Polar Curves

Polar Curves

1. Polar Coordinates

A point $P$ is given by $(r,θ)$ where $r$ = signed distance from pole $O$ and $θ$ = angle from polar axis ($Ox$). Conversion:

A polar curve is $r = f(θ)$ (or $F(r,θ)=0$). Examples: $r = a(1-cosθ)$ (cardioid), $r = a cos 2θ$ (four-petalled rose), $r = aθ$ (spiral of Archimedes), $r = a(1+cosθ)$ variations.

2. Symmetry — Quick Tests

For $r = f(θ)$:

• Symmetric about initial line ($θ=0$): $f(-θ)=f(θ)$ or $f(2π-θ)=f(θ)$ — replace $θ→-θ$, equation unchanged.
• Symmetric about pole ($O$): $f(θ+π)= -f(θ)$ or equation in $r²$.
• Symmetric about $θ=π/2$: $f(π-θ)=f(θ)$.
  Symmetry halves sketching work; examiners accept a symmetry statement before plotting.

3. Tangent in Polar Form

In Cartesian terms $x(θ)=f(θ)cosθ, y(θ)=f(θ)sinθ$. The slope is:

Let $φ$ be angle between radius vector $OP$ and tangent at $P$:

• $φ$ is measured from $OP$ (produced) towards tangent. Then the angle the tangent makes with initial line is $ψ = θ + φ$.
• At the pole ($r=0$): tangents are $θ = α$ where $f(α)=0$ — set $r=0$ and solve for $θ$; these are the pole tangents (e.g., rose $r=a cos2θ$ gives $4$ tangents at $45°$ intervals).

4. Angle Between Radius Vector and Tangent — Derivation Sketch

In $Δ$ with $r, r+Δr, Δθ$, $tan φ = r Δθ/Δr → r dθ/dr$ as $Δθ→0$.

5. Polar Subtangent / Subnormal and Pedal Equation

• Polar subtangent (perp to $OP$): $= r² / [dr/dθ] = r · tan φ$
• Polar subnormal (along $OP$): $= dr/dθ = r cot φ$
• Pedal distance $p$ (perp from pole to tangent): $1/p² = 1/r² + 1/r⁴ (dr/dθ)²$ or $p = r sinφ$. The pedal equation $p = p(r)$ eliminates $θ$.

6. Worked Examples

(a) Cardioid r = a(1 - cosθ):
    r' = a sinθ ⇒ tan φ = r/r' = (1-cosθ)/sinθ = tan(θ/2) ⇒ φ = θ/2
    At θ=π/2 : r=a, φ=π/4, ψ=3π/4 ; at pole r=0 ⇒ 1-cosθ=0 ⇒ θ=0 (cusp) ✓

(b) Circle r = 2a cosθ (diameter on initial line):
    r' = -2a sinθ ⇒ tan φ = 2a cosθ/(-2a sinθ)= -cotθ = tan(π/2+θ) ⇒ φ=π/2+θ
    ψ = θ+φ = π/2+2θ ; subtangent = r²/r' = -2a cos²θ/sinθ                ✓

(c) Rose r = a cos 2θ — pole tangents: r=0 ⇒ cos2θ=0 ⇒ 2θ=±π/2, ±3π/2 ⇒ θ=±π/4, ±3π/4 ✓

(d) Spiral r = aθ : r' = a ⇒ tan φ = aθ/a = θ ⇒ φ = arctan θ → grows with radius ✓

(e) Find φ for r = a e^{θ cotα} (equiangular spiral):
    r' = a cotα e^{θ cotα}= r cotα ⇒ tan φ = r/r' = tanα ⇒ φ = α (constant!) ⇒ spiral cuts every radius vector at same angle α ✓

7. Sketching Tip for Exams

Make table $θ$: $0, π/6, π/3, π/2,…$ → $r$ → plot; mark pole tangents and axis symmetry; the curve is closed if $r(θ+2π)=r(θ)$.

8. Exam Points

• Derive $tan φ = r / [dr/dθ]$ and find $φ$, $ψ$ at given $θ$ — 5 marks, very frequent.
• Find pole tangents ($r=0$) — 3 marks, frequent.
• Polar subtangent / subnormal / pedal equation — 5 marks, frequent.
• Symmetry and sketch $r=a(1±cosθ), r=a cos nθ, r=aθ$ — 5-7 marks.
• Angle between two polar curves = $|φ_1-φ_2|$ or $|ψ_1-ψ_2|$ — 5 marks.