Explore 3D graphing techniques to trace points, constrain regions, and find intersections between two complex multivariable functions | Step-by-Step Solution

What We're Solving:

We need to visualize two 3D surfaces: z₁(x,y) = 2x² - 1.05x⁴ + (1/6)x⁶ - xy + y² and z₂(x,y) = x² + 4y + (1/8)y³, then find where these surfaces intersect in 3D space.

The Approach:

We want to:

• 1. Understand each surface's shape individually
• 2. Find where they "meet" (intersect)
• 3. Trace meaningful points and curves on these surfaces

The key insight is that intersections occur where z₁(x,y) = z₂(x,y), which gives us a curve (or curves) in the xy-plane that we can then lift up to the 3D surfaces.

Step-by-Step Solution:

Step 1: Analyze Each Surface Individually

• For z₁: It's a 6th-degree polynomial in x with quadratic y terms. The -xy term creates a "saddle-like" twist.
• For z₂: This is simpler - quadratic in x, cubic in y, with a strong linear y component (4y term).

Step 2: Set Up the Intersection Condition To find intersections, set z₁(x,y) = z₂(x,y): 2x² - 1.05x⁴ + (1/6)x⁶ - xy + y² = x² + 4y + (1/8)y³

Rearranging: x² - 1.05x⁴ + (1/6)x⁶ - xy + y² - 4y - (1/8)y³ = 0

Step 3: Choose Your Visualization Strategy You have several approaches:

• Parametric approach: Fix x values, solve for y, then compute z
• Contour approach: Create level curves and build up the surface
• Grid approach: Evaluate both functions on a rectangular grid

Step 4: Implement Systematic Point Tracing For a grid approach:

• Choose a reasonable domain (try x ∈ [-2, 2], y ∈ [-3, 3] initially)
• Create a mesh: x_vals = linspace(-2, 2, 50), y_vals = linspace(-3, 3, 50)
• Compute z₁ and z₂ at each grid point
• Plot both surfaces with different colors/transparency

Step 5: Find Intersection Curves The intersection equation is complex, so use numerical methods:

• For each x value, solve the resulting equation in y
• Or use contour plotting where z₁ - z₂ = 0
• The intersection will appear as curves where the surfaces meet

The Framework for Implementation:

Visualization Setup: ``` Domain: x ∈ [-2, 2], y ∈ [-3, 3] (adjust as needed) Grid resolution: 50×50 points minimum Surface 1: z₁ (use warm colors - red/orange) Surface 2: z₂ (use cool colors - blue/green) Intersection: Highlight in contrasting color (black/white) ```

Key Features to Identify:

• Critical points (where gradients = 0)
• Intersection curves
• Boundary behavior
• Regions where one surface is above/below the other

Memory Tip:

Think "S.E.T." for 3D surface problems:

• Set up your domain and grid
• Evaluate functions systematically
• Trace the intersections where surfaces meet

Remember: intersections of 3D surfaces typically form curves, not just points! These curves live in 3D space but can be projected down to the xy-plane for easier analysis.

The beauty of this problem is seeing how two completely different mathematical expressions create surfaces that dance around each other in 3D space. Take your time with the visualization - the intersection curves will reveal the elegant relationship between these functions!