Common tangent using python

Question

I am trying to find a common tangent to two curves using python but I am not able to solve it.
The equations to the two curves are complicated that involve logarithms.

Is there a way in python to compute the x coordinates of a tangent that is common to both the curves in general. If I have 2 curves f(x) and g(x), I want to find the x-coordinates x1 and x2 on a common tangent where x1 lies on f(x) and x2 on g(x). I am trying f'(x1) = g'(x2) and f'(x1) = f(x1) - f(x2) / (x1 - x2) to get x1 and x2 but I am not able to get values using nonlinsolve as the equations are too complicated.

I want to just find x-coordinates of the common tangent

Can anyone suggest a better way?

import numpy as np
import sympy
from sympy import *
from matplotlib import pyplot as plt


x = symbols('x')
a, b, c, d, e, f = -99322.50019502985, -86864.87072433547, -96876.05627516498, -89703.35055202093, -3390.863799999999, -20942.518


def func(x):
    y1_1 = a - a*x + b*x
    y1_2 = c - c*x + d*x

    c1 = (1 - x) ** (1 - x)
    c2 = (x ** x)
    y2 = 12471 * (sympy.log((c1*c2)))

    y3 = 2*f*x**3 - x**2*(e + 3*f) + x*(e + f)
    eqn1 = y1_1 + y2 + y3
    eqn2 = y1_2 + y2 + y3
    return eqn1, eqn2


val = np.linspace(0, 1)
f1 = sympy.lambdify(x, func(x)[0])(val)
f2 = sympy.lambdify(x, func(x)[1])(val)

plt.plot(val, f1)
plt.plot(val, f2)
plt.show()

I am trying this

x1, x2 = sympy.symbols('x1 x2')

fun1 = func(x1)[0]
fun2 = func(x2)[0]
diff1 = diff(fun1,x1)
diff2 = diff(fun2,x2)
eq1 = diff1 - diff2
eq2 = diff1 - ((fun1 - fun2) / (x1 - x2))

sol = nonlinsolve([eq1, eq2], [x1, x2])

Answer 1

the first thing that needs to be done is to reduce the formulas

for example the first formula is actually this:

formula = x*(1 - x)*(17551.6542 - 41885.036*x) + x*(1 - x)*(41885.036*x - 24333.3818) + 12457.6294706944*x + log((x/(1 - x))**(12000*x)*(1 - x)**12000) - 99322.5001950298
formula = (x-x^2)*(17551.6542 - 41885.036*x) + (x-x^2)*(41885.036*x - 24333.3818) + 12457.6294706944*x + log((x/(1 - x))**(12000*x)*(1 - x)**12000) - 99322.5001950298

# constants
a = 41885.036
b = 17551.6542
c = 24333.3818
d = 12457.6294706944
e = 99322.5001950298
f = 12000

formula = (x-x^2)*(b - a*x) + (x-x^2)*(a*x - c) + d*x + log((x/(1 - x))**(f*x)*(1 - x)**f) - e
formula = (ax^3 -bx^2 + bx - ax^2) + (x-x^2)*(a*x - c) + d*x + log((x/(1 - x))**(f*x)*(1 - x)**f) - e
formula = ax^3 -bx^2 + bx - ax^2 -ax^3 + ax^2 + cx^2 -cx + d*x + log((x/(1 - x))**(f*x)*(1 - x)**f) - e

# collect x terms by power (note how the x^3 tern drops out, so its easier).
formula =  (c-b)*x^2 + (b-c+d)*x  + log((x/(1 - x))**(f*x)*(1 - x)**f) - e

which is much cleaner and is a quadratic with a log term. i expect that you can do some work on the log term too, but this is an excercise for the original poster .

likewise the second formula can be reduced in the same way, which is again an excercise for the original poster .

From this, both equations need to be differentiated with respect to x to find the tangent. Then set both formulas to be equal to each other (for a common tangent).

This would completely solve the question.

I actually wonder if this is a python question at all or actually a pure maths question.....

Answer 2

The important point to note is that, since the derivatives are monotonic, for any value of derivative of fun1 , there is a solution for fun2 . This can be easily seen if you plot both derivatives.

Thus, we want a function that, given an x1 , returns an x2 that matches it. I'll use numerical solution because the system is too cumbersome for numerical solution.

import scipy.optimize

def find_equal_value(f1, f2, x, x1):
    goal = f1.subs(x, x1)
    to_solve = sympy.lambdify(x, (f2 - goal)**2)  # Quadratic functions tend to be better behaved, and the result is the same
    sol = scipy.optimize.fmin(func=to_solve, x0=x1, ftol=1e-8, disp=False)  # The value for f1 is a good starting guess
    return sol[0]

I used fmin as the solver above because it worked and I knew how to use it by heart. Maybe root_scalar can give better results.

Using the function above, let's get some pairs (x1, x2) where the derivatives are equal:

df1 = sympy.diff(func(x)[0])
df2 = sympy.diff(func(x)[1])

x1 = 0.25236537  # Close to the zero derivative
x2 = find_equal_value(df1, df2, x, x1)
print(f'Derivative of f1 in x1: {df1.subs(x, x1)}')
print(f'Derivative of f2 in x2: {df2.subs(x, x2)}')
print(f'Error: {df1.subs(x, x1) - df2.subs(x, x2)}')

This results is:

Derivative of f1 in x1: 0.0000768765858083498
Derivative of f2 in x2: 0.0000681969431752805
Error: 0.00000867964263306931

If you want a x2 for several x1 s (beware that in some cases the solver hits a value where the logs are invalid. Always check your result for validity):

x1s = np.linspace(0.2, 0.8, 50)
x2s = [find_equal_value(df1, df2, x, x1) for x1 in x1s]
plt.plot(x1s, x2s); plt.grid(); plt.show()