🤖 Feeling the Surface: Tangent Planes & Taylor Approximations in SageMath

""Exploring Tangent Planes & Taylor Approximations: How SageMath Models Surfaces for Robotics & AI""

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🤖 Feeling the Surface: Tangent Planes & Taylor Approximations in SageMath

Imagine a robot’s fingertip gently brushing a curved surface.

Its sensors feel the rise and fall of the terrain—trying to make sense of subtle hills and dips beneath. But how does a machine truly "understand" a surface?

It starts with mathematics—specifically, something called the tangent plane, the first building block in modeling how a surface behaves near a point. While robots physically feel surfaces, mathematical models help them predict and react—sometimes even before contact or beyond their immediate sensing range.

To refine this understanding, we turn to Taylor polynomials, which capture not just the slope but also the curvature of the surface.

Let’s dive in with SageMath to:

• Visualize tangent planes as the flattest possible fit to a curved surface.
• Explore Taylor approximations that get progressively closer to the real surface.
• Discover how this is used in robotic sensing and drone terrain mapping.

🧭 Part 1: Intuition Behind the Tangent Plane

Just like a tangent line approximates a curve at a point, a tangent plane is the best linear approximation of a surface at a specific point.

📌 Think of balancing a flat tray on a curved dome at a single point—the perfectly balanced tray is your tangent plane.

The partial derivatives f_x(a,b) and f_y(a,b) represent how steep the surface feels in the x and y directions—like a fingertip detecting tilt as it moves.

🔧 Tangent Plane Formula

T(x,y) = f(a,b) + f_x(a,b) (x - a) + f_y(a,b) (y - b)

Let’s visualize this for:

$$ f(x,y) = 4xy e^{-(x^2 + y^2)} \quad \text{at} \quad (a,b) = (1,1) $$

SageMath Code – Tangent Plane

🌍 Real-World Applications: Why It Matters

We’ll plot:

• Blue:

  the original surface

• Green:

  the tangent plane

• Red point:

  the point of tangency

SageMath Code Tangent Plane

Visualization Tangent Plane

🤖 Real-Life Application: Robotic Touch Sensors

In robotics, touch sensors often detect small variations in pressure or displacement across a contact surface. These changes correlate with:

• f_x(a,b)-

  how the surface tilts in the x-direction

• f_y(a,b)-

  how it slopes in the y-direction

The tangent plane helps the robot model the contact area and adjust its response accordingly—for instance, adapting its grip or posture.

👉 For even more nuanced interpretation, such as understanding curvature or twist, higher-order approximations like Taylor polynomials are used. These give a more detailed picture of how the surface bends under pressure.

Part 2: Taylor Polynomials – Adding Curvature

ITangent planes are great—but what if we want even more accuracy?

That's where Taylor polynomials come in. They’re like upgraded tangent planes:

• 1st-degree:

  Linear (just like the tangent plane)

• 2nd-degree:

  Adds curvature in both x and y

• 3rd-degree:

  Adds twisting and bending accuracy!

Let’s use:

$$f(x,y) = \cos(x) \cos(y)$$

and expand it at (0,0)

🧠 Intuition:

Taylor polynomials build layer by layer—each term adds finer detail to better match the original surface around the point.

🔢 SageMath Code – Taylor Approximations

Visual Comparison– Taylor Approximations

🎨 Visual Comparison

We’ll plot:

• Blue:

  Original surface

• Red:

  1st-degree (linear, tangent plane)

• Green:

  2nd-degree

• Yellow:

  3rd-degree (most accurate)

SageMath Code – Taylor Approximations

Visualization Taylor Approximations

🚁 Real-Life Application: Drone Terrain Mapping

Drones use sensors like:

• Lidar (light detection and ranging)
• Sonar
• Stereo/depth cameras

These sensors collect point data about terrain elevation. To move smoothly and safely, drones need a local surface model. Taylor approximations:

• Capture both slope and curvature
• Provide smoother paths for altitude adjustment
• Help avoid jerky movement over hills, dips, or bumps

In short: Taylor polynomials make the digital terrain feel real to drones.

🔍 Reflection & Exploration

💬 What to Explore:

• Try different expansion points (a,b)
• Explore more complex surfaces like:
• $$\ln(1 + x^2 + y^2)$$
• $$\sin(xy)$$
• $$\frac{x^2 - y^2}{1 + x^2 + y^2}$$

🧩 Guiding Questions:

• How does the accuracy change as you move further from the expansion point?
• Where might a linear model be sufficient? Where does curvature matter?
• How might higher-degree models impact real-time decision-making (e.g., for robots, drones, or simulations)?

🔜 What's Next?

Up next, we’ll explore:

• Gradient vectors and directional derivatives – key tools in navigating surfaces, climbing ridges, and optimizing paths.

📌 Tinker with the code, visualize different surfaces, and share your insights below! Where do you see tangent planes or curvature in your world?