Draw a parametrized curve using pyplot.plot() in Matplotlib

A parametrized curve is defined by equations where both x and y coordinates are expressed as functions of a parameter (usually t). Matplotlib's pyplot.plot() can easily visualize these curves by plotting the computed x and y coordinates.

Basic Parametrized Curve

Let's create a simple parametrized curve using trigonometric functions ?

import numpy as np
import matplotlib.pyplot as plt

# Set figure size
plt.rcParams["figure.figsize"] = [7.50, 3.50]
plt.rcParams["figure.autolayout"] = True

# Number of sample points
N = 400

# Parameter t from 0 to 2?
t = np.linspace(0, 2 * np.pi, N)

# Parametric equations for a cardioid
r = 0.5 + np.cos(t)
x = r * np.cos(t)
y = r * np.sin(t)

# Create plot
fig, ax = plt.subplots()
ax.plot(x, y, 'b-', linewidth=2)
ax.set_title('Parametrized Curve (Cardioid)')
ax.grid(True)
ax.axis('equal')

plt.show()

The output shows a heart-shaped curve called a cardioid ?

[A plot displaying a cardioid curve - heart-shaped parametric curve]

Different Parametrized Curves

Here are examples of various parametrized curves with different equations ?

import numpy as np
import matplotlib.pyplot as plt

fig, axes = plt.subplots(2, 2, figsize=(10, 8))
fig.suptitle('Different Parametrized Curves')

t = np.linspace(0, 2 * np.pi, 1000)

# Spiral
x1 = t * np.cos(t)
y1 = t * np.sin(t)
axes[0,0].plot(x1, y1, 'r-')
axes[0,0].set_title('Spiral')
axes[0,0].grid(True)

# Lissajous curve
x2 = np.cos(3*t)
y2 = np.sin(4*t)
axes[0,1].plot(x2, y2, 'g-')
axes[0,1].set_title('Lissajous Curve')
axes[0,1].grid(True)

# Rose curve
k = 5
x3 = np.cos(k*t) * np.cos(t)
y3 = np.cos(k*t) * np.sin(t)
axes[1,0].plot(x3, y3, 'm-')
axes[1,0].set_title('Rose Curve')
axes[1,0].grid(True)

# Cycloid
R = 1
x4 = R * (t - np.sin(t))
y4 = R * (1 - np.cos(t))
axes[1,1].plot(x4, y4, 'c-')
axes[1,1].set_title('Cycloid')
axes[1,1].grid(True)

plt.tight_layout()
plt.show()

[A 2x2 subplot showing four different parametric curves: spiral, Lissajous curve, rose curve, and cycloid]

Key Steps for Parametrized Curves

Animating Parametrized Curves

You can create animated parametrized curves by plotting partial parameter ranges ?

import numpy as np
import matplotlib.pyplot as plt

# Create a figure for animation-like effect
fig, ax = plt.subplots(figsize=(8, 6))

# Full parameter range
t_full = np.linspace(0, 4 * np.pi, 1000)

# Plot curve in segments with different colors
segments = 5
colors = ['red', 'blue', 'green', 'orange', 'purple']

for i in range(segments):
    start = i * len(t_full) // segments
    end = (i + 1) * len(t_full) // segments
    t_segment = t_full[start:end]
    
    # Parametric equations for a spiral
    x = t_segment * np.cos(t_segment) / 4
    y = t_segment * np.sin(t_segment) / 4
    
    ax.plot(x, y, color=colors[i], linewidth=2, alpha=0.8)

ax.set_title('Multi-colored Parametric Spiral')
ax.grid(True)
ax.axis('equal')

plt.show()

[A colorful spiral curve with different colored segments]

Conclusion

Parametrized curves allow you to create complex mathematical shapes by defining x and y as functions of a parameter t. Use np.linspace() to create the parameter range and plt.plot() to visualize the resulting curve.