SymPy求解含绝对值的复变量不等式及绘图问题求助

Hey there! Let’s break down how to fix this issue you’re facing with SymPy and complex number point sets—since you’re trying to plot the set of complex numbers ( z = a + bi ) satisfying ( |z - i| < 10 ), here are some practical alternatives to get past those SymPy roadblocks:

1. Skip SymPy Solving Entirely (Use Geometric Logic + Matplotlib) #

First off, your inequality ( \sqrt{a^2 + (b-1)^2} < 10 ) is just an open disk on the complex plane: center at ( (0,1) ) (since that’s ( 0 + 1i )), radius 10. You don’t need to mess with solveset at all—just plot it directly with matplotlib:

import matplotlib.pyplot as plt
import numpy as np

# Create a grid of real (a) and imaginary (b) values
a_vals = np.linspace(-10, 10, 400)
b_vals = np.linspace(-9, 11, 400)
a_grid, b_grid = np.meshgrid(a_vals, b_vals)

# Check which points satisfy the inequality
in_disk = np.sqrt(a_grid**2 + (b_grid - 1)**2) < 10

# Plot the region
plt.figure(figsize=(6,6))
plt.contourf(a_grid, b_grid, in_disk, cmap='coolwarm', levels=[0,1])
plt.scatter(0, 1, color='black', marker='x', label='Center (0, 1)')
plt.xlabel('Real Part (a)')
plt.ylabel('Imaginary Part (b)')
plt.title('Open Disk: |z - i| < 10')
plt.legend()
plt.axis('equal')
plt.show()

This is fast, intuitive, and gives you a clean plot right away.

2. Use SymPy’s Implicit Plotting (No Solveset Needed) #

If you want to stick with SymPy for the symbolic side, use plot_implicit to render the inequality directly. This bypasses the ConditionSet problem entirely:

from sympy import symbols, sqrt, Lt, plot_implicit

# Define real variables for the complex parts
a, b = symbols('a b', real=True)
# Your inequality rewritten symbolically
disk_inequality = Lt(sqrt(a**2 + (b - 1)**2), 10)

# Plot it directly
plot_implicit(
    disk_inequality,
    (a, -10, 10),
    (b, -9, 11),
    title='|z - i| < 10',
    xlabel='Real Part (a)',
    ylabel='Imaginary Part (b)'
)

SymPy handles the real-domain constraint internally here, so you don’t have to wrestle with solveset errors.

3. Working with ConditionSet (If You Must) #

If you really need to extract usable points from the ConditionSet that solveset returns, you can generate random points that satisfy the condition. It’s not the most efficient, but it works:

from sympy import symbols, sqrt, Lt, solveset, S, random

a, b = symbols('a b', real=True)
disk_inequality = Lt(sqrt(a**2 + (b - 1)**2), 10)
solution_set = solveset(disk_inequality, (a, b), domain=S.Reals)

# Function to generate a valid point in the set
def get_valid_point():
    while True:
        a_val = random.uniform(-10, 10)
        b_val = random.uniform(-9, 11)
        if sqrt(a_val**2 + (b_val - 1)**2) < 10:
            return (a_val, b_val)

# Generate 50 sample points
sample_points = [get_valid_point() for _ in range(50)]
print("Sample valid points:", sample_points[:5])

Just keep in mind—this is only useful if you need discrete points, not the full continuous region.

Why You Ran Into That Error #

The core issue here is that solveset is designed for finding symbolic solutions, but your inequality describes a continuous region, not a finite set of points. SymPy returns a ConditionSet because there’s no way to write this solution as a simple interval or list of points. Direct plotting tools are made exactly for this kind of problem, so they’re always the better bet.

内容的提问来源于stack exchange，提问作者Moritz Lehner