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Lagrange Example - 3d Surface with a constraint curve and lifted constraint curve

5 min read

Description

This Python script uses matplotlib to visualize a 3D surface (a paraboloid) with a squiggled constraint curve. The curve is lifted onto the surface, with two points highlighted: one at the maximum and one at the minimum of the curve. Below is an explanation of the main steps:

Image

Code

import micropip
await micropip.install("numpy")
await micropip.install("matplotlib")
 
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
 
# Define the surface z = f(x, y) (a paraboloid)
def f(x, y):
    return x**2 + y**2
 
# Set up grid for plotting
x = np.linspace(-1.5, 1.5, 400)
y = np.linspace(-1.5, 1.5, 400)
X, Y = np.meshgrid(x, y)
Z = f(X, Y)
 
# Create figure and 3D axis
fig = plt.figure(figsize=(6, 4))
ax = fig.add_subplot(111, projection='3d')
 
# Plot the surface (paraboloid)
ax.plot_surface(X, Y, Z, rstride=5, cstride=5, alpha=0.3, cmap='coolwarm')
 
# Plot the constraint curve at z = 0 (a circle)
theta = np.linspace(0, 2 * np.pi, 100)
r = 1  # Radius of the constraint (constant for regular curve)
x_constraint = r * np.cos(theta)
y_constraint = r * np.sin(theta)
z_constraint = np.zeros_like(x_constraint)
 
ax.plot(x_constraint, y_constraint, z_constraint, color='red', lw=2, label='Constraint curve')
 
# Modify the constraint curve to introduce "squiggle" (move radially in/out)
def squiggled_constraint(theta, base_r=1, amplitude=0.1, frequency=5):
    # Adjust radius based on sinusoidal squiggle
    r_squiggled = base_r + amplitude * np.sin(frequency * theta)
    x_squiggled = r_squiggled * np.cos(theta)
    y_squiggled = r_squiggled * np.sin(theta)
    return x_squiggled, y_squiggled
 
# Get the squiggled constraint curve points
x_squiggled, y_squiggled = squiggled_constraint(theta)
 
# Compute the corresponding z values for the squiggled curve from the surface
z_squiggled = f(x_squiggled, y_squiggled)
 
# Plot the squiggled lifted constraint curve
ax.plot(x_squiggled, y_squiggled, z_squiggled, color='orange', lw=3, label='Squiggled lifted constraint curve')
 
# Find the indices for max and min points on the squiggled curve
max_idx = np.argmax(z_squiggled)  # Index of the maximum z value
min_idx = np.argmin(z_squiggled)  # Index of the minimum z value
 
# Plot and label the max point
ax.scatter(x_squiggled[max_idx], y_squiggled[max_idx], z_squiggled[max_idx], color='green', s=100)
ax.text(x_squiggled[max_idx], y_squiggled[max_idx], z_squiggled[max_idx] + 0.2, 'Maximum', color='green', fontsize=12)
 
# Plot and label the min point
ax.scatter(x_squiggled[min_idx], y_squiggled[min_idx], z_squiggled[min_idx], color='blue', s=100)
ax.text(x_squiggled[min_idx], y_squiggled[min_idx], z_squiggled[min_idx] + 0.2, 'Minimum', color='blue', fontsize=12)
 
# Add labels and title
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('Z')
ax.set_title('3D Surface with Squiggled Constraint Curve and Max/Min Points')
 
# Customizing view angle
ax.view_init(elev=30, azim=120)
 
plt.legend()
plt.show()

Explanation Step By Step

import micropip
await micropip.install("numpy")
await micropip.install("matplotlib")
 
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
  • Imports, using micropip for the virtual env stuff
# Define the surface z = f(x, y) (a paraboloid)
def f(x, y):
    return x**2 + y**2
  • Function f(x, y): This function defines the surface of a paraboloid. It calculates ( z = x^2 + y^2 ) for given ( x ) and ( y ) values.
# Set up grid for plotting
x = np.linspace(-1.5, 1.5, 400)
y = np.linspace(-1.5, 1.5, 400)
X, Y = np.meshgrid(x, y)
Z = f(X, Y)
  • Grid Setup: Here, a grid of points is created in the ( x )- and ( y )-directions using np.linspace. The corresponding ( z )-values for each ( (x, y) ) pair are calculated using the paraboloid function f(x, y).
# Create figure and 3D axis
fig = plt.figure(figsize=(6, 4))
ax = fig.add_subplot(111, projection='3d')
 
# Plot the surface (paraboloid)
ax.plot_surface(X, Y, Z, rstride=5, cstride=5, alpha=0.3, cmap='coolwarm')
  • Surface Plot: A 3D plot of the paraboloid is created using ax.plot_surface(). The rstride and cstride parameters control the resolution of the mesh, and alpha sets the transparency.
# Modify the constraint curve to introduce "squiggle" (move radially in/out)
def squiggled_constraint(theta, base_r=1, amplitude=0.1, frequency=5):
    r_squiggled = base_r + amplitude * np.sin(frequency * theta)
    x_squiggled = r_squiggled * np.cos(theta)
    y_squiggled = r_squiggled * np.sin(theta)
    return x_squiggled, y_squiggled
  • Squiggled Constraint Curve: The function squiggled_constraint generates a modified version of the constraint curve by adjusting the radius ( r ) based on a sinusoidal term ( r = r_0 + A \sin(f \theta) ). This creates the squiggling effect.
    • amplitude controls how much the curve moves in or out.
    • frequency determines how many squiggles appear around the curve.
# Get the squiggled constraint curve points
x_squiggled, y_squiggled = squiggled_constraint(theta)
  • Generate Squiggled Curve: This line computes the new ( x )- and ( y )-coordinates for the squiggled curve using the squiggled_constraint function.
# Compute the corresponding z values for the squiggled curve from the surface
z_squiggled = f(x_squiggled, y_squiggled)
  • Lift the Curve to the Surface: The ( z )-coordinates for the squiggled curve are calculated by passing the squiggled ( x )- and ( y )-coordinates into the surface function ( z = f(x, y) ).
# Find the indices for max and min points on the squiggled curve
max_idx = np.argmax(z_squiggled)  # Index of the maximum z value
min_idx = np.argmin(z_squiggled)  # Index of the minimum z value
  • Find Maximum and Minimum Points:
    • np.argmax(z_squiggled) finds the index of the point on the squiggled curve with the highest ( z )-value (maximum).
    • np.argmin(z_squiggled) finds the index of the point with the lowest ( z )-value (minimum).
# Plot and label the max point
ax.scatter(x_squiggled[max_idx], y_squiggled[max_idx], z_squiggled[max_idx], color='green', s=100)
ax.text(x_squiggled[max_idx], y_squiggled[max_idx], z_squiggled[max_idx] + 0.2, 'Maximum', color='green', fontsize=12)
  • Plot Maximum Point: The maximum point is plotted with a green dot (ax.scatter) and labeled “Maximum” slightly above the point (ax.text).
# Plot and label the min point
ax.scatter(x_squiggled[min_idx], y_squiggled[min_idx], z_squiggled[min_idx], color='blue', s=100)
ax.text(x_squiggled[min_idx], y_squiggled[min_idx], z_squiggled[min_idx] + 0.2, 'Minimum', color='blue', fontsize=12)
  • Plot Minimum Point: Similarly, the minimum point is plotted with a blue dot and labeled “Minimum.”
# Add labels and title
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('Z')
ax.set_title('3D Surface with Squiggled Constraint Curve and Max/Min Points')
  • Labels and Titles: Standard axis labels and a plot title are added for clarity.
# Customizing view angle
ax.view_init(elev=30, azim=120)
  • View Angle: The ax.view_init function adjusts the viewing angle of the 3D plot to give a good perspective of both the surface and the squiggled curve.
plt.legend()
plt.show()
  • Displaying the graph

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