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How can I find the area inside two unsorted x,y arrays?

I want to find the are inside the region limited by two unsorted arrays (x and y).
If they were sorted, I could just follow this example:
theta = np.linspace(0, 2 * np.pi, num=1000, endpoint=True)
x = np.sin(theta)
y = np.cos(theta)
answer = np.trapz(y, x=x)
In which the x and y array are correctly sorted in a way that allows trapz to correctly function (even if endpoint=False).
However, in my data x and y are not sorted. I would like to find the area enclosed by x and y as given in the following example:
theta = np.linspace(0, 2 * np.pi, num=1000, endpoint=True)
ii = np.arange(len(x))
np.random.shuffle(ii)
x = np.sin(theta)[ii]
y = np.cos(theta)[ii]
answer = np.trapz(y, x=x) #This no longer gives the correct integral.
Is there a way to find the area enclosed by the arrays without sorting by angular position? It doesn't have to be using trampz. Thank you

I have yet to try Convex Hull, but ended up simply sorting the points by angular position and (under the assumption that points actually enclose a surface) just integrated using trapz. I include in this answer the method I used.
def integrate_contours(x,y):
xx = x - x.mean()
yy = y - y.mean()
theta_2 = np.arctan2(xx,yy)
yy = yy[np.argsort(theta_2)]
xx = xx[np.argsort(theta_2)]
return np.abs(np.trapz(yy, x=xx))

Evaluate function in points inside half sphere and plot slides in Python

I am trying to evaluate a function that depends on the radius from the center of a sphere to any point inside half a sphere.
I start by defining three arrays corresponding to the points along the radius, the elevation and azimuthal angles. In a for loop I compute the x, y and z coordinates to evaluate the function.
I am not sure if I am doing the mapping properly. I need to store the values of the evaluated function in a 3D matrix corresponding to the x, y, and z coordinates to plot slices in a postprocessing step, but I am stuck identifying how I can define the size of my function matrix.
In cartesian coordinates is really easy since one can link every coordinate with the dimension of the matrix. That's why I need some guidance in how I can slide the matrix since I don't have a 3D matrix with the cartesian coordinates. How I can construct this matrix from the spherical coordintaes?
Any help will be more than appreciated!
Here is my (unfruitful) attempt:
import numpy as np
beta = 1
rho = np.linspace(0, 1, 20)
phi = np.linspace(0, 2*np.pi, 20)
theta = np.linspace(0, np.pi/2, 10)
f = np.empty([len(theta), len(theta), len(phi)], dtype=complex)
for i in range(len(rho)):
for j in range(len(phi)):
for k in range(len(theta)):
x = rho[i] * np.sin(theta[k]) * np.cos(phi[j])
y = rho[i] * np.sin(theta[k]) * np.sin(phi[j])
z = rho[i] * np.cos(theta[k])
R = np.sqrt(x**2 + y**2 + z**2)
f[k, i, j] = -1j*((z/R)/(z/R + beta)) * (np.exp(1j*k*R)/R)

You just have a typo, the second dimension is again len(theta) isntead of len(rho). It should be
f = np.empty([len(theta), len(rho), len(phi)], dtype=complex)
Note also that, if I am not mistaken, you don't need R at all, it's just rho[i].

How to use mgrid to interpolate between a rectangle and a circle

I am trying to create a 3D surface that has a 1/4 rectangle for the exterior and 1/4 circle for the interior. I had help before to create the 3D surface with an ellipse as an exterior but I cannot do this for a rectangle for some reason. I have done the math by hand which makes sense, but my code does not. I would greatly appreciate any help with this.
import numpy as np
import pyvista as pv
# parameters for the waveguide
# diameter of the inner circle
waveguide_throat = 30
# axes of the outer ellipse
ellipse_x = 250
ellipse_y = 170
# shape parameters for the z profile
depth_factor = 4
angle_factor = 40
# number of grid points in radial and angular direction
array_length = 100
phase_plug = 0
phase_plug_dia = 20
plug_offset = 5
dome_dia = 28
# theta is angle where x and y intersect
theta = np.arctan(ellipse_x / ellipse_y)
# chi is for x direction and lhi is for y direction
chi = np.linspace(0, theta, 100)
lhi = np.linspace(theta, np.pi/2, 100)
# mgrid to create structured grid
r, phi = np.mgrid[0:1:array_length*1j, 0:np.pi/2:array_length*1j]
# Rectangle exterior, circle interior
x = (ellipse_y * np.tan(chi)) * r + ((waveguide_throat / 2 * (1 - r)) * np.cos(phi))
y = (ellipse_x / np.tan(lhi)) * r + ((waveguide_throat / 2 * (1 - r)) * np.sin(phi))
# compute z profile
angle_factor = angle_factor / 10000
z = (ellipse_x / 2 * r / angle_factor) ** (1 / depth_factor)
plotter = pv.Plotter()
waveguide_mesh = pv.StructuredGrid(x, y, z)
plotter.add_mesh(waveguide_mesh)
plotter.show()

The linear interpolation you're trying to use is a general tool that should work (with one small caveat). So the issue is first with your rectangular edge.
Here's a sanity check which plots your interior and exterior lines:
# debugging: plot interior and exterior
exterior_points = np.array([
ellipse_y * np.tan(chi),
ellipse_x / np.tan(lhi),
np.zeros_like(chi)
]).T
phi_aux = np.linspace(0, np.pi/2, array_length)
interior_points = np.array([
waveguide_throat / 2 * np.cos(phi_aux),
waveguide_throat / 2 * np.sin(phi_aux),
np.zeros_like(phi_aux)
]).T
plotter = pv.Plotter()
plotter.add_mesh(pv.wrap(exterior_points))
plotter.add_mesh(pv.wrap(interior_points))
plotter.show()
The bottom left is your interior circle, looks good. The top right is what's supposed to be a rectangle, but isn't.
To see why your original surface looks the way it does, we have to note one more thing (this is the small caveat I mentioned): the orientation of your curves is also the opposite. This implies that you interpolate the "top" (in the screenshot) point of your interior curve with the "bottom" point of the exterior curve. This explains the weird fan shape.
So you need to fix the exterior curve, and make sure the orientation of the two edges is the same. Note that you can just create the two 1d arrays for the two edges, and then interpolate them. You don't have to come up with a symbolic formula that you plug into the interpolation step. If you have 1d arrays of the same shape x_interior, y_interior, x_exterior, y_exterior then you can then do x_exterior * r + x_interior * (1 - r) and the same for y. This means removing the mgrid call, only using an array r of shape (n, 1), and making use of array broadcasting to do the interpolation. This means doing r = np.linspace(0, 1, array_length)[:, None].
So the question is how to define your rectangle. You need to have the same number of points on the rectangular curve than what you have on the circle (I would strongly recommend using the array_length parameter everywhere to ensure this!). Since you want to span the whole rectangle, I believe you have to choose an array index (i.e. a certain angle in the circular arc) which will map to the corner of the rectangle. Then it's a simple matter of varying only y for the points until that index, and x for the rest (or vice versa).
Here's what I mean: you know that the rectangle's corner is at angle theta in your code (although I think you have x and y mixed up if we assume the conventional relationship between "x", "y" and the tangent of the angle). Since theta goes from 0 to pi/2, and your phi values also go from 0 to pi/2, you should choose index (array_length * (2*theta/np.pi)).round().astype(int) - 1 (or something similar) that will map to the rectangle's corner. If you have a square, this gives you theta = pi/4, and consequently (array_length / 2).round().astype(int) - 1. For array_length = 3 this is index (2 - 1) == 1, which is the middle index for 3-length arrays. (The more points you have along the edge, the less it will matter if you commit an off-by-one error here.)
The only remaining complication then is that we have to explicitly broadcast the 1d z array to the common shape. And we can use the same math you used to get a rectangular edge that is equidistant in angles.
Your code fixed with this suggestion (note that I've added 1 to the corner index because I'm using it as a right-exclusive range index):
import numpy as np
import pyvista as pv
# parameters for the waveguide
# diameter of the inner circle
waveguide_throat = 30
# axes of the outer ellipse
ellipse_x = 250
ellipse_y = 170
# shape parameters for the z profile
depth_factor = 4
angle_factor = 40
# number of grid points in radial and angular direction
array_length = 100
# quarter circle interior line
phi = np.linspace(0, np.pi/2, array_length)
x_interior = waveguide_throat / 2 * np.cos(phi)
y_interior = waveguide_throat / 2 * np.sin(phi)
# theta is angle where x and y intersect
theta = np.arctan2(ellipse_y, ellipse_x)
# find array index which maps to the corner of the rectangle
corner_index = (array_length * (2*theta/np.pi)).round().astype(int)
# construct rectangular coordinates manually
x_exterior = np.zeros_like(x_interior)
y_exterior = x_exterior.copy()
phi_aux = np.linspace(0, theta, corner_index)
x_exterior[:corner_index] = ellipse_x
y_exterior[:corner_index] = ellipse_x * np.tan(phi_aux)
phi_aux = np.linspace(np.pi/2, theta, array_length - corner_index, endpoint=False)[::-1] # mind the reverse!
x_exterior[corner_index:] = ellipse_y / np.tan(phi_aux)
y_exterior[corner_index:] = ellipse_y
# interpolate between two curves
r = np.linspace(0, 1, array_length)[:, None] # shape (array_length, 1) for broadcasting
x = x_exterior * r + x_interior * (1 - r)
y = y_exterior * r + y_interior * (1 - r)
# debugging: plot interior and exterior
exterior_points = np.array([
x_exterior,
y_exterior,
np.zeros_like(x_exterior),
]).T
interior_points = np.array([
x_interior,
y_interior,
np.zeros_like(x_interior),
]).T
plotter = pv.Plotter()
plotter.add_mesh(pv.wrap(exterior_points))
plotter.add_mesh(pv.wrap(interior_points))
plotter.show()
# compute z profile
angle_factor = angle_factor / 10000
z = (ellipse_x / 2 * r / angle_factor) ** (1 / depth_factor)
# explicitly broadcast to the shape of x and y
z = np.broadcast_to(z, x.shape)
plotter = pv.Plotter()
waveguide_mesh = pv.StructuredGrid(x, y, z)
plotter.add_mesh(waveguide_mesh, style='wireframe')
plotter.show()
The curves look reasonable:
As does the interpolated surface:

How to use a smooth curve to link points approximately distributing in a circle?

I have a set of twelve points, which center at (0, 0) and distribute approximately in a circle, at the interval of 30 degrees, shown in the image.
The twelve points
I want to use a smooth curve to link (go through) them like the image below (I draw the red line by hand).
a hand-drawn curve in red
I want to make it in python or matlab. I have tried some interpolation methods for the upper half and lower half separately, and wanted to combine them as a complete curve. However, the results always overshoot.
Thank you for any suggestions!

I think the key here is to note that you have to consider it as a parametrized curve in 2d, not just a 1d to 2d function. Furthermore since it should be something like a circle, you need an interpolation method that supports periodic boundaries. Here are two methods for which this applies:
% set up toy data
t = linspace(0, 2*pi, 10);
t = t(1:end-1);
a = 0.08;
b = 0.08;
x = cos(t+a*randn(size(t))) + b*randn(size(t));
y = sin(t+a*randn(size(t))) + b*randn(size(t));
plot(x, y, 'ok');
% fourier interpolation
z = x+1i*y;
y = interpft(z, 200);
hold on
plot(real(y), imag(y), '-.r')
% periodic spline interpolation
z = [z, z(1)];
n = numel(z);
t = 1:n;
pp = csape(t, z, 'periodic');
ts = linspace(1, n, 200);
y = ppval(pp, ts);;
plot(real(y), imag(y), ':b');

Thank for suggestions from #flawr. According to the answer from #flawr, I implemented the periodic spline interpolation in python (still working on implementing fourier interpolation in python.). Here is the code:
import numpy as np
import matplotlib.pyplot as plt
from scipy.interpolate import CubicSpline
# set up toy data
t = np.linspace(0, 2*np.pi, 10)
t = t[0:-1]
a = 0.08
b = 0.08
x = np.cos(t + a * np.random.normal(size=len(t))) + b * np.random.normal(size=len(t))
y = np.sin(t + a * np.random.normal(size=len(t))) + b * np.random.normal(size=len(t))
plt.scatter(x, y)
# periodic spline interpolation
z = []
for idx in range(len(x)):
z.append(complex(x[idx], y[idx]))
z.append(complex(x[0], y[0]))
len_z = len(z)
t = [i for i in range(len_z)]
cs = CubicSpline(t, z, bc_type='periodic')
xs = np.linspace(0, len_z, 200)
y_new = cs(xs)
plt.plot(y_new.real, y_new.imag)
plt.show()

Buggy behavior in matplotlib pcolormesh with FFT arrays?

I am trying to reproduce the example of the Gabor transform that it is in his wikipedia entry, and I do not know if it is a bug or I am missing something. The example is the calculate the Gabor transform of a sinusoidal signal:
To plot the frequencies sorted, I create an unsorted axis. Then I use mesh grid to create 2D axes and plot with pcolormesh. Here is the piece of the code:
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.gridspec as gridsp
dt = 0.05
x = np.arange(-50.0,50.0,dt)
y = np.sin(2.0 * np.pi * x)
Nx = len(x)
w = np.fft.fftfreq(Nx,dt)
sigma = 1.0 / 3.0
neg = np.where (x <= 0.0)
pos = np.where (x > 0.0)
T,W = np.meshgrid(x,w)
func = np.zeros(Nx)
tmp = np.zeros(Nx,dtype='complex64')
gabor = np.zeros((Nx,Nx))
func[neg] = np.sin(2.0 * np.pi * x[neg])
func[pos] = np.sin(4.0 * np.pi * x[pos])
for it in range(Nx):
tmp[:] = np.fft.fft(func[:] * np.exp( - ( x[it] - x[:] ) * ( x[it] - x[:] ) / 2.0 / sigma / sigma ) )
gabor[:,it] = np.real(np.conj(tmp) * tmp)
fig = plt.figure(figsize=(20,10),facecolor='white')
gs = gridsp.GridSpec(2, 1)
ax1 = plt.subplot(gs[0,0])
ax1.plot(x,func,'r',linewidth=2)
ax1.axis('tight')
ax1.set_xticks(np.arange(min(x),max(x),1.) )
ax1.set_xlabel('time',fontsize=20)
ax1.set_ylabel(r'$\sin{time}$',fontsize=20)
ax1.set_xlim([-6.0,6.0])
ax2 = plt.subplot(gs[1,0])
surf1 = ax2.pcolormesh(T,W,gabor,shading='gouraud')
ax2.axis('tight')
ax2.set_xticks(np.arange(min(x),max(x),2.) )
ax2.set_yticks(np.arange(min(w),max(w),2.) )
ax2.set_xlabel('time',fontsize=20)
ax2.set_ylabel('frequency',fontsize=20)
ax2.set_xlim([-6.0,6.0])
ax2.set_ylim([-4.0,4.0])
gs.tight_layout(fig)
plt.show()
Here is the figure I get,
It seems that the upper part of the plot is reduced to zero. If I try it using fftshift when I create the transform and the axis,
for it in range(Nx):
tmp[:] = np.fft.fftshift(np.fft.fft(func[:] * np.exp( - ( x[it] - x[:] ) * ( x[it] - x[:] ) / 2.0 / sigma / sigma ) ) )
gabor[:,it] = np.real(np.conj(tmp) * tmp)
T,W = np.meshgrid(x,np.fft.fftshift(w))
Then I get this figure:
!
It seems that pcolormesh routine can not flip upside down the array as it is usually done in 1D plots. does anybody know exactly why it is doing this?
Thanks,
Alex

The problem lies in W. Or actually in w. When w is plotted:
Thus pcolormesh receives non-monotonic Y coordinates and gets confused. If you look at the description of pcolor or pcolormesh it is clear they cannot do anything reasonable with non-monotonic data.
So, your gabor is fine:
ax.imshow(gabor)
as you can see:
There are several possibilities how to fix this. One of them is to feed both W and gabor to fftshift that way the frequencies will roll back to monotonic. Or - if you want to have the figure as above (negative frequencies on the top), just add the maximum frequency to all negative values of W.
It might also be cleaner to supply pcolormesh with x and w instead of T and W.
If you want performance, you might be better of with imshow (it can be used when the data is equispaced in both dimensions. The only slight problem is the calculation of extents (which actually may be slightly off even in the question). The extents tell the outer limets of the highest, lowest, leftmost and rightmost pixels. However, the pixel vectors only tell the centers of the pixels.
We need to know the following:
number of points in X direction (num_x)
number of points in Y direction (num_y)
value of the first and last x sample (x0, x1)
value of the first and last y sample (y0, y1)
After that we can use imshow to show the data with correct scaling:
dx = 1. * (x1 - x0) / (num_x-1)
dy = 1. * (y1 - y0) / (num_y-1)
ax.imshow(img, extent=[x0 - dx/2, x1 + dx/2, y0 - dy/2, y1 + dy/2], origin='lower', interpolation='nearest')
So, applied to the question's data:
gabor_shifted = np.fft.fftshift(gabor, axes=0)
w_shifted = np.fft.fftshift(w)
x0 = x[0]
x1 = x[-1]
w0 = w_shifted[0]
w1 = w_shifted[-1]
dx = 1.*(x1-x0) / (len(x) - 1)
dw = 1.*(w1-w0) / (len(w) - 1)
ax2.imshow(gabor_shifted, extent=[x0-dx/2, x1+dx/2, w0-dw/2, w1+dw/2], interpolation='nearest', origin='lower', aspect='auto')
ax2.grid('on', color='w')
ax2.ylim(-4,4)
which gives: