Maths › Polar coordinates › Polar curves
Polar curves
Describe a point by how far and in what direction instead of across and up, and curves that torment cartesian algebra collapse into one short equation in r and θ.
Builds on Radians, arcs and small angles and Circles.
IN THIS TOPIC
- Plot points from (r, θ) and convert both ways with x = r cos θ, y = r sin θ.
- Convert polar equations to cartesian form and recognise the result.
- Sketch the standard families: r = a, θ = α, cardioids and roses.
WHAT YOU PROBABLY THINK
Polar coordinates are only a novelty; anything they describe is just as easy in x and y.
How far, which way
A point's polar coordinates are (r, θ): distance from the pole and anticlockwise angle from the initial line. The dictionary between the systems is immediate from a right triangle:
WORKED EXAMPLE
A polar equation in cartesian clothing
Convert r = 4 cos θ to cartesian form and identify the curve.
Multiply both sides by r: r² = 4r cos θ.
Substitute: x² + y² = 4x.
Complete the square: (x − 2)² + y² = 4, a circle of radius 2 centred at (2, 0).
Spot check: θ = π/3 gives r = 2, the point (1, √3), and indeed (1 − 2)² + 3 = 4.
The standard sketches
Sketching runs on a short repertoire. r = a is a circle round the pole; θ = α is a half-line at fixed bearing; r = a(1 + cos θ) is the cardioid, a heart shape swelling to 2a on the initial line and pinching to zero opposite; r = a cos 2θ is a four-petal rose. Tabulating r at the compass angles θ = 0, π/2, π, 3π/2 pins each shape in seconds, which is quicker than any cartesian attack on the same curves; the opening claim has clearly never met a cardioid in x and y.
WORKED EXAMPLE
Reading a cardioid's vital points
For r = 1 + cos θ, find r at θ = 0, π/2 and π, and describe the curve.
θ = 0: r = 2. θ = π/2: r = 1. θ = π: r = 0.
The curve bulges to 2 rightwards, passes 1 unit above the pole, and closes onto the pole itself at the back: a heart lying on its side.
The pinch at the pole happens because r reaches exactly zero there; curves with r = a + b cos θ and a > b never touch the pole.
YOUR TURN
Cartesian to polar
Find polar coordinates for the cartesian point (−3, 3), with 0 ≤ θ < 2π.
Show the working
r = √(9 + 9) = 3√2.
The point sits in the second quadrant: θ = π − π/4 = 3π/4.
So (r, θ) = (3√2, 3π/4). Quoting θ = arctan(−1) = −π/4 without checking the quadrant is the standard trap.
THE EXAM BIT
- To convert an equation, manufacture r², r cos θ and r sin θ; multiplying through by r is the usual first move.
- Check the quadrant before writing θ; arctan alone cannot tell (−3, 3) from (3, −3).
- Sketch from a table of r at θ = 0, π/2, π, 3π/2, and mark where r = 0.
- State the range of θ that traces the curve once; examiners look for it.
CHECK YOURSELF
Convert the polar point (2, π/6) to cartesian coordinates, exactly.
Show a hint
x = r cos θ, y = r sin θ.
Show the answer
x
=
2
c
o
s
(
π
/
6
)
=
√
3
a
n
d
y
=
2
s
i
n
(
π
/
6
)
=
1
:
t
h
e
p
o
i
n
t
(
√
3
,
1
)
.
x = r cos θ and y = r sin θ; build r² and r cos θ to convert equations.
Sketch polar curves from r at the four compass angles, marking any pass through the pole.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
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- Plot points from (r, θ) and convert both ways with x = r cos θ, y = r sin θ.
- Convert polar equations to cartesian form and recognise the result.
- Sketch the standard families: r = a, θ = α, cardioids and roses.
Open the full revision checklist to see every objective in the course in one place.
No animated video for this topic yet; these notes stand alone.