Intersection of two graphs in Python, find the x value

There may be multiple intersections, you can find the (x,y) point at every intersection by the following list comprehension

intersections = [(x[i], f[i]) for i,_ in enumerate(zip(f,g)) if f[i] == g[i]]

As a simple example

>>> x = [1,2,3,4,5]
>>> f = [2,4,6,8,10]
>>> g = [10,8,6,4,2]
>>> [(x[i], f[i]) for i,_ in enumerate(zip(f,g)) if f[i] == g[i]]
[(3, 6)]

So this found one intersection point at x = 3, y = 6. Note that if you are using float the two values may not be exactly equal, so you could use some tolerance instead of ==.

Even if f and g intersect, you cannot be sure that f[i]== g[i] for integer i (the intersection probably occurs between points).

You should instead test like

# detect intersection by change in sign of difference
d = f - g
for i in range(len(d) - 1):
    if d[i] == 0. or d[i] * d[i + 1] < 0.:
        # crossover at i
        x_ = x[i]

For arrays f and g, we could simply do the following:

np.pad(np.diff(np.array(f > g).astype(int)), (1,0), 'constant', constant_values = (0,))

This will give the array of all the crossover points. Every 1 is a crossover from below to above and every -1 a crossover from above to below.

Well, I was looking for a matplotlib for two curves which were different in size and had not the same x values. Here is what I come up with:

import numpy as np
import matplotlib.pyplot as plt
import sys

fig = plt.figure()
ax = fig.add_subplot(111)

# x1 = [1,2,3,4,5,6,7,8]
# y1 = [20,100,50,120,55,240,50,25]
# x2 = [3,4,5,6,7,8,9]
# y2 = [25,200,14,67,88,44,120]

x1=[1.4,2.1,3,5.9,8,9,12,15]
y1=[2.3,3.1,1,3.9,8,9,11,9]
x2=[1,2,3,4,6,8,9,12,14]
y2=[4,12,7,1,6.3,7,5,6,11]

ax.plot(x1, y1, color='lightblue',linewidth=3, marker='s')
ax.plot(x2, y2, color='darkgreen', marker='^')

y_lists = y1[:]
y_lists.extend(y2)
y_dist = max(y_lists)/200.0

x_lists = x1[:]
x_lists.extend(x2)  
x_dist = max(x_lists)/900.0
division = 1000
x_begin = min(x1[0], x2[0])     # 3
x_end = max(x1[-1], x2[-1])     # 8

points1 = [t for t in zip(x1, y1) if x_begin<=t[0]<=x_end]  # [(3, 50), (4, 120), (5, 55), (6, 240), (7, 50), (8, 25)]
points2 = [t for t in zip(x2, y2) if x_begin<=t[0]<=x_end]  # [(3, 25), (4, 35), (5, 14), (6, 67), (7, 88), (8, 44)]
# print points1
# print points2

x_axis = np.linspace(x_begin, x_end, division)
idx = 0
id_px1 = 0
id_px2 = 0
x1_line = []
y1_line = []
x2_line = []
y2_line = []
xpoints = len(x_axis)
intersection = []
while idx < xpoints:
    # Iterate over two line segments
    x = x_axis[idx]
    if id_px1>-1:
        if x >= points1[id_px1][0] and id_px1<len(points1)-1:
            y1_line = np.linspace(points1[id_px1][1], points1[id_px1+1][1], 1000) # 1.4 1.401 1.402 etc. bis 2.1
            x1_line = np.linspace(points1[id_px1][0], points1[id_px1+1][0], 1000)
            id_px1 = id_px1 + 1
            if id_px1 == len(points1):
                x1_line = []
                y1_line = []
                id_px1 = -1
    if id_px2>-1:
        if x >= points2[id_px2][0] and id_px2<len(points2)-1:
            y2_line = np.linspace(points2[id_px2][1], points2[id_px2+1][1], 1000)
            x2_line = np.linspace(points2[id_px2][0], points2[id_px2+1][0], 1000)
            id_px2 = id_px2 + 1
            if id_px2 == len(points2):
                x2_line = []
                y2_line = []
                id_px2 = -1
    if x1_line!=[] and y1_line!=[] and x2_line!=[] and y2_line!=[]:
        i = 0
        while abs(x-x1_line[i])>x_dist and i < len(x1_line)-1:
            i = i + 1
        y1_current = y1_line[i]
        j = 0
        while abs(x-x2_line[j])>x_dist and j < len(x2_line)-1:
            j = j + 1
        y2_current = y2_line[j]
        if abs(y2_current-y1_current)<y_dist and i != len(x1_line) and j != len(x2_line):
            ymax = max(y1_current, y2_current)
            ymin = min(y1_current, y2_current)
            xmax = max(x1_line[i], x2_line[j])
            xmin = min(x1_line[i], x2_line[j])
            intersection.append((x, ymin+(ymax-ymin)/2))
            ax.plot(x, y1_current, 'ro') # Plot the cross point
    idx += 1    
print "intersection points", intersection
plt.show()

Intersection probably occurs between points. Let’s explore the example bellow.

import numpy as np
import matplotlib.pyplot as plt
xs=np.arange(0, 20)
y1=np.arange(0, 20)*2
y2=np.array([1, 1.5, 3,  8,  9,  20, 23, 21, 13, 23, 18, 20, 23, 24, 31, 28, 30, 33, 37, 36])

plotting the 2 curves above, along with their intersections, using as intersection the average coordinates before and after proposed from idx intersection, all points are closer to the first curve.

idx=np.argwhere(np.diff(np.sign(y1 - y2 )) != 0).reshape(-1) + 0
plt.plot(xs, y1)
plt.plot(xs, y2)
for i in range(len(idx)):
    plt.plot((xs[idx[i]]+xs[idx[i]+1])/2.,(y1[idx[i]]+y1[idx[i]+1])/2., 'ro')
plt.legend(['Y1', 'Y2'])
plt.show()   

using as intersection the average coordinates before and after but for both y1 and y2 curves usually are closer to true intersection

plt.plot(xs, y1)
plt.plot(xs, y2)
for i in range(len(idx)):
    plt.plot((xs[idx[i]]+xs[idx[i]+1])/2.,(y1[idx[i]]+y1[idx[i]+1]+y2[idx[i]]+y2[idx[i]+1])/4., 'ro')
plt.legend(['Y1', 'Y2'])
plt.show()   

For an even more accurate intersection estimation we could use interpolation.

Here’s a solution which:

• Works with N-dimensional data
• Uses Euclidean distance rather than merely finding cross-overs in the y-axis
• Is more efficient with lots of data (it queries a KD-tree, which should query in logarathmic time instead of linear time).
• You can change the distance_upper_bound in the KD-tree query to define how close is close enough.
• You can query the KD-tree with many points at the same time, if needed. Note: if you need to query thousands of points at once, you can get dramatic performance increases by querying the KD-tree with another KD-tree.

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from scipy.spatial import cKDTree
from scipy import interpolate

fig = plt.figure()
ax = fig.add_axes([0, 0, 1, 1], projection='3d')
ax.axis('off')

def upsample_coords(coord_list):
    # s is smoothness, set to zero
    # k is degree of the spline. setting to 1 for linear spline
    tck, u = interpolate.splprep(coord_list, k=1, s=0.0)
    upsampled_coords = interpolate.splev(np.linspace(0, 1, 100), tck)
    return upsampled_coords

# target line
x_targ = [1, 2, 3, 4, 5, 6, 7, 8]
y_targ = [20, 100, 50, 120, 55, 240, 50, 25]
z_targ = [20, 100, 50, 120, 55, 240, 50, 25]
targ_upsampled = upsample_coords([x_targ, y_targ, z_targ])
targ_coords = np.column_stack(targ_upsampled)

# KD-tree for nearest neighbor search
targ_kdtree = cKDTree(targ_coords)

# line two
x2 = [3,4,5,6,7,8,9]
y2 = [25,35,14,67,88,44,120]
z2 = [25,35,14,67,88,44,120]
l2_upsampled = upsample_coords([x2, y2, z2])
l2_coords = np.column_stack(l2_upsampled)

# plot both lines
ax.plot(x_targ, y_targ, z_targ, color='black', linewidth=0.5)
ax.plot(x2, y2, z2, color='darkgreen', linewidth=0.5)

# find intersections
for i in range(len(l2_coords)):
    if i == 0:  # skip first, there is no previous point
        continue

    distance, close_index = targ_kdtree.query(l2_coords[i], distance_upper_bound=.5)

    # strangely, points infinitely far away are somehow within the upper bound
    if np.isinf(distance):
        continue

    # plot ground truth that was activated
    _x, _y, _z = targ_kdtree.data[close_index]
    ax.scatter(_x, _y, _z, 'gx')
    _x2, _y2, _z2 = l2_coords[i]
    ax.scatter(_x2, _y2, _z2, 'rx')  # Plot the cross point


plt.show()

For those who are using or open to use the Shapely library for geometry-related computations, getting the intersection will be much easier. You just have to construct LineString from each line and get their intersection as follows:

import numpy as np
import matplotlib.pyplot as plt
from shapely.geometry import LineString

x = np.arange(0, 1000)
f = np.arange(0, 1000)
g = np.sin(np.arange(0, 10, 0.01) * 2) * 1000

plt.plot(x, f)
plt.plot(x, g)

first_line = LineString(np.column_stack((x, f)))
second_line = LineString(np.column_stack((x, g)))
intersection = first_line.intersection(second_line)

if intersection.geom_type == 'MultiPoint':
    plt.plot(*LineString(intersection).xy, 'o')
elif intersection.geom_type == 'Point':
    plt.plot(*intersection.xy, 'o')

And to get the x and y values as NumPy arrays you would just write:

x, y = LineString(intersection).xy
# x: array('d', [0.0, 149.5724669847373, 331.02906176584617, 448.01182730277833, 664.6733061190541, 743.4822641140581])
# y: array('d', [0.0, 149.5724669847373, 331.02906176584617, 448.01182730277833, 664.6733061190541, 743.4822641140581])

or if an intersection is only one point:

x, y = intersection.xy

I had a similar problem, but with one discontinue function, like the tangent function. To avoid get points on the discontinuity, witch i didn’t want to consider a intersection, i added a tolerance parameter on the previous solutions that use np.diff and np.sign. I set the tolerance parameter as the mean of the differences between the two data points, witch suffices in my case.

import numpy as np
import matplotlib.pyplot as plt

fig,ax = plt.subplots(nrows = 1,ncols = 2)

x = np.arange(0, 1000)
f = 2*np.arange(0, 1000)
g = np.tan(np.arange(0, 10, 0.01) * 2) * 1000


#here we set a threshold to decide if we will consider that point as a intersection
tolerance = np.abs(np.diff(f-g)).mean()
idx = np.argwhere((np.diff(np.sign(f - g)) != 0) & (np.abs(np.diff(f-g)) <= tolerance)).flatten()

#general case (tolerance = infinity)
tolerance = np.inf
idx2 = np.argwhere((np.diff(np.sign(f - g)) != 0) & (np.abs(np.diff(f-g)) <= tolerance)).flatten()

ax1,ax2 = ax

ax1.plot(x,f); ax1.plot(x,g)
ax2.plot(x,f); ax2.plot(x,g)

ax1.plot(x[idx], f[idx], 'o'); ax1.set_ylim(-3000,3000)
ax2.plot(x[idx2],f[idx2], 'o'); ax2.set_ylim(-3000,3000)
plt.show()

As a documented and tested function (credit for the algorithm goes to @Matt, I only changed the example to something simpler and used linspace instead of arange to handle non-integers better):

from typing import Iterable, Tuple
import numpy as np
import doctest

def intersect(x: np.array, f: np.array, g: np.array) -> Iterable[Tuple[(int, int)]]:
    """
    Finds the intersection points between `f` and `g` on the domain `x`.
    Given:
        - `x`: The discretized domain.
        - `f`: The discretized values of the first function calculated on the
               discretized domain.
        - `g`: The discretized values of the second function calculated on the
               discretized domain.
    Returns:
        An iterable containing the (x,y) points of intersection.


    Test case, line-parabola intersection:
        
    >>> x = np.linspace(0, 10, num=10000)
    >>> f = 3 * x
    >>> g = np.square(x)
    >>> list(intersect(x, f, g))
    [(0.0, 0.0), (2.999299929992999, 8.997899789978998)]
    """
    idx = np.argwhere(np.diff(np.sign(f - g))).flatten()
    return zip(x[idx], f[idx])

if __name__ == "__main__":
    doctest.testmod()

In Python 2, just remove the type hints.