Equations of Curves in Polar Coordinates

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In polar coordinates, the curve of an equation $F (ρ, θ) = 0$ is the set of points that satisfy it, and a point is on the curve as soon as one of its polar names satisfies the equation. The most common form is $ρ = f (θ)$ : let a ray start on the polar axis and turn counterclockwise; when it reaches $θ$ , mark the point on the ray with $ρ = f (θ)$ .

Choose “Circle through the pole”, $ρ = 4 cos θ$ , and turn the ray. Why is the curve finished once $θ$ reaches $π$ ? For $θ$ in $(\frac{π}{2}, π)$ , which side of the ray is the point on?
Choose “Four-petal rose”. In what order do the four petals appear? Which two are dashed ( $ρ < 0$ )?
Choose “Limaçon with inner loop”. For which range of $θ$ is the inner loop drawn? What is the sign of $ρ$ then?
Choose “Archimedean spiral”. By how much does $ρ$ grow with each turn?
For the cardioid, press the three symmetry checks one after another. Which one shows ✓? Does the way the dashed curve (the mirror image) sits on the original agree with that verdict?

A few basic equations can be read straight off the picture:

The line through the pole with angle of inclination $α$ : $θ = α$ ( $ρ$ can be positive or negative, so one ray becomes the whole line).
The line whose perpendicular from the pole has length $p$ and direction angle $α$ : $ρ cos (θ - α) = p$ .
The circle with center at the pole and radius $r$ : $ρ = r$ ; the circle through the pole with center $(a, 0)$ on the polar axis: $ρ = 2 a cos θ$ .

Predict first

How many “petals” does the graph of $ρ = 2 cos 3 θ$ have?

3

6

9

Choose an answer first; then you can see the result.

Why

In rectangular coordinates the equation of a “circle through the origin” has three parameters and does not look nice. In polar coordinates all it takes is one right triangle.

Example: the polar equation of the circle with center (2, 0) and radius 2

Step 1 / 5

Find a special segment
The circle passes through the pole $O$ and meets the polar axis again at $A (4, 0)$ , so $O A$ is a diameter. Take any point $P (ρ, θ)$ on the circle.

$∣ O A ∣ = 2 a = 4$
The angle in a semicircle
Turn to a second-quadrant direction
Half a turn finishes it
Check in rectangular coordinates

To graph a general $ρ = f (θ)$ , analyze first and plot afterwards. The book’s order is: symmetry → range (the largest $∣ ρ ∣$ , the directions where $ρ = 0$ ) → period → table of values and plotting. Be careful with the symmetry tests:

Analysis: is ρ = 2 sin 2θ symmetric about the polar axis?

Step 1 / 3

The first substitution
The mirror image of $(ρ, θ)$ in the polar axis can be written as $(ρ, - θ)$ . Substituting: $ρ = 2 sin (- 2 θ) = - 2 sin 2 θ$ . The equation has changed.

$(ρ, θ) \to (ρ, - θ) : ρ = - 2 sin 2 θ$
Same mirror point, another name
Confirm with the picture

Remove one condition

Break it

Counterexample lab: remove one condition

Conclusion: $ρ = 2 sin 2 θ$ is a four-petal rose

Draw by the rule: for $θ \in (\frac{π}{2}, π)$ , $ρ < 0$ and the point lands in the fourth quadrant. After a full turn, all four petals are there.

Practice

Exercise 1Basic

Write the line $x + 3 y = 4$ in the form $ρ cos (θ - α) = p$ ( $p > 0$ , $0 \leq α < 2 π$ ). If $α = k π$ , find $k$ .

Exercise 2Basic

The curve $ρ = 6 sin θ$ is a circle. What is its radius?

Exercise 3Advanced

How many petals does the graph of $ρ = 3 cos 4 θ$ have?

Exercise 4Challenge

Every chord of the cardioid $ρ = 2 (1 + cos θ)$ that passes through the pole (with both ends on the curve) has the same length. What is it?

Try this

Why

Example: the polar equation of the circle with center (2, 0) and radius 2

Analysis: is ρ = 2 sin 2θ symmetric about the polar axis?

Remove one condition

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

One substitution fails, so you conclude there is no symmetry

Letting θ run only from 0 to 2π and thinking the curve is complete