numpy polynomial.Polynomial.fit() gives different coefficients than polynomial.polyfit()

I do not understand why polynomial.Polynomial.fit() gives coefficients very different from the expected coefficients :

import numpy as np

x = np.linspace(0, 10, 50)
y = x**2 + 5 * x + 10

print(np.polyfit(x, y, 2))
print(np.polynomial.polynomial.polyfit(x, y, 2))
print(np.polynomial.polynomial.Polynomial.fit(x, y, 2))

Gives :

[ 1.  5. 10.]
[10.  5.  1.]
poly([60. 75. 25.])

The two first results are OK, and thanks to this answer I understand why the two arrays are in reversed order.

However, I do not understand the signification of the third result. The coefficients looks wrong, though the polynomial that I got this way seems to give correct predicted values.

Answer

The answer is slightly hidden in the docs, of course. Looking at the class numpy.polynomial.polynomial.Polynomial(coef, domain=None, window=None) It is clear that in general the coefficients [a, b, c, …] are for the polynomial a + b * x + c * x**2 + .... However, there are the keyword parameters domain and window both with default [-1,1]. I am not into that class, so I am not sure about the purpose, but it is clear that a remapping takes place. Now in the case of polynomial.Polynomial.fit() one has a class method that automatically takes the x data as domain, but still makes the mapping to the window. Hence, in the OP [0-10] is mapped onto [-1,1]. This is done by x = x' / 5 - 1 or x' -> 5 * x + 5. Putting the latter in the OP polynomial we get

( 5 x' + 5 )**2 + 5 * ( 5 * x' + 5 ) + 10 = 25 * x'**2 + 75 * x' + 60

Voila.

To get the expected result one has to put

print(np.polynomial.polynomial.Polynomial.fit(x, y, 2, window=[0, 10] ) )

wich gives

poly([10.  5.  1.])