Parametric and Polar

🎢 The Roller Coaster Adventure: Parametric & Polar Coordinates

Imagine you’re designing the world’s most amazing roller coaster. You need to describe every twist, turn, and loop. That’s exactly what parametric and polar coordinates help us do!

🎯 The Big Picture

Think of coordinates like giving directions to a friend:

• Regular (Cartesian): “Go 3 blocks east, then 4 blocks north”
• Parametric: “At minute 1, be here. At minute 2, be there. At minute 3, be over there…”
• Polar: “Face north, turn 45°, then walk 5 steps”

Each system has its superpower. Let’s discover them!

📍 Part 1: Parametric Equations

What Are They?

Imagine a GPS tracker on an ant crawling across your desk. Every second, you record where the ant is.

• At t = 0: ant is at (1, 0)
• At t = 1: ant is at (2, 3)
• At t = 2: ant is at (5, 4)

Parametric equations describe BOTH the x-position AND y-position using a third variable (usually time, t).

x = f(t)    ← where you are horizontally
y = g(t)    ← where you are vertically

🌟 Simple Example: A Circle

To draw a circle with radius 3:

x = 3·cos(t)
y = 3·sin(t)
where t goes from 0 to 2π

Why it works: As t moves from 0 to 2π, the point traces out a perfect circle! It’s like a clock hand sweeping around.

🎡 Real-World Example: Ferris Wheel

You’re on a Ferris wheel with radius 10 meters:

x = 10·cos(t)
y = 10·sin(t) + 12

The “+12” means the center is 12 meters up (so you don’t go underground!).

graph TD
    A["t = 0"] --> B["Bottom of wheel"]
    B --> C["t = π/2"]
    C --> D["Right side"]
    D --> E["t = π"]
    E --> F["Top of wheel"]
    F --> G["t = 3π/2"]
    G --> H["Left side"]
    H --> A

Why Use Parametric?

📐 Part 2: Parametric Curve Derivatives

The Big Question: How Fast Is It Going?

When the ant crawls along its path, we want to know:

1. How fast is x changing? → dx/dt
2. How fast is y changing? → dy/dt
3. What’s the slope at any point? → dy/dx

🧮 The Magic Formula

To find the slope dy/dx of a parametric curve:

$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$

Think of it like this: The slope equals “how fast y is changing” divided by “how fast x is changing.”

🌟 Example: Circle Slope

For our circle x = 3cos(t), y = 3sin(t):

dx/dt = -3sin(t)
dy/dt = 3cos(t)

dy/dx = 3cos(t) / (-3sin(t))
      = -cot(t)

At t = π/4 (45°):

dy/dx = -cot(π/4) = -1

The slope is -1, meaning the tangent line goes downward at 45°!

🚀 Second Derivative (Acceleration)

To find how the slope is changing (curvature):

$\frac{d^2y}{dx^2} = \frac{d(dy/dx)/dt}{dx/dt}$

Why care? This tells you if the curve is bending upward or downward!

📏 Part 3: Parametric Arc Length

How Long Is the Path?

If you walk along a curvy path, how far did you travel?

Not just the straight-line distance—the actual distance along the curve!

🧵 The Arc Length Formula

For a parametric curve from t = a to t = b:

$L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} , dt$

Think of it as: At each tiny moment, find how far you moved, then add up all those tiny distances.

🌟 Example: Quarter Circle

For x = 3cos(t), y = 3sin(t) from t = 0 to t = π/2:

dx/dt = -3sin(t)
dy/dt = 3cos(t)

(dx/dt)² + (dy/dt)² = 9sin²(t) + 9cos²(t)
                     = 9(sin²(t) + cos²(t))
                     = 9

√9 = 3

So L = ∫₀^(π/2) 3 dt = 3 · (π/2) = 3π/2

Check: A full circle has circumference 2π·3 = 6π. A quarter is 6π/4 = 3π/2 ✓

🎯 Part 4: Vector-Valued Functions

Parametrics, But Fancier

Instead of writing two separate equations, we bundle them into a vector:

$\vec{r}(t) = \langle x(t), y(t) \rangle$

Or in 3D: $\vec{r}(t) = \langle x(t), y(t), z(t) \rangle$

🌟 Example: Helix

A helix (like a spring or DNA):

r(t) = ⟨cos(t), sin(t), t⟩

• x and y make a circle
• z keeps increasing
• Result: a spiral going upward!

Velocity and Acceleration Vectors

Velocity vector: How fast position is changing $\vec{v}(t) = \vec{r}‘(t) = \langle x’(t), y’(t) \rangle$

Acceleration vector: How fast velocity is changing $\vec{a}(t) = \vec{v}‘(t) = \langle x’‘(t), y’'(t) \rangle$

🚗 Example: Car on Circle

Position: r(t) = ⟨4cos(t), 4sin(t)⟩

Velocity: v(t) = ⟨-4sin(t), 4cos(t)⟩

Acceleration: a(t) = ⟨-4cos(t), -4sin(t)⟩

Notice: Acceleration points toward the center! That’s centripetal acceleration.

graph TD
    A["Position r"] --> B["Derivative"]
    B --> C["Velocity v"]
    C --> D["Derivative"]
    D --> E["Acceleration a"]

🧭 Part 5: Polar Coordinates

A Different Way to Locate Points

Cartesian: “Go right 3, up 4” → (3, 4)

Polar: “Turn 53°, walk 5 steps” → (5, 53°)

In polar, every point is described by:

• r = distance from origin (how far)
• θ = angle from positive x-axis (which direction)

🔄 Converting Between Systems

Polar → Cartesian:

x = r·cos(θ)
y = r·sin(θ)

Cartesian → Polar:

r = √(x² + y²)
θ = arctan(y/x)  [adjust for quadrant!]

🌟 Example: Convert (3, 4) to Polar

r = √(3² + 4²) = √25 = 5
θ = arctan(4/3) ≈ 53.13°

So (3, 4) in Cartesian = (5, 53.13°) in polar!

Why Use Polar?

🌹 Part 6: Polar Curves

Beautiful Shapes from Simple Equations

Circles: r = a (constant)

r = 4  → circle with radius 4

Lines through origin: θ = constant

θ = π/4  → line at 45° angle

Cardioids: r = a(1 + cos(θ)) or r = a(1 + sin(θ))

r = 2(1 + cos(θ))  → heart-like shape

Rose curves: r = a·cos(nθ) or r = a·sin(nθ)

r = 3cos(2θ)  → 4-petal rose
r = 3cos(3θ)  → 3-petal rose

General rule: n petals if n is odd, 2n petals if n is even!

🌟 Example: Plot r = 2cos(θ)

This makes a circle passing through the origin with diameter 2!

Slope of Polar Curves

To find dy/dx for polar curves, use:

$\frac{dy}{dx} = \frac{r\cos(\theta) + \sin(\theta) \cdot dr/d\theta}{-r\sin(\theta) + \cos(\theta) \cdot dr/d\theta}$

Or convert to parametric first:

x = r(θ)·cos(θ)
y = r(θ)·sin(θ)

📐 Part 7: Area in Polar Coordinates

How to Find Area Under a Polar Curve

With Cartesian, we slice into vertical rectangles.

With polar, we slice into pizza slices (sectors)!

🍕 The Area Formula

Area of a polar region from θ = α to θ = β:

$A = \frac{1}{2}\int_\alpha^\beta r^2 , d\theta$

Why 1/2? Area of a sector = (1/2)r²θ. The integral adds up infinitely many tiny sectors.

🌟 Example: Area of Circle r = 3

A = (1/2)∫₀^(2π) 3² dθ
  = (1/2)∫₀^(2π) 9 dθ
  = (9/2) · 2π
  = 9π

Check: πr² = π(3)² = 9π ✓

🌹 Example: Area of One Petal of r = cos(2θ)

One petal goes from θ = -π/4 to θ = π/4:

A = (1/2)∫_{-π/4}^{π/4} cos²(2θ) dθ

Using cos²(u) = (1 + cos(2u))/2:

A = (1/2)∫_{-π/4}^{π/4} (1/2)(1 + cos(4θ)) dθ
  = (1/4)[θ + sin(4θ)/4]_{-π/4}^{π/4}
  = π/8

Area Between Two Polar Curves

If r₁ is the outer curve and r₂ is inner:

$A = \frac{1}{2}\int_\alpha^\beta (r_1^2 - r_2^2) , d\theta$

🎯 Quick Summary

graph TD
    A["Parametric"] --> B["x = f/t/, y = g/t/"]
    A --> C["Derivative: dy/dx = dy/dt ÷ dx/dt"]
    A --> D["Arc Length: ∫√dx² + dy² dt"]
    A --> E["Vectors: r = ⟨x,y⟩"]

    F["Polar"] --> G["Point: /r, θ/"]
    F --> H["Convert: x=rcosθ, y=rsinθ"]
    F --> I["Curves: circles, roses, cardioids"]
    F --> J["Area: ½∫r² dθ"]

💡 Key Takeaways

1. Parametric = positions at different times. Great for motion!

2. Parametric derivatives = slope by dividing dy/dt by dx/dt

3. Arc length = integrate the speed (√sum of squared derivatives)

4. Vectors = bundle x, y (and z) into one neat package

5. Polar = distance + angle instead of x + y

6. Polar curves = circles, roses, spirals made simple!

7. Polar area = pizza slices, formula: ½∫r²dθ

🌟 You Did It!

You now understand TWO powerful new ways to describe curves and positions. Whether it’s a roller coaster, a planet’s orbit, or a beautiful rose curve—you have the tools to analyze it!

Remember: Cartesian is like a grid. Parametric is like a stopwatch. Polar is like a compass. Each has its perfect use! 🚀