Converting Between Coordinate Systems

Converting Between Coordinate Systems

Converting between Cartesian (x, y), polar (r, θ), and parametric forms is a core skill. Each system has strengths: Cartesian for algebra, polar for curves with rotational features, and parametric for motion.

Cartesian to Polar

Given a Cartesian equation in x and y, use the substitutions:

x = r cos θ, y = r sin θ, x² + y² = r²

Example 1: Convert x² + y² = 9 to polar.

• r² = 9 → r = 3 (a circle of radius 3)

Example 2: Convert x² + y² = 4x to polar.

• r² = 4r cos θ → r = 4 cos θ (dividing by r, valid since r = 0 is just the origin, which is on the curve anyway)
• r = 4 cos θ is a circle of diameter 4, centred at (2, 0)

Example 3: Convert y = x to polar.

• r sin θ = r cos θ → tan θ = 1 → θ = π/4 (a half-line from the origin)

Example 4: Convert xy = 4 to polar.

• (r cos θ)(r sin θ) = 4 → r² cos θ sin θ = 4 → r² = 8/sin 2θ or r² = 8 cosec 2θ

Polar to Cartesian

Given r = f(θ), multiply through or use r² = x² + y²:

Example 1: Convert r = 6 cos θ to Cartesian.

• Multiply both sides by r: r² = 6r cos θ
• x² + y² = 6x → (x − 3)² + y² = 9
• Circle centred (3, 0), radius 3

Example 2: Convert r = 2/(1 + cos θ) to Cartesian.

• r(1 + cos θ) = 2 → r + r cos θ = 2 → r + x = 2 → r = 2 − x
• Square: r² = (2 − x)² → x² + y² = 4 − 4x + x²
• y² = 4 − 4x → y² = −4(x − 1) (a parabola with vertex at (1, 0))

Conics in Polar Form

The general polar equation of a conic with focus at the pole is:

r = l/(1 + e cos θ) or r = l/(1 + e sin θ)

where:

• e = eccentricity: e < 1 (ellipse), e = 1 (parabola), e > 1 (hyperbola)
• l = semi-latus rectum

e	Curve	Cartesian equivalent
0	Circle	x² + y² = l²
0 < e < 1	Ellipse	x²/a² + y²/b² = 1
1	Parabola	y² = 4ax
e > 1	Hyperbola	x²/a² − y²/b² = 1

Parametric to Polar

If x = x(t), y = y(t):

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

Example: The parametric curve x = a cos t, y = a sin t gives:

• r = √(a²cos²t + a²sin²t) = a
• θ = t
• Polar form: r = a (circle)

Polar to Parametric

A polar curve r = f(θ) is already parametric with θ as the parameter:

• x(θ) = f(θ) cos θ
• y(θ) = f(θ) sin θ

This is useful for finding gradients, arc lengths, and plotting.

Arc Length in Polar Coordinates

The arc length of r = f(θ) from θ = α to θ = β is:

L = ∫_α^β √(r² + (dr/dθ)²) dθ

Example: Arc length of the cardioid r = a(1 + cos θ) for 0 ≤ θ ≤ 2π.

dr/dθ = −a sin θ

r² + (dr/dθ)² = a²(1 + cos θ)² + a² sin²θ = a²(1 + 2cos θ + cos²θ + sin²θ) = a²(2 + 2cos θ) = 2a²(1 + cos θ)

Using 1 + cos θ = 2cos²(θ/2):

= 4a² cos²(θ/2)

√(...) = 2a |cos(θ/2)|

By symmetry: L = 2 ∫₀^π 2a cos(θ/2) dθ = 4a [2 sin(θ/2)]₀^π = 4a × 2 = 8a

Tangent Lines Revisited

For the tangent at a point on r = f(θ):

• Horizontal tangent: dy/dθ = 0, i.e., f'(θ) sin θ + f(θ) cos θ = 0
• Vertical tangent: dx/dθ = 0, i.e., f'(θ) cos θ − f(θ) sin θ = 0

At the pole (r = 0), if r = 0 at θ = α, then the tangent direction is θ = α.

Common Conversion Pitfalls

1. The sign of r:

In polar, r can be negative. The point (−r, θ) is the same as (r, θ + π). When converting, check that your θ is in the right range.

2. Multiple representations:

The point (1, 0) in Cartesian can be written as (1, 0), (1, 2π), (−1, π), etc. in polar. When finding intersections, you may miss solutions if you only solve r₁(θ) = r₂(θ).

3. Dividing by r:

When converting, dividing by r loses the solution r = 0 (the origin). Check separately whether the origin lies on both curves.

4. The argument function:

θ = arctan(y/x) only gives the correct angle in the first and fourth quadrants. Add π for points in the second and third quadrants:

Quadrant	x	y	θ correction
I	+	+	θ = arctan(y/x)
II	−	+	θ = arctan(y/x) + π
III	−	−	θ = arctan(y/x) + π
IV	+	−	θ = arctan(y/x) + 2π (or just arctan gives negative angle)

Exam Tips

• When converting polar → Cartesian, the first move is usually to multiply both sides by r to create r² or r cos θ / r sin θ terms
• For Cartesian → polar, replace x, y, and x² + y² systematically — don't mix approaches
• Always verify by substituting a known point (e.g., the intercepts)
• The polar form of a conic r = l/(1 + e cos θ) is heavily tested — know how to read off e and l
• When finding the arc length, simplify r² + (dr/dθ)² fully before integrating — half-angle identities often help
• Sketch the curve first — the visual check catches sign and range errors