In a programming language (Python, C#, etc) I need to determine how to calculate the angle between a line and the horizontal axis?
I think an image describes best what I want:
Given (P1x,P1y) and (P2x,P2y) what is the best way to calculate this angle? The origin is in the topleft and only the positive quadrant is used.
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Answer
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:
- Treat
(deltaX, deltaY)
as a vector. - Normalize that vector to a unit vector. To do so, divide
deltaX
anddeltaY
by the vector’s length (sqrt(deltaX*deltaX+deltaY*deltaY)
), unless the length is 0. - 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 atP1
). - And
deltaY
will now be the sine of that angle. - 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
anddeltaY
will tell you which quadrant the angle is in, in relation to the positive X axis atP1
:+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 https://en.wikipedia.org/wiki/Unit_circle