[math] Using atan2 to find angle between two vectors

I understand that:

atan2(vector.y, vector.x) = the angle between the vector and the X axis.

But I wanted to know how to get the angle between two vectors using atan2. So I came across this solution:

atan2(vector1.y - vector2.y, vector1.x - vector2.x)

My question is very simple:

Will the two following formulas produce the same number?

• atan2(vector1.y - vector2.y, vector1.x - vector2.x)

• atan2(vector2.y - vector1.y, vector2.x - vector1.x)

If not: How do I know what vector comes first in the subtractions?

I think a better formula was posted here: http://www.mathworks.com/matlabcentral/answers/16243-angle-between-two-vectors-in-3d

angle = atan2(norm(cross(a,b)), dot(a,b))

So this formula works in 2 or 3 dimensions. For 2 dimensions this formula simplifies to the one stated above.

If you care about accuracy for small angles, you want to use this:

angle = 2*atan2(|| ||b||a - ||a||b ||, || ||b||a + ||a||b ||)

Where "||" means absolute value, AKA "length of the vector". See https://math.stackexchange.com/questions/1143354/numerically-stable-method-for-angle-between-3d-vectors/1782769

However, that has the downside that in two dimensions, it loses the sign of the angle.

As a complement to the answer of @martin-r one should note that it is possible to use the sum/difference formula for arcus tangens.

angle = atan2(vec2.y, vec2.x) - atan2(vec1.y, vec1.x);
angle = -atan2(vec1.x * vec2.y - vec1.y * vec2.x, dot(vec1, vec2))
        where dot = vec1.x * vec2.x  + vec1.y * vec2.y

• Caveat 1: make sure the angle remains within -pi ... +pi
• Caveat 2: beware when the vectors are getting very similar, you might get extinction in the first argument, leading to numerical inaccuracies

angle(vector.b,vector.a)=pi/2*((1+sgn(xa))*(1-sgn(ya^2))-(1+sgn(xb))*(1-sgn(yb^2)))

+pi/4*((2+sgn(xa))*sgn(ya)-(2+sgn(xb))*sgn(yb))

+sgn(xa*ya)*atan((abs(xa)-abs(ya))/(abs(xa)+abs(ya)))

-sgn(xb*yb)*atan((abs(xb)-abs(yb))/(abs(xb)+abs(yb)))

xb,yb and xa,ya are the coordinates of the two vectors

Here a little program in Python that uses the angle between vectors to determine if a point is inside or outside a certain polygon

import sys
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.patches as patches
from shapely.geometry import Point, Polygon
from pprint import pprint

# Plot variables
x_min, x_max = -6, 12
y_min, y_max = -3, 8
tick_interval = 1
FIG_SIZE = (10, 10)
DELTA_ERROR = 0.00001
IN_BOX_COLOR = 'yellow'
OUT_BOX_COLOR = 'black'


def angle_between(v1, v2):
    """ Returns the angle in radians between vectors 'v1' and 'v2'
        The sign of the angle is dependent on the order of v1 and v2
        so acos(norm(dot(v1, v2))) does not work and atan2 has to be used, see:
        https://stackoverflow.com/questions/21483999/using-atan2-to-find-angle-between-two-vectors
    """
    arg1 = np.cross(v1, v2)
    arg2 = np.dot(v1, v2)
    angle = np.arctan2(arg1, arg2)
    return angle


def point_inside(point, border):
    """ Returns True if point is inside border polygon and False if not
        Arguments:
        :point: x, y in shapely.geometry.Point type
        :border: [x1 y1, x2 y2, ... , xn yn] in shapely.geomettry.Polygon type
    """    
    assert len(border.exterior.coords) > 2,\
        'number of points in the polygon must be > 2'

    point = np.array(point)
    side1 = np.array(border.exterior.coords[0]) - point
    sum_angles = 0
    for border_point in border.exterior.coords[1:]:
        side2 = np.array(border_point) - point
        angle = angle_between(side1, side2)
        sum_angles += angle
        side1 = side2

    # if wn is 1 then the point is inside
    wn = sum_angles / 2 / np.pi
    if abs(wn - 1) < DELTA_ERROR:
        return True
    else:
        return False


class MainMap():

    @classmethod
    def settings(cls, fig_size):
        # set the plot outline, including axes going through the origin
        cls.fig, cls.ax = plt.subplots(figsize=fig_size)
        cls.ax.set_xlim(-x_min, x_max)
        cls.ax.set_ylim(-y_min, y_max)
        cls.ax.set_aspect(1)
        tick_range_x = np.arange(round(x_min + (10*(x_max - x_min) % tick_interval)/10, 1),
            x_max + 0.1, step=tick_interval)
        tick_range_y = np.arange(round(y_min + (10*(y_max - y_min) % tick_interval)/10, 1), 
            y_max + 0.1, step=tick_interval)
        cls.ax.set_xticks(tick_range_x)
        cls.ax.set_yticks(tick_range_y)
        cls.ax.tick_params(axis='both', which='major', labelsize=6)
        cls.ax.spines['left'].set_position('zero')
        cls.ax.spines['right'].set_color('none')
        cls.ax.spines['bottom'].set_position('zero')
        cls.ax.spines['top'].set_color('none')

    @classmethod
    def get_ax(cls):
        return cls.ax

    @staticmethod
    def plot():
        plt.tight_layout()
        plt.show()


class PlotPointandRectangle(MainMap):

    def __init__(self, start_point, rectangle_polygon, tolerance=0):

        self.current_object = None
        self.currently_dragging = False
        self.fig.canvas.mpl_connect('key_press_event', self.on_key)
        self.plot_types = ['o', 'o-']
        self.plot_type = 1
        self.rectangle = rectangle_polygon

        # define a point that can be moved around
        self.point = patches.Circle((start_point.x, start_point.y), 0.10,
            alpha=1)
        if point_inside(start_point, self.rectangle):
            _color = IN_BOX_COLOR
        else:
            _color = OUT_BOX_COLOR
        self.point.set_color(_color)
        self.ax.add_patch(self.point)
        self.point.set_picker(tolerance)
        cv_point = self.point.figure.canvas
        cv_point.mpl_connect('button_release_event', self.on_release)
        cv_point.mpl_connect('pick_event', self.on_pick)
        cv_point.mpl_connect('motion_notify_event', self.on_motion)

        self.plot_rectangle()

    def plot_rectangle(self):
        x = [point[0] for point in self.rectangle.exterior.coords]
        y = [point[1] for point in self.rectangle.exterior.coords]
        # y = self.rectangle.y
        self.rectangle_plot, = self.ax.plot(x, y,
            self.plot_types[self.plot_type], color='r', lw=0.4, markersize=2)

    def on_release(self, event):
        self.current_object = None
        self.currently_dragging = False

    def on_pick(self, event):
        self.currently_dragging = True
        self.current_object = event.artist

    def on_motion(self, event):
        if not self.currently_dragging:
            return
        if self.current_object == None:
            return

        point = Point(event.xdata, event.ydata)
        self.current_object.center = point.x, point.y
        if point_inside(point, self.rectangle):
            _color = IN_BOX_COLOR
        else:
            _color = OUT_BOX_COLOR
        self.current_object.set_color(_color)

        self.point.figure.canvas.draw()

    def remove_rectangle_from_plot(self):
        try:
            self.rectangle_plot.remove()
        except ValueError:
            pass

    def on_key(self, event):
        # with 'space' toggle between just points or points connected with
        # lines
        if event.key == ' ':
            self.plot_type = (self.plot_type + 1) % 2
            self.remove_rectangle_from_plot()
            self.plot_rectangle()
            self.point.figure.canvas.draw()


def main(start_point, rectangle):

    MainMap.settings(FIG_SIZE)
    plt_me = PlotPointandRectangle(start_point, rectangle)  #pylint: disable=unused-variable
    MainMap.plot()

if __name__ == "__main__":
    try:
        start_point = Point([float(val) for val in sys.argv[1].split()])
    except IndexError:
        start_point= Point(0, 0)

    border_points = [(-2, -2),
                     (1, 1),
                     (3, -1),
                     (3, 3.5),
                     (4, 1),
                     (5, 1),
                     (4, 3.5),
                     (5, 6),
                     (3, 4),
                     (3, 5),
                     (-0.5, 1),
                     (-3, 1),
                     (-1, -0.5),
                    ]               

    border_points_polygon = Polygon(border_points)
    main(start_point, border_points_polygon)

The formula, angle(vector.b,vector.a), that I sent, give results

in the four quadrants and for any coordinates xa,ya and xb,yb.

For coordinates xa=ya=0 and or xb=yb=0 is undefined.

The angle can be bigger or smaller than pi, and can be positive

or negative.

Nobody pointed out that if you have a single vector, and want to find the angle of the vector from the X axis, you can take advantage of the fact that the argument to atan2() is actually the slope of the line, or (delta Y / delta X). So if you know the slope, you can do the following:

given:

A = angle of the vector/line you wish to determine (from the X axis).

m = signed slope of the vector/line.

then:

A = atan2(m, 1)

Very useful!

The proper way to do it is by find the sine of the angle using the cross product, and the cosine of the angle using the dot product and combine the two with the Atan2() function.

In C# this is

public struct Vector2
{
    public double X, Y;

    /// <summary>
    /// Returns the angle between two vectos
    /// </summary>
    public static double GetAngle(Vector2 A, Vector2 B)
    {
        // |A·B| = |A| |B| COS(?)
        // |A×B| = |A| |B| SIN(?)

        return Math.Atan2(Cross(A,B), Dot(A,B));
    }

    public double Magnitude { get { return Math.Sqrt(Dot(this,this)); } }

    public static double Dot(Vector2 A, Vector2 B)
    {
        return A.X*B.X+A.Y*B.Y;
    }
    public static double Cross(Vector2 A, Vector2 B)
    {
        return A.X*B.Y-A.Y*B.X;
    }
}
class Program
{
    static void Main(string[] args)
    {
        Vector2 A=new Vector2() { X=5.45, Y=1.12};
        Vector2 B=new Vector2() { X=-3.86, Y=4.32 };

        double angle=Vector2.GetAngle(A, B) * 180/Math.PI;
        // angle = 120.16850967865749
    }
}

See test case above in GeoGebra.

You don't have to use atan2 to calculate the angle between two vectors. If you just want the quickest way, you can use dot(v1, v2)=|v1|*|v2|*cos A to get

A = Math.acos( dot(v1, v2)/(v1.length()*v2.length()) );