## How to check when a vector has made one turns in python

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I am actually working on data that represents a (roughly) noisy circle in a 2d space. i acquire data one point at a time and the goal is know when the points have made a circle.

To do so, i considered each successive points as only one vector that has turned a bit. And to know when one turn has been made (meaning when the circle is formed), I check when both x and y coordinate has change there sign twice (meaning the vector has made 2*0.5 turn = 1 turn). then i wait one more half turn to compensate the starting error. Indeed, depending on where it has started in the quarter of the space it was at first, it may have not do a whole turn.

I don't need to be extremely precise. So this is kind of fine for me, but i wonder if there is another method thas is more efficient and that tells the real number of turns. This could speed up a bit the process as the points arrives quit slowly (avoiding me to wait one more useless half turn)

One important point is that i can only use Numpy.

EDIT : more precision, the distance between each point is NOT regular. At first, the circle starts to be formed slowly and it then speed up. So, at the beginning the points are more dense than at the end. Another thing is that the (0,0) point may even not be contained in the circle. Finally, i said roughly circular shaped because it tends to be ellipsis shaped, but not badly formed, just noisy.

And sorry but i can't provide data, at least for now. I'll tell you if it is possible during the week.

You can monitor the distance of each point to the first point and when this distance reaches a minimum it means the circle has closed. The following shows a plot of point distances to the first point along the circle: This is the relevant code for the algorithm:

```distances = []
points = [next_point()]
while True:  # break below
points.append(next_point())
distances.append(np.linalg.norm(points[-1] - points))
if len(distances) >= 3:
left = distances[-2] - distances[-3]
right = distances[-1] - distances[-2]
if left < 0 and right > 0:  # minimum detected
break
del points[-1], distances[-1]  # optionally delete the last point in order to leave the circle open
```

Testing on a data set which varies both the angle difference and the radius the following result is obtained: This is the full example code:

```from math import pi
import random

import matplotlib.pyplot as plt
import numpy as np

def generate():
angle = pi/4
angle_upper_lim = 0.002
while True:
angle += 2*pi * random.uniform(0.001, angle_upper_lim)
angle_upper_lim *= 1.03  # make the circle fill faster

generator = generate()

def next_point(n=1):
"""n: number of points per group"""
return sum(next(generator) for __ in range(n)) / n

distances = []
points = [next_point()]
while True:  # break below
points.append(next_point())
distances.append(np.linalg.norm(points[-1] - points))
if len(distances) >= 3:
left = distances[-2] - distances[-3]
right = distances[-1] - distances[-2]
if left < 0 and right > 0:  # minimum detected
break
del points[-1], distances[-1]  # optionally delete the last point in order to leave the circle open

fig, (ax1, ax2) = plt.subplots(ncols=2, figsize=(10.8, 4.8))
ax1.set_title('Data points')
ax1.scatter(*np.stack(points, axis=1), s=5, c=np.arange(len(points)))
ax1.plot(*points[ 0], 's', ms=8, label='First point', color='#2ca02c')
ax1.plot(*points[-1], '*', ms=12, label='Last point', color='#ff7f0e')
ax1.legend(loc='center')

ax2.set(title='Distance of circle points to first point', xlabel='Point #', ylabel='Distance')
ax2.yaxis.tick_right()
ax2.yaxis.set_label_position('right')
ax2.plot(distances, '-o', ms=4)
ax2.plot(len(distances)-1, distances[-1], '*', ms=10, label='circle closed')
ax2.legend()

plt.show()
```

In case the radius of data points varies as well it is important to choose a window of sufficient size which will group and average consecutive data points for greater stability. The function `next_point` can be adjusted by using `n=5` for example. The following result is obtained by uncommenting the radius variation in the above code and using a window size of `n=5`: Data science with Python: Turn your conditional loops to Numpy , Numpy, short for Numerical Python, is the fundamental package required heavy use of linear algebra operations on a long list/vector/matrix of numbers). in Python, pointer indirection and per-element dynamic type checking. And it turns out one can easily vectorize simple blocks of conditional loops� For example, maybe you want to plot column 1 vs column 2, or you want the integral of data between x = 4 and x = 6, but your vector covers 0 < x < 10. Indexing is the way to do these things. A key point to remember is that in python array/vector indices start at 0. Unlike Matlab, which uses parentheses to index a array, we use brackets in python.

If each new data point is guaranteed to have a greater polar angle than the previous one, i.e. the circle is incrementally formed without any point "stepping-back" in the procedure, then for each pair of consecutive points you can compute the angle between them and then you can stop when the sum reaches two pi. For example:

```angle = 0
points = [next(generator)]  # 'generator' produces the data points
while angle < 2*pi:
points.append(next(generator))
angle += np.arccos(
np.dot(points[-2], points[-1]) /
(np.linalg.norm(points[-2]) * np.linalg.norm(points[-1]))
)
del points[-1]  # optionally delete the last point in order to stay below 2 pi
```

Here's an example plot using the above method: And the example code:

```from math import pi
import random

import matplotlib.pyplot as plt
import numpy as np

def generate():
angle = 0
angle_upper_lim = 0.002
while True:
angle += 2*pi * random.uniform(0.001, angle_upper_lim)
angle_upper_lim *= 1.03  # make the circle fill faster

generator = generate()

angle = 0
points = [next(generator)]  # 'generator' produces the data points
while angle < 2*pi:
points.append(next(generator))
angle += np.arccos(
np.dot(points[-2], points[-1]) /
(np.linalg.norm(points[-2]) * np.linalg.norm(points[-1]))
)
del points[-1]  # optionally delete the last point in order to stay below 2 pi

fig, ax = plt.subplots()
ax.scatter(*np.stack(points, axis=1), s=5)
ax.set_title(f'Total angle: {angle/pi:.2f} pi')
ax.plot([0, points[ 0]], [0, points[ 0]], '--s', ms=8, label='First point', color='#2ca02c')
ax.plot([0, points[-1]], [0, points[-1]], '--*', ms=12, label='Last point', color='#ff7f0e')
ax.legend()

plt.show()
```

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You can use the first point as an offset which will be subtracted from all the points. This shifts the edge of the circle to the origin. Now imagine a tangent to the circle (at any point, but specifically we will be using the origin), then the circle lies completely on one side of the tangent. The tangent itself spans 180 degrees and if we walk along the circle starting from the origin, always measuring the angle between consecutive vectors, we will have measured 180 degrees in total once we arrive back to the origin (in case the points on the circle are infinitesimally spaced). This allows to compute the accumulated sum of angles and stop when it reaches 180 degrees (= pi). Now since in reality the points have a finite spacing we will miss some fraction of the 180 degrees at the beginning and the end of the circle (w.r.t. the origin). This implies that when we reach 180 degrees we will have collected slightly more points than are necessary in order to close the circle; the OP indicates that this is desired behavior, i.e. the circle must be closed (better some overlap than not closed).

This is the relevant code for the algorithm:

```angle = 0
offset = next(generator)  # 'generator' produces the data points
points = [next(generator) - offset]
while angle <= pi:
points.append(next(generator) - offset)
angle += np.arccos(
np.dot(points[-2], points[-1]) /
(np.linalg.norm(points[-2]) * np.linalg.norm(points[-1]))
)
del points[-1]  # optionally delete the last point in order to stay below pi
```

The following plot shows an example where the radius of each point is exactly the same, only the polar angle varies: This is the complete example code:

```from math import pi
import random

import matplotlib.pyplot as plt
import numpy as np

def generate():
angle = pi/4
angle_upper_lim = 0.002
while True:
angle += 2*pi * random.uniform(0.001, angle_upper_lim)
angle_upper_lim *= 1.03  # make the circle fill faster

generator = generate()

angle = 0
offset = next(generator)  # 'generator' produces the data points
points = [next(generator) - offset]
while angle <= pi:
points.append(next(generator) - offset)
angle += np.arccos(
np.dot(points[-2], points[-1]) /
(np.linalg.norm(points[-2]) * np.linalg.norm(points[-1]))
)
del points[-1]  # optionally delete the last point in order to stay below pi

fig, ax = plt.subplots(figsize=(4.8, 4.8))
ax.scatter(*np.stack(points, axis=1), s=5, c=np.arange(len(points)))
ax.set_title(f'Total angle: {angle/pi:.2f} pi')
ax.plot(*points[ 0], 's', ms=8, label='First point', color='#2ca02c')
ax.plot(*points[-1], '*', ms=12, label='Last point', color='#ff7f0e')
ax.legend()

plt.show()
```

If however the data points also have varying radius the method suffers from these variations as viewed from the origin. In this case consecutive data points can be grouped together and then using the mean value for greater stability: This is the code using a grouping of data points for greater stability:

```from math import pi
import random

import matplotlib.pyplot as plt
import numpy as np

def generate():
angle = pi/4
angle_upper_lim = 0.002
while True:
angle += 2*pi * random.uniform(0.001, angle_upper_lim)
angle_upper_lim *= 1.03  # make the circle fill faster

generator = generate()

def next_point(n=5):
"""n: number of points per group"""
return sum(next(generator) for __ in range(n)) / n

angle = 0
offset = next(generator)  # 'generator' produces the data points
points = [next_point() - offset]
while angle <= pi:
points.append(next_point() - offset)
angle += np.arccos(
np.dot(points[-2], points[-1]) /
(np.linalg.norm(points[-2]) * np.linalg.norm(points[-1]))
)
del points[-1]  # optionally delete the last point in order to stay below pi

fig, ax = plt.subplots(figsize=(4.8, 4.8))
ax.scatter(*np.stack(points, axis=1), s=5, c=np.arange(len(points)))
ax.set_title(f'Total angle: {angle/pi:.2f} pi')
ax.plot(*points[ 0], 's', ms=8, label='First point', color='#2ca02c')
ax.plot(*points[-1], '*', ms=12, label='Last point', color='#ff7f0e')
ax.legend(loc='center')

plt.show()
```

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