Polar Coordinates and Polar Curves

Polar Curves

By the end of this lesson, you will be able to convert between polar and rectangular coordinates, sketch polar curves, and analyze polar curves using calculus.

The Polar Coordinate System

In the polar coordinate system, a point is described by (r, θ):

• r: distance from the origin (pole)
• θ: angle measured from the positive x-axis (polar axis)

Conversion formulas:

Polar → Rectangular: $$x = r\cos\theta, \quad y = r\sin\theta$$

Rectangular → Polar: $$r = \sqrt{x^2 + y^2}, \quad \tan\theta = \frac{y}{x}$$

Common Polar Curves

Circles

• r = a: Circle of radius |a| centered at origin
• r = 2a·cos(θ): Circle through origin, diameter 2a, centered at (a, 0)
• r = 2a·sin(θ): Circle through origin, diameter 2a, centered at (0, a)

Lines

• θ = c: A line through the origin making angle c with the polar axis
• r·cos(θ) = a: Vertical line x = a
• r·sin(θ) = b: Horizontal line y = b

Cardioids

$$r = a(1 \pm \cos\theta) \quad \text{or} \quad r = a(1 \pm \sin\theta)$$

Heart-shaped curve; extends from r = 0 to r = 2a.

Roses

$$r = a\cos(n\theta) \quad \text{or} \quad r = a\sin(n\theta)$$

• If n is odd: n petals
• If n is even: 2n petals

Example: r = 3cos(2θ) has 4 petals; r = 2sin(3θ) has 3 petals.

Limaçons

$$r = a \pm b\cos\theta$$

• If a < b: inner loop
• If a = b: cardioid
• If a > b: dimple (a < 2b) or convex (a ≥ 2b)

Lemniscate

$$r^2 = a^2\cos(2\theta) \quad \text{or} \quad r^2 = a^2\sin(2\theta)$$

Figure-eight shaped; extends to r = a.

Sketching Polar Curves

Strategy:

1. Create a table of (θ, r) values
2. Plot and connect; watch for r < 0 (reflect through origin)