The objective is to find the point of intersection of two linear equations. These two linear equation are derived using the Numpy polyfit functions.
Given two time series (xLeft, yLeft) and (xRight, yRight), the linear least suqares fit to each of them was calculated using polyfit as shown below:
xLeft = [
6168, 6169, 6170, 6171, 6172, 6173, 6174, 6175, 6176, 6177,
6178, 6179, 6180, 6181, 6182, 6183, 6184, 6185, 6186, 6187
]
yLeft = [
0.98288751, 1.3639959, 1.7550986, 2.1539073, 2.5580614,
2.9651523, 3.3727503, 3.7784295, 4.1797948, 4.5745049,
4.9602985, 5.3350167, 5.6966233, 6.0432272, 6.3730989,
6.6846867, 6.9766307, 7.2477727, 7.4971657, 7.7240791
]
xRight = [
6210, 6211, 6212, 6213, 6214, 6215, 6216, 6217, 6218, 6219,
6220, 6221, 6222, 6223, 6224, 6225, 6226, 6227, 6228, 6229,
6230, 6231, 6232, 6233, 6234, 6235, 6236, 6237, 6238, 6239,
6240, 6241, 6242, 6243, 6244, 6245, 6246, 6247, 6248, 6249,
6250, 6251, 6252, 6253, 6254, 6255, 6256, 6257, 6258, 6259,
6260, 6261, 6262, 6263, 6264, 6265, 6266, 6267, 6268, 6269,
6270, 6271, 6272, 6273, 6274, 6275, 6276, 6277, 6278, 6279,
6280, 6281, 6282, 6283, 6284, 6285, 6286, 6287, 6288]
yRight=[
7.8625913, 7.7713094, 7.6833806, 7.5997391, 7.5211883,
7.4483986, 7.3819046, 7.3221073, 7.2692747, 7.223547,
7.1849418, 7.1533613, 7.1286001, 7.1103559, 7.0982385,
7.0917811, 7.0904517, 7.0936642, 7.100791, 7.1111741,
7.124136, 7.1389918, 7.1550579, 7.1716633, 7.1881566,
7.2039142, 7.218349, 7.2309117, 7.2410989, 7.248455,
7.2525721, 7.2530937, 7.249711, 7.2421637, 7.2302341,
7.213747, 7.1925621, 7.1665707, 7.1356878, 7.0998487,
7.0590014, 7.0131001, 6.9621005, 6.9059525, 6.8445964,
6.7779589, 6.7059474, 6.6284504, 6.5453324, 6.4564347,
6.3615761, 6.2605534, 6.1531439, 6.0391097, 5.9182019,
5.7901659, 5.6547484, 5.5117044, 5.360805, 5.2018456,
5.034656, 4.8591075, 4.6751242, 4.4826899, 4.281858,
4.0727611, 3.8556159, 3.6307325, 3.3985188, 3.1594861,
2.9142516, 2.6635408, 2.4081881, 2.1491354, 1.8874279,
1.6242117,1.3607255,1.0982931,0.83831298
]
left_line = np.polyfit(xleft, yleft, 1)
right_line = np.polyfit(xRight, yRight, 1)
In this case, polyfit outputs the coeficients m and b for y = mx + b, respectively.
The intersection of the two linear equations then can be calculated as follows:
x0 = -(left_line[1] - right_line[1]) / (left_line[0] - right_line[0])
y0 = x0 * left_line[0] + left_line[1]
However, I wonder whether there exist Numpy build-in approach to calculate the last two steps?
Not exactly a built-in approach, but you can simplify the problem. Say I have lines given my y = m1 * x + b1 and y = m2 * x + b2. You can trivially find an equation for the difference, which is also a line:
y = (m1 - m2) * x + (b1 - b2)
Notice that this line will have a root at the intersection of the two original lines, if they intersect. You can use the numpy.polynomial.Polynomial class to perform these operations:
>>> (np.polynomial.Polynomial(left_line[::-1]) - np.polynomial.Polynomial(right_line[::-1])).roots()
array([6192.0710885])
Notice that I had to swap the order of the coefficients, since Polynomial expects smallest to largest, while np.polyfit returns the opposite. In fact, np.polyfit is not recommended. Instead, you can get Polynomial obejcts directly using np.polynomial.Polynomial.fit class method. Your code would then look like:
left_line = np.polynomial.Polynomial.fit(xLeft, yLeft, 1, domain=[-1, 1])
right_line = np.polynomial.Polynomial.fit(xRight, yRight, 1, domain=[-1, 1])
x0 = (left_line - right_line).roots()
y0 = left_line(x0)
The domain is mapped to the window [-1, 1]. If you do not specify a domain, the peak-to-peak of the x-values will be used instead. You do not want this, since it will result in a mapping of the input values. Instead, we explicitly specify that the domain [-1, 1] maps to the same window. An alternative would be to use the default domain and set e.g. window=[xLeft.min(), xLeft.max()]. The problem with this approach is that it would then create different domains for the polynomials, preventing the operation left_line - right_line.
See https://numpy.org/doc/stable/reference/routines.polynomials.classes.html for more information.