I would like to use the nice circle that the polar projection gives, but I have cartesian data. The best I can do now is the following, but it has a notch thing.
fig, ax = plt.subplots(subplot_kw={'projection': 'polar'})
xs, ys = np.meshgrid(np.linspace(-1,1,50),np.linspace(-1,1,50),)
parab = xs**2+ys**2
rs = np.sqrt(xs**2+ys**2)
ths = np.arctan2(ys, xs)
ax.pcolormesh(ths, rs, parab, shading='nearest')
The parameter shading='nearest' looks like unsupported in polar projection. I manualy center the color C[i,j] on (ths[i,j], rs[i,j]) by using an other meshgrid xs_ ys_
fig, ax = plt.subplots(subplot_kw={'projection': 'polar'})
n = 50
xs, ys = np.meshgrid(np.linspace(-1,1,n),np.linspace(-1,1,n))
dx, dy = 0.5*np.ptp(xs)/(n-1), 0.5*np.ptp(ys)/(n-1)
xs_,ys_ = np.meshgrid(np.linspace(-1-dx,1+dx,n+1),np.linspace(-1-dx,1+dx,n+1))
parab = (xs_+dx)**2+(ys_+dx)**2
rs = np.sqrt(xs**2+ys**2)
ths = np.arctan2(ys, xs)
rs_ = np.sqrt(xs_**2+ys_**2)
ths_ = np.arctan2(ys_, xs_)
ax.pcolormesh(ths_, rs_, parab)
plt.show()
output :
