Python atan2 () function

atan2 | Python Methods and Functions

Syntax :

 atan2 (y, x) 

Parameter :

  (y, x) -  Both must be a numeric value. 

radians. The double value is from polar coordinate (r, theta). 

Type Error:

 Returns a TypeError if anything other than float is passed. 

Code # 1:

# Python3 demo
# atan2 () method

# import math

import math

# prints theta value
# two negative coordinates

theta1 = math.atan2 ( - 0.9 , - 0.9

print (  "atan2 (-0.9, -0.9):" , theta1) 

# prints theta value
# two positive coordinates

theta2 = math.atan2 ( 1.2 , 1.5

print ( "atan2 (1.2, 1.5):" , theta2) 

# prints the theta value of one
# positive and one negative coordinates

theta3 = math.atan2 ( 1.2 , - 1.5

print ( " atan2 (1.2, -1.5): " , theta3)


 atan2 (-0.5, -0.5): -2.356194490192345 atan2 (1.2, 1.5): 0.6747409422235526 atan2 (1.2, -1.5): 2.4668517113662407 

Code # 2:

# Python3 program to demonstrate the atan () method

# import math

import math

# list containing x and y coordinates

y = [ 1 , 2 , 3 , 4

x = [ 6 , 3 , 7 , 8

# move in range to get theta
# for all y and x coordinates

for i in range ( len (x)):

  theta = math.atan2 (y [i], x [i]) 

  print (theta) 


 0.16514867741462683 0.5880026035475675 0.40489178628508343 0.4636476090008061 

Code # 3: Program showing error

# Python3 demo program
# TypeError in method atan ()

# import math

import math


y, x = 3 , 6

# when sent integer values ​​
# returns a TypeError

theta = math.atan2 ([y], [x]) 

print (theta)


 Traceback (most recent call last): File "/home/", line 9, in theta = math.atan2 ([y], [x]) TypeError: a float is required 

Practical Use:
This function is used to determine the slope in radians when given Y and X coordinates.

Code # 4:

# Let`s find the slope when X
# and Y coordinates given

# import math

import math

# X and Y coordinates

X = 2 ; Y = 2  

# bias in radians

theta1 = math.atan2 (Y, X) 

# print the slope in radians

print (theta1)



Python atan2 () function: StackOverflow Questions

Answer #1

For people (like me) coming here via search engine and just looking for a solution which works out of the box, I recommend installing mpu. Install it via pip install mpu --user and use it like this to get the haversine distance:

import mpu

# Point one
lat1 = 52.2296756
lon1 = 21.0122287

# Point two
lat2 = 52.406374
lon2 = 16.9251681

# What you were looking for
dist = mpu.haversine_distance((lat1, lon1), (lat2, lon2))
print(dist)  # gives 278.45817507541943.

An alternative package is gpxpy.

If you don"t want dependencies, you can use:

import math

def distance(origin, destination):
    Calculate the Haversine distance.

    origin : tuple of float
        (lat, long)
    destination : tuple of float
        (lat, long)

    distance_in_km : float

    >>> origin = (48.1372, 11.5756)  # Munich
    >>> destination = (52.5186, 13.4083)  # Berlin
    >>> round(distance(origin, destination), 1)
    lat1, lon1 = origin
    lat2, lon2 = destination
    radius = 6371  # km

    dlat = math.radians(lat2 - lat1)
    dlon = math.radians(lon2 - lon1)
    a = (math.sin(dlat / 2) * math.sin(dlat / 2) +
         math.cos(math.radians(lat1)) * math.cos(math.radians(lat2)) *
         math.sin(dlon / 2) * math.sin(dlon / 2))
    c = 2 * math.atan2(math.sqrt(a), math.sqrt(1 - a))
    d = radius * c

    return d

if __name__ == "__main__":
    import doctest

The other alternative package is haversine

from haversine import haversine, Unit

lyon = (45.7597, 4.8422) # (lat, lon)
paris = (48.8567, 2.3508)

haversine(lyon, paris)
>> 392.2172595594006  # in kilometers

haversine(lyon, paris, unit=Unit.MILES)
>> 243.71201856934454  # in miles

# you can also use the string abbreviation for units:
haversine(lyon, paris, unit="mi")
>> 243.71201856934454  # in miles

haversine(lyon, paris, unit=Unit.NAUTICAL_MILES)
>> 211.78037755311516  # in nautical miles

They claim to have performance optimization for distances between all points in two vectors

from haversine import haversine_vector, Unit

lyon = (45.7597, 4.8422) # (lat, lon)
paris = (48.8567, 2.3508)
new_york = (40.7033962, -74.2351462)

haversine_vector([lyon, lyon], [paris, new_york], Unit.KILOMETERS)

>> array([ 392.21725956, 6163.43638211])

Answer #2

Update: User cphyc has kindly created a Github repository for the code in this answer (see here), and bundled the code into a package which may be installed using pip install matplotlib-label-lines.

Pretty Picture:

semi-automatic plot-labeling

In matplotlib it"s pretty easy to label contour plots (either automatically or by manually placing labels with mouse clicks). There does not (yet) appear to be any equivalent capability to label data series in this fashion! There may be some semantic reason for not including this feature which I am missing.

Regardless, I have written the following module which takes any allows for semi-automatic plot labelling. It requires only numpy and a couple of functions from the standard math library.


The default behaviour of the labelLines function is to space the labels evenly along the x axis (automatically placing at the correct y-value of course). If you want you can just pass an array of the x co-ordinates of each of the labels. You can even tweak the location of one label (as shown in the bottom right plot) and space the rest evenly if you like.

In addition, the label_lines function does not account for the lines which have not had a label assigned in the plot command (or more accurately if the label contains "_line").

Keyword arguments passed to labelLines or labelLine are passed on to the text function call (some keyword arguments are set if the calling code chooses not to specify).


  • Annotation bounding boxes sometimes interfere undesirably with other curves. As shown by the 1 and 10 annotations in the top left plot. I"m not even sure this can be avoided.
  • It would be nice to specify a y position instead sometimes.
  • It"s still an iterative process to get annotations in the right location
  • It only works when the x-axis values are floats


  • By default, the labelLines function assumes that all data series span the range specified by the axis limits. Take a look at the blue curve in the top left plot of the pretty picture. If there were only data available for the x range 0.5-1 then then we couldn"t possibly place a label at the desired location (which is a little less than 0.2). See this question for a particularly nasty example. Right now, the code does not intelligently identify this scenario and re-arrange the labels, however there is a reasonable workaround. The labelLines function takes the xvals argument; a list of x-values specified by the user instead of the default linear distribution across the width. So the user can decide which x-values to use for the label placement of each data series.

Also, I believe this is the first answer to complete the bonus objective of aligning the labels with the curve they"re on. :)

from math import atan2,degrees
import numpy as np

#Label line with line2D label data
def labelLine(line,x,label=None,align=True,**kwargs):

    ax = line.axes
    xdata = line.get_xdata()
    ydata = line.get_ydata()

    if (x < xdata[0]) or (x > xdata[-1]):
        print("x label location is outside data range!")

    #Find corresponding y co-ordinate and angle of the line
    ip = 1
    for i in range(len(xdata)):
        if x < xdata[i]:
            ip = i

    y = ydata[ip-1] + (ydata[ip]-ydata[ip-1])*(x-xdata[ip-1])/(xdata[ip]-xdata[ip-1])

    if not label:
        label = line.get_label()

    if align:
        #Compute the slope
        dx = xdata[ip] - xdata[ip-1]
        dy = ydata[ip] - ydata[ip-1]
        ang = degrees(atan2(dy,dx))

        #Transform to screen co-ordinates
        pt = np.array([x,y]).reshape((1,2))
        trans_angle = ax.transData.transform_angles(np.array((ang,)),pt)[0]

        trans_angle = 0

    #Set a bunch of keyword arguments
    if "color" not in kwargs:
        kwargs["color"] = line.get_color()

    if ("horizontalalignment" not in kwargs) and ("ha" not in kwargs):
        kwargs["ha"] = "center"

    if ("verticalalignment" not in kwargs) and ("va" not in kwargs):
        kwargs["va"] = "center"

    if "backgroundcolor" not in kwargs:
        kwargs["backgroundcolor"] = ax.get_facecolor()

    if "clip_on" not in kwargs:
        kwargs["clip_on"] = True

    if "zorder" not in kwargs:
        kwargs["zorder"] = 2.5


def labelLines(lines,align=True,xvals=None,**kwargs):

    ax = lines[0].axes
    labLines = []
    labels = []

    #Take only the lines which have labels other than the default ones
    for line in lines:
        label = line.get_label()
        if "_line" not in label:

    if xvals is None:
        xmin,xmax = ax.get_xlim()
        xvals = np.linspace(xmin,xmax,len(labLines)+2)[1:-1]

    for line,x,label in zip(labLines,xvals,labels):

Test code to generate the pretty picture above:

from matplotlib import pyplot as plt
from scipy.stats import loglaplace,chi2

from labellines import *

X = np.linspace(0,1,500)
A = [1,2,5,10,20]
funcs = [np.arctan,np.sin,loglaplace(4).pdf,chi2(5).pdf]

for a in A:


for a in A:


for a in A:

xvals = [0.8,0.55,0.22,0.104,0.045]

for a in A:

lines = plt.gca().get_lines()
labelLine(l1,0.6,label=r"$Re=${}".format(l1.get_label()),ha="left",va="bottom",align = False)

Answer #3

First find the difference between the start point and the end point (here, this is more of a directed line segment, not a "line", since lines extend infinitely and don"t start at a particular point).

deltaY = P2_y - P1_y
deltaX = P2_x - P1_x

Then calculate the angle (which runs from the positive X axis at P1 to the positive Y axis at P1).

angleInDegrees = arctan(deltaY / deltaX) * 180 / PI

But arctan may not be ideal, because dividing the differences this way will erase the distinction needed to distinguish which quadrant the angle is in (see below). Use the following instead if your language includes an atan2 function:

angleInDegrees = atan2(deltaY, deltaX) * 180 / PI

EDIT (Feb. 22, 2017): In general, however, calling atan2(deltaY,deltaX) just to get the proper angle for cos and sin may be inelegant. In those cases, you can often do the following instead:

  1. Treat (deltaX, deltaY) as a vector.
  2. Normalize that vector to a unit vector. To do so, divide deltaX and deltaY by the vector"s length (sqrt(deltaX*deltaX+deltaY*deltaY)), unless the length is 0.
  3. After that, deltaX will now be the cosine of the angle between the vector and the horizontal axis (in the direction from the positive X to the positive Y axis at P1).
  4. And deltaY will now be the sine of that angle.
  5. If the vector"s length is 0, it won"t have an angle between it and the horizontal axis (so it won"t have a meaningful sine and cosine).

EDIT (Feb. 28, 2017): Even without normalizing (deltaX, deltaY):

  • The sign of deltaX will tell you whether the cosine described in step 3 is positive or negative.
  • The sign of deltaY will tell you whether the sine described in step 4 is positive or negative.
  • The signs of deltaX and deltaY will tell you which quadrant the angle is in, in relation to the positive X axis at P1:
    • +deltaX, +deltaY: 0 to 90 degrees.
    • -deltaX, +deltaY: 90 to 180 degrees.
    • -deltaX, -deltaY: 180 to 270 degrees (-180 to -90 degrees).
    • +deltaX, -deltaY: 270 to 360 degrees (-90 to 0 degrees).

An implementation in Python using radians (provided on July 19, 2015 by Eric Leschinski, who edited my answer):

from math import *
def angle_trunc(a):
    while a < 0.0:
        a += pi * 2
    return a

def getAngleBetweenPoints(x_orig, y_orig, x_landmark, y_landmark):
    deltaY = y_landmark - y_orig
    deltaX = x_landmark - x_orig
    return angle_trunc(atan2(deltaY, deltaX))

angle = getAngleBetweenPoints(5, 2, 1,4)
assert angle >= 0, "angle must be >= 0"
angle = getAngleBetweenPoints(1, 1, 2, 1)
assert angle == 0, "expecting angle to be 0"
angle = getAngleBetweenPoints(2, 1, 1, 1)
assert abs(pi - angle) <= 0.01, "expecting angle to be pi, it is: " + str(angle)
angle = getAngleBetweenPoints(2, 1, 2, 3)
assert abs(angle - pi/2) <= 0.01, "expecting angle to be pi/2, it is: " + str(angle)
angle = getAngleBetweenPoints(2, 1, 2, 0)
assert abs(angle - (pi+pi/2)) <= 0.01, "expecting angle to be pi+pi/2, it is: " + str(angle)
angle = getAngleBetweenPoints(1, 1, 2, 2)
assert abs(angle - (pi/4)) <= 0.01, "expecting angle to be pi/4, it is: " + str(angle)
angle = getAngleBetweenPoints(-1, -1, -2, -2)
assert abs(angle - (pi+pi/4)) <= 0.01, "expecting angle to be pi+pi/4, it is: " + str(angle)
angle = getAngleBetweenPoints(-1, -1, -1, 2)
assert abs(angle - (pi/2)) <= 0.01, "expecting angle to be pi/2, it is: " + str(angle)

All tests pass. See

Answer #4

Edit: Just as a note, if you just need a quick and easy way of finding the distance between two points, I strongly recommend using the approach described in Kurt"s answer below instead of re-implementing Haversine -- see his post for rationale.

This answer focuses just on answering the specific bug OP ran into.

It"s because in Python, all the trig functions use radians, not degrees.

You can either convert the numbers manually to radians, or use the radians function from the math module:

from math import sin, cos, sqrt, atan2, radians

# approximate radius of earth in km
R = 6373.0

lat1 = radians(52.2296756)
lon1 = radians(21.0122287)
lat2 = radians(52.406374)
lon2 = radians(16.9251681)

dlon = lon2 - lon1
dlat = lat2 - lat1

a = sin(dlat / 2)**2 + cos(lat1) * cos(lat2) * sin(dlon / 2)**2
c = 2 * atan2(sqrt(a), sqrt(1 - a))

distance = R * c

print("Result:", distance)
print("Should be:", 278.546, "km")

The distance is now returning the correct value of 278.545589351 km.

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