Algorithm to detect left or right turn from x,y co-ordinates

Question

I have a data set of x,y co-ordinates, starting from origin, recorded each second. I can detect distance, speed,acceleration, modulus of displacement . Is there any algorithm to detect whether a left or right turn ?

I am currently calculating distance and modulus of displacement for every 10 seconds, if the displacement is approximately equal to distance, then the vehicle is on straight path, but of the values change then there is a turn involved.

IS there an algorithm to decide whether the turn was left or right ? My data looks like this

Time   x     y
 0     0     0
 1    -0.2  -0.1
 2    -0.7   0.9
 3    -0.8   0.9
 4    -1     0.8
 5    -1.1   0.8
 6    -1.2   0.7
 7    -1.4   0.7
 8    -1.9   1.7
 9    -2     1.7
10    -2.2   1.6
11    -2.3   1.6
12    -2.5   1.5
13    -2.6   1.5
14    -2.7   1.5
15    -2.9   1.4
16    -3.6   1.2
17    -4.1  -0.1
18    -4.7  -1.5
19    -4.7  -2.6
20    -4.3  -3.7
21    -4.3  -3.7
22    -4.7  -3.8
23    -6.2  -3.1
24    -9.9  -1.9
25   -13.7  -1.9
26   -17.9  -2
27   -21.8  -0.8
28   -25.1  -0.6
29   -28.6   1.8

Answer 1

Looking at 3 points p0, p1 and p2, you can look at the relative orientation of the two vectors p1 - p0 and p2 - p1. An easy way to do this is to calculate the cross product between the two vectors. The x- and y-components of the cross product are 0 because both vectors are in the xy-plane. So only the z-component of the cross product needs to be calculated.

If the z-component of the cross product is positive, you know that the second vector points left relative to the first one, because the first vector, second vector, and a vector in the positive z-direction are right handed. If the cross product is negative, the second vector points to the right relative to the first one.

I used my mad Python skills (I use Python about once a year...) to put this into the code below. There's a little logic so that the Left/Right designation can be printed at the middle point, even though it can only be calculated after the next point was read. To enable that, a couple of previous lines are saved away, with their printing delayed. The actual calculation is in the calcDir() function.

import sys

fileName = sys.argv[1]
dataFile = open(fileName, 'r')

def calcDir(p0, p1, p2):
    v1x = float(p1[0]) - float(p0[0])
    v1y = float(p1[1]) - float(p0[1])
    v2x = float(p2[0]) - float(p1[0])
    v2y = float(p2[1]) - float(p1[1])
    if v1x * v2y - v1y * v2x > 0.0:
        return 'Left'
    else:
        return 'Right'

lineIdx = 0
for line in dataFile:
    line = line.rstrip()
    lineIdx += 1
    if lineIdx == 1:
        print line
    elif lineIdx == 2:
        line0 = line
        print line0
    elif lineIdx == 3:
        line1 = line
    else:
        line2 = line
        dir = calcDir(line0.split()[1:], line1.split()[1:], line2.split()[1:])
        print line1 + ' ' + dir
        line0 = line1
        line1 = line2

print line2

Answer 2

Yes: you want to calculate the dot product of the previous direction and the new direction.

If you start by normalising the two vectors (giving each one a length of 1) then the dot product will be the cosine of the angle between the two vectors, and this will allow you to determine whether it's a left turn or a right turn, and by how much you've turned.

You might also find the further explanation here to be handy.