y'(t)^2 \right)^{3/2}}$$

Let : $$ x(t) = t R_0 \cos\frac{\pi}{4}$$ $$ y(t) = 2 R_0 \mathrm{sech}\left( \frac{x(t)}{R_0} \right) = \frac{2 R_0}{\cosh\left( \frac{x(t)}{R_0} \right)}$$



In [11]:

    
from sympy import *

For a nice $\LaTeX$ printing :



In [12]:

    
init_printing()

Defining the symbolic notations :



In [13]:

    
t, r, a, b = symbols('t r a b')



In [14]:

    
x = -r * sin(t*pi/4)
y = a/cosh(sin(t*pi/4))+b



In [25]:

    
(diff(y,t)+1).subs(t,-1)









    Out[25]:





$$\frac{\sqrt{2} \pi a \sinh{\left (\frac{\sqrt{2}}{2} \right )}}{8 \cosh^{2}{\left (\frac{\sqrt{2}}{2} \right )}} + 1$$



In [26]:

    
solve(_, a)









    Out[26]:





$$\begin{bmatrix}- \frac{4 \sqrt{2} \cosh^{2}{\left (\frac{\sqrt{2}}{2} \right )}}{\pi \sinh{\left (\frac{\sqrt{2}}{2} \right )}}\end{bmatrix}$$

x,y

Calculating the differential expressions



In [49]:

    
xp = diff(x, t)
yp = diff(y, t).simplify()
xpp = diff(xp, t)
ypp = diff(yp, t).simplify()



In [50]:

    
xp, yp, xpp, ypp









    Out[50]:





$$\begin{pmatrix}- \frac{\pi r}{4} \cos{\left (\frac{\pi t}{4} \right )}, & - \frac{\pi a \cos{\left (\frac{\pi t}{4} \right )} \sinh{\left (\sin{\left (\frac{\pi t}{4} \right )} \right )}}{4 \cosh^{2}{\left (\sin{\left (\frac{\pi t}{4} \right )} \right )}}, & \frac{\pi^{2} r}{16} \sin{\left (\frac{\pi t}{4} \right )}, & \frac{\pi^{2} a}{16 \cosh{\left (\sin{\left (\frac{\pi t}{4} \right )} \right )}} \left(\sin{\left (\frac{\pi t}{4} \right )} \tanh{\left (\sin{\left (\frac{\pi t}{4} \right )} \right )} + \cos^{2}{\left (\frac{\pi t}{4} \right )} - \frac{2 \cos^{2}{\left (\frac{\pi t}{4} \right )}}{\cosh^{2}{\left (\sin{\left (\frac{\pi t}{4} \right )} \right )}}\right)\end{pmatrix}$$



In [51]:

    
curvature = (xp * ypp - yp * xpp) / (xp**2 + yp**2)**Rational(3,2)



In [52]:

    
curvature









    Out[52]:





$$\frac{1}{\left(\frac{\pi^{2} a^{2} \cos^{2}{\left (\frac{\pi t}{4} \right )} \sinh^{2}{\left (\sin{\left (\frac{\pi t}{4} \right )} \right )}}{16 \cosh^{4}{\left (\sin{\left (\frac{\pi t}{4} \right )} \right )}} + \frac{\pi^{2} r^{2}}{16} \cos^{2}{\left (\frac{\pi t}{4} \right )}\right)^{\frac{3}{2}}} \left(- \frac{\pi^{3} a r \cos{\left (\frac{\pi t}{4} \right )}}{64 \cosh{\left (\sin{\left (\frac{\pi t}{4} \right )} \right )}} \left(\sin{\left (\frac{\pi t}{4} \right )} \tanh{\left (\sin{\left (\frac{\pi t}{4} \right )} \right )} + \cos^{2}{\left (\frac{\pi t}{4} \right )} - \frac{2 \cos^{2}{\left (\frac{\pi t}{4} \right )}}{\cosh^{2}{\left (\sin{\left (\frac{\pi t}{4} \right )} \right )}}\right) + \frac{\pi^{3} a r \sin{\left (\frac{\pi t}{4} \right )} \cos{\left (\frac{\pi t}{4} \right )}}{64 \cosh^{2}{\left (\sin{\left (\frac{\pi t}{4} \right )} \right )}} \sinh{\left (\sin{\left (\frac{\pi t}{4} \right )} \right )}\right)$$



In [53]:

    
curvature.simplify()









    Out[53]:





$$- \frac{a r \left(1 - \frac{2}{\cosh^{2}{\left (\sin{\left (\frac{\pi t}{4} \right )} \right )}}\right) \cos^{3}{\left (\frac{\pi t}{4} \right )}}{\left(\left(\frac{a^{2} \sinh^{2}{\left (\sin{\left (\frac{\pi t}{4} \right )} \right )}}{\cosh^{4}{\left (\sin{\left (\frac{\pi t}{4} \right )} \right )}} + r^{2}\right) \cos^{2}{\left (\frac{\pi t}{4} \right )}\right)^{\frac{3}{2}} \cosh{\left (\sin{\left (\frac{\pi t}{4} \right )} \right )}}$$



In [54]:

    
equa = r*cosh(a*t) - 1/curvature



In [55]:

    
equa.simplify()









    Out[55]:





$$\frac{a r^{2} \left(- \left(\sin{\left (\frac{\pi t}{4} \right )} \tanh{\left (\sin{\left (\frac{\pi t}{4} \right )} \right )} + \cos^{2}{\left (\frac{\pi t}{4} \right )}\right) \cosh^{2}{\left (\sin{\left (\frac{\pi t}{4} \right )} \right )} + \sin{\left (\frac{\pi t}{4} \right )} \sinh{\left (\sin{\left (\frac{\pi t}{4} \right )} \right )} \cosh{\left (\sin{\left (\frac{\pi t}{4} \right )} \right )} + 2 \cos^{2}{\left (\frac{\pi t}{4} \right )}\right) \cos{\left (\frac{\pi t}{4} \right )} \cosh{\left (a t \right )} - \left(\left(\frac{a^{2} \sinh^{2}{\left (\sin{\left (\frac{\pi t}{4} \right )} \right )}}{\cosh^{4}{\left (\sin{\left (\frac{\pi t}{4} \right )} \right )}} + r^{2}\right) \cos^{2}{\left (\frac{\pi t}{4} \right )}\right)^{\frac{3}{2}} \cosh^{3}{\left (\sin{\left (\frac{\pi t}{4} \right )} \right )}}{a r \left(- \left(\sin{\left (\frac{\pi t}{4} \right )} \tanh{\left (\sin{\left (\frac{\pi t}{4} \right )} \right )} + \cos^{2}{\left (\frac{\pi t}{4} \right )}\right) \cosh^{2}{\left (\sin{\left (\frac{\pi t}{4} \right )} \right )} + \sin{\left (\frac{\pi t}{4} \right )} \sinh{\left (\sin{\left (\frac{\pi t}{4} \right )} \right )} \cosh{\left (\sin{\left (\frac{\pi t}{4} \right )} \right )} + 2 \cos^{2}{\left (\frac{\pi t}{4} \right )}\right) \cos{\left (\frac{\pi t}{4} \right )}}$$



In [56]:

    
u = symbols('u')
diff(1/cosh(a*u), u)









    Out[56]:





$$- \frac{a \sinh{\left (a u \right )}}{\cosh^{2}{\left (a u \right )}}$$



In [23]:

    
sinh(-u)









    Out[23]:





$$- \sinh{\left (u \right )}$$



In [27]:

    
cosh(-u)









    Out[27]:





$$\cosh{\left (u \right )}$$



In [58]:

    
1/cosh(-sqrt(2)/2)









    Out[58]:





$$\frac{1}{\cosh{\left (\frac{\sqrt{2}}{2} \right )}}$$



In [64]:

    
f = Function('f')(t)
diff(a/cosh(f/r)+b, t)









    Out[64]:





$$- \frac{a \sinh{\left (\frac{1}{r} f{\left (t \right )} \right )} \frac{d}{d t} f{\left (t \right )}}{r \cosh^{2}{\left (\frac{1}{r} f{\left (t \right )} \right )}}$$



In [70]:

    
diff(-r*sin(t*pi/4),t)









    Out[70]:





$$- \frac{\pi r}{4} \cos{\left (\frac{\pi t}{4} \right )}$$



In [71]:

    
cosh(a)**2/sinh(a)









    Out[71]:





$$\frac{\cosh^{2}{\left (a \right )}}{\sinh{\left (a \right )}}$$



In [74]:

    
expand(_)









    Out[74]:





$$\frac{\cosh^{2}{\left (a \right )}}{\sinh{\left (a \right )}}$$



In [77]:

    
cosh(sqrt(2)/2)









    Out[77]:





$$\cosh{\left (\frac{\sqrt{2}}{2} \right )}$$



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