如何从 SymPy 中提取椭圆的中心坐标、高度、宽度和 phi 以绘制拟合椭圆？

Question

I have been working with lsq-ellipse package where I get the coordinates of ellipse with the following code below:

from ellipse import LsqEllipse
from matplotlib.patches import Ellipse
coords_D0 = np.array(coords_D0)
reg = LsqEllipse().fit(coords_D0)
center_D0, width_D0, height_D0, phi_D0 = reg.as_parameters()
print(f'center: {center_D0[0]:.3f}, {center_D0[1]:.3f}')
print(f'width: {width_D0:.3f}')
print(f'height: {height_D0:.3f}')
print(f'phi: {phi_D0:.3f}')

However, my coords_D0 variable consists of three coordinates which caused the following error:

ValueError: Received too few samplesGot 3 features, 5 or more required.

But, after looking into some packages and online, I found that sympy also can do Ellipse and I understand that you can extract the centre, vradius and hradius from sympy . But, I would like to know how to get the width, height and phi from sympy and will it be the same as the lsq-ellipse package to be used in Ellipse of matplotlib? I use the values from lsq-ellipse package in matplotlib to form the ellipse part and it can be found in the following code line:

Code:

ellipse_D0 = Ellipse(xy=center_D0, width=2*width_D0, height=2*height_D0, angle=np.rad2deg(phi_D0),edgecolor='b', fc='None', lw=2, label='Fit', zorder=2)

My coordinates are the following:

coords_D0 =
-1.98976     -1.91574
-0.0157721    2.5438
2.00553      -0.628061

# another points
coords_D1 =
-0.195518   0.0273673
-0.655686   -1.45848
-0.447061   -0.168108

# another points
coords_D2 =
-2.28529    0.91896
-2.43207    0.446211
-2.23044    0.200087

Side Question:

Is there a way to fit an ellipse to these coordinates (in general, 3 coordinates or more)?

Answer 1

Assuming that the OP is about the Minimum Volume Enclosing Ellipse, I'd suggest the following solution.

#! /usr/bin/python3
# coding=utf-8
import matplotlib.pyplot as plt
from matplotlib.patches import Ellipse

import numpy as np

from mymodules3.mvee import mvee

coords= list()
coords.append( np.array([
    -1.98976,     -1.91574,
    -0.0157721,    2.5438,
    2.00553,      -0.628061
]).reshape(3,-1) )

coords.append(  np.array([
    -0.195518,   0.0273673,
    -0.655686,   -1.4584,8
    -0.447061,   -0.168108,
]).reshape(3,-1)
)

coords.append( np.array([
    -2.28529,    0.91896,
    -2.43207,    0.446211,
    -2.23044,    0.200087
]).reshape(3,-1)
)


fig = plt.figure()
ax = fig.add_subplot( 1, 1, 1 )

for i, col in enumerate( ['k', 'g', 'm'] ):
    sol = mvee( coords[i] )

    e =Ellipse(
        sol[0],
        width=2 * sol[1][0],
        height=2 * sol[1][1],
        angle=sol[2] *  180/3.1415926,
        color=col, alpha=0.5,
        zorder = -1000 + i
    )
    ax.scatter( coords[i][:,0], coords[i][:,1], c=col, zorder=10 * i )
    ax.add_artist( e )
plt.show()

providing

The mvee is is based on an SE answer on a similar question.

"""
NMI : 2021-11-11 
Minimum Volume Enclosing Ellipsoids, see e.g.
NIMA MOSHTAGH : MINIMUM VOLUME ENCLOSING ELLIPSOIDS
or 
Linus Källberg : Minimum_Enclosing_Balls_and_Ellipsoids (Thesis)
"""
from warnings import warn

from numpy import pi
from numpy import sqrt
from numpy import arccos
from numpy import dot, outer
from numpy import diag, transpose
from numpy import append
from numpy import asarray
from numpy import ones
from numpy import argmax

from numpy.linalg import inv
from numpy.linalg import norm
from numpy.linalg import eig


def mvee( data, tolerance=1e-4, maxcnt=1000 ):
    """
    param data: list of xy data points
    param tolerance: termination condition for iterative approximation
    param maxcnt: maximum number of iterations
    type data: iterable of float
    type tolerance: float
    return: (offset, semiaxis, angle)
    return type: ( (float, float), (float, float), float )
    """
    locdata = asarray( data )
    N = len( locdata )
    if not locdata.shape == ( N, 2):
        raise ValueError ( " data must be of shape( n, 2 )" )
    if tolerance >= 1 or tolerance <= 0:
        raise ValueError (" 0 < tolerance < 1 required")
    if not isinstance( maxcnt, int ):
        raise TypeError
    if not maxcnt > 0:
        raise ValueError
    count = 1
    err = 1
    d = 2
    d1 = d + 1
    u = ones( N ) / N
    P = transpose( locdata )
    Q = append( P, ones( N ) ).reshape( 3, -1 )
    while ( err > tolerance):
        X = dot( Q, dot( diag( u ), transpose( Q ) ) )
        M = diag( 
            dot( 
                transpose( Q ),
                dot(
                    inv( X ),
                    Q
                )
            )
        )
        maximum = max( M )
        j = argmax( M )
        step_size = ( maximum - d1 ) / ( d1 * ( maximum - 1 ) )
        new_u = ( 1 - step_size ) * u
        new_u[ j ] += step_size
        err = norm( new_u - u )
        count = count + 1
        u = new_u
        if count > maxcnt:
            warn(
                "Process did not converge in {} steps".format(
                    count - 1
                ),
                UserWarning
            )
            break
    U = diag( u )
    c = dot( P,  u )
    A = inv(
        dot(
            P,
            dot( U, transpose( P ) )
        ) - outer( c, c )
    ) / d
    E, V = eig( A )
    phiopt = arccos( V[ 0, 0 ] )
    if V[ 0, 1 ] < 0: 
        phiopt = 2 * pi - phiopt
    ### cw vs ccw and periodicity of pi
    phiopt = -phiopt % pi
    sol =  (  c, sqrt( 1.0 / E ), phiopt)
    return sol