Polar Coordinates and Parametric Equations

Polar Coordinate System

Different from Cartesian (x, y)

Polar: Point located by (r, θ)

• r = distance from origin (radius)
• θ = angle from positive x-axis

Advantages: Natural for circular/spiral patterns

Example: Plot Polar Points

(3, π/4): Distance 3, angle 45°

(2, π): Distance 2, angle 180° (on negative x-axis)

(4, 0): Distance 4, angle 0° (on positive x-axis)

Negative r: (r, θ) = (-r, θ + π)

Converting: Polar to Cartesian

Formulas:

x = r cos(θ)
y = r sin(θ)

Derived from right triangle

Example: Polar to Cartesian

(r, θ) = (4, π/3)

Convert:

x = 4 cos(π/3) = 4(1/2) = 2
y = 4 sin(π/3) = 4(√3/2) = 2√3

Cartesian: (2, 2√3)

Converting: Cartesian to Polar

Formulas:

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

Example: Cartesian to Polar

(x, y) = (3, 3)

Convert:

r = √(3² + 3²) = √18 = 3√2
θ = arctan(3/3) = arctan(1) = π/4

Polar: (3√2, π/4)

Example: Quadrant Adjustment

(-1, 1)

Calculate:

r = √(1 + 1) = √2
θ = arctan(1/-1) = arctan(-1)

Point in quadrant II, so θ = 3π/4

Polar: (√2, 3π/4)

Polar Equations

Relate r and θ

Common forms:

• Circles: r = a
• Lines through origin: θ = constant
• Spirals: r = aθ
• Rose curves: r = a sin(nθ) or r = a cos(nθ)

Example: Circle

r = 4

All points distance 4 from origin

Circle of radius 4 centered at origin

Example: Line

θ = π/6

All points at angle 30°

Line through origin at 30° angle

Rose Curves

r = a sin(nθ) or r = a cos(nθ)

Number of petals:

• n odd → n petals
• n even → 2n petals

Example: Four-Petal Rose

r = 3 cos(2θ)

n = 2 (even) → 2(2) = 4 petals

Maximum r = 3

Limaçons

r = a ± b sin(θ) or r = a ± b cos(θ)

Types depend on ratio a/b:

• a/b < 1: Inner loop
• a/b = 1: Cardioid (heart-shaped)
• 1 < a/b < 2: Dimpled
• a/b ≥ 2: Convex

Example: Cardioid

r = 1 + cos(θ)

a = b = 1, so a/b = 1

Heart-shaped curve

Spiral of Archimedes

r = aθ

As angle increases, distance increases linearly

Example: Spiral

r = θ

At θ = π: r = π At θ = 2π: r = 2π

Outward spiral

Lemniscate

r² = a² sin(2θ) or r² = a² cos(2θ)

Figure-eight shape

Example: Lemniscate

r² = 9 cos(2θ)

Take square root: r = ±3√(cos(2θ))

Defined when cos(2θ) ≥ 0

Figure-eight centered at origin

Converting Polar Equations

To Cartesian: Use x = r cos θ, y = r sin θ, r² = x² + y²

Example: Polar to Cartesian Equation

r = 4 sin(θ)

Multiply by r:

r² = 4r sin(θ)
x² + y² = 4y
x² + y² - 4y = 0
x² + (y - 2)² = 4

Circle: center (0, 2), radius 2

Parametric Equations

Express x and y as functions of parameter t

x = f(t)
y = g(t)

Advantage: Describe motion/direction along curve

Example: Circle

Parametric equations for circle radius 3:

x = 3 cos(t)
y = 3 sin(t)

As t goes from 0 to 2π, traces circle

Eliminating the Parameter

Convert parametric to Cartesian

Methods:

• Solve for t, substitute
• Use trig identities
• Algebraic manipulation

Example: Eliminate Parameter

x = t + 1, y = 2t - 1

Solve first for t:

t = x - 1

Substitute into second:

y = 2(x - 1) - 1
y = 2x - 3

Cartesian: y = 2x - 3 (line)

Example: Trig Identity

x = cos(t), y = sin(t)

Use identity:

x² + y² = cos²(t) + sin²(t) = 1

Cartesian: x² + y² = 1 (unit circle)

Parametric Curves

Can represent more complex motions

Example: Ellipse

x = 4 cos(t), y = 2 sin(t)

Eliminate parameter:

x/4 = cos(t)
y/2 = sin(t)

(x/4)² + (y/2)² = 1
x²/16 + y²/4 = 1

Ellipse with semi-axes 4 and 2

Cycloid

Path traced by point on rolling circle

Parametric:

x = r(t - sin t)
y = r(1 - cos t)

Famous curve with interesting properties

Applications: Projectile Motion

Horizontal and vertical motion:

x = (v₀ cos α)t
y = (v₀ sin α)t - (1/2)gt²

v₀ = initial velocity, α = launch angle, g = gravity

Example: Projectile

Launch at 30 m/s, 45° angle

Parametric:

x = 30 cos(45°)·t ≈ 21.2t
y = 30 sin(45°)·t - 4.9t² ≈ 21.2t - 4.9t²

Eliminate t to get parabola

Derivatives in Parametric Form

dy/dx = (dy/dt)/(dx/dt)

Use chain rule

Example: Parametric Derivative

x = t², y = t³

Find dy/dx:

dx/dt = 2t
dy/dt = 3t²

dy/dx = 3t²/(2t) = (3/2)t

Applications: Computer Graphics

Bezier curves: Parametric polynomials

Animation: Position as function of time

Game development: Character paths

Polar Area

Area enclosed by r = f(θ) from θ = α to θ = β:

A = (1/2)∫[α to β] r² dθ

Example: Area of Circle

r = 2, full circle (0 to 2π)

Calculate:

A = (1/2)∫[0 to 2π] 4 dθ
  = 2∫[0 to 2π] dθ
  = 2[θ]₀^(2π)
  = 4π

Matches πr² = π(2)² = 4π ✓

Practice