Polar Coordinates — Complete Interactive Guide

1. What polar coordinates are

A point in the plane is described by (r, θ) instead of (x, y).

r: signed distance from the origin (the pole). r can be negative.
θ: angle from the positive x-axis, usually in radians. Counterclockwise is positive.

The same point has infinitely many polar names: (r, θ) = (r, θ + 2πk) = (−r, θ + π).

2. Conversion with Cartesian

Polar to Cartesian

x = r * cos(θ)
y = r * sin(θ)

Cartesian to Polar

r = sqrt(x^2 + y^2)
θ = atan2(y, x) (gives correct quadrant)

Interactive converter

r: θ:

x: y:

3. Plotting points and the meaning of negative r

To plot (r, θ): rotate θ from the x-axis, then move |r| along that ray. If r is negative, move in the opposite direction.

Example: (3, π/4) is in first quadrant. (−3, π/4) is the same as (3, 5π/4), in third quadrant.

4. Classic polar curves

Name	Equation	Shape notes
Circle centered at pole	r = a	radius \|a\|
Circle off-center	r = 2a cos θ	center (a,0), radius a
Line through pole	θ = α	straight line at angle α
Cardioid	r = a(1 ± cos θ) or a(1 ± sin θ)	heart shape, cusp at pole
Limaçon	r = a ± b cos θ	inner loop if a < b, dimple if a ≈ b
Rose	r = a cos(kθ) or a sin(kθ)	k petals if k odd, 2k if k even
Archimedean spiral	r = aθ	constant spacing between turns
Lemniscate	r^2 = a^2 cos 2θ	figure-eight, two lobes

Interactive polar plotter

Curve:
a: b: k:
θ max: rad

5. Symmetry tests

About x-axis: replace θ with −θ. If r unchanged, symmetric.
About y-axis: replace θ with π − θ.
About pole: replace r with −r, or θ with θ + π.

Example: r = a cos θ is symmetric about x-axis because cos(−θ)=cos θ.

6. Calculus in polar

Area

Area swept from θ = α to β: A = (1/2) ∫_{α}^{β} r(θ)^2 dθ

For a full rose r = a cos(kθ), compute over one petal then multiply.

Arc length

L = ∫_{α}^{β} sqrt(r^2 + (dr/dθ)^2) dθ

Slope

Parametric form x = r cosθ, y = r sinθ gives dy/dx = (r' sinθ + r cosθ)/(r' cosθ − r sinθ)

Tangent at pole

Find θ where r=0. The line θ = constant is tangent.

7. Polar and complex numbers

Euler's formula: e^{iθ} = cosθ + i sinθ. A complex number becomes z = r e^{iθ} = r(cosθ + i sinθ).

Multiplication adds angles and multiplies radii: r1 e^{iθ1} * r2 e^{iθ2} = r1 r2 e^{i(θ1+θ2)}. This is why polar form is natural for rotations and phasors.

8. Why polar matters

Physics: central forces, orbits, angular momentum.
Engineering: antenna radiation patterns, polar plots of frequency response.
Navigation: bearing and range.
Mathematics: integrals with circular symmetry are simpler in polar.

9. GNU Octave lab

Octave has polarplot (modern) and polar (legacy), plus pol2cart and cart2pol.

% Basic conversion
theta = pi/4; r = 5;
[x, y] = pol2cart(theta, r)   % x=3.5355 y=3.5355
[r2, th2] = cart2pol(3, 4)    % r=5 th=0.9273

% Plot a cardioid
theta = linspace(0, 2*pi, 1000);
r = 2*(1 + cos(theta));
polarplot(theta, r); title('Cardioid r = 2(1+cos θ)');

% Rose with 5 petals
figure; k=5; a=3;
r = a*cos(k*theta);
polarplot(theta, r); title('Rose r = 3 cos 5θ');

% Limaçon with inner loop
figure; a=1; b=2;
r = a + b*cos(theta);
polarplot(theta, r); title('Limaçon r = 1 + 2 cos θ');

% Archimedean spiral
figure; a=0.2;
theta = linspace(0, 4*pi, 2000);
r = a*theta;
polarplot(theta, r); title('Spiral r = 0.2 θ');

% Lemniscate r^2 = a^2 cos 2θ
figure; a=2;
theta = linspace(0, 2*pi, 2000);
r = a*sqrt(max(0, cos(2*theta)));  % plot positive branch
polarplot(theta, r); hold on;
polarplot(theta, -r);  % negative branch
title('Lemniscate r^2 = 4 cos 2θ');

% Area of cardioid r = 2(1+cos θ)
rfunc = @(t) 2*(1+cos(t));
area = 0.5 * integral(@(t) rfunc(t).^2, 0, 2*pi)  % = 6*pi

% Arc length of circle r = 3
L = integral(@(t) sqrt(3^2 + 0), 0, 2*pi)  % = 6*pi

% Complex numbers in polar
z = 5 * exp(1i*pi/3);
abs(z), angle(z)   % r and θ

Octave tips

Use atan2(y,x) not atan(y/x) for correct quadrant.
For filled area, use fill after converting to Cartesian.
polarplot accepts degrees if you set thetalim, but radians are default.

10. Practice problems

Convert (−2, 3π/4) to Cartesian. Then convert (x,y)=(−1,−1) back to polar with r>0, θ in [0,2π).
Sketch r = 1 − sin θ. Identify symmetry and find area of one loop.
Show that r = 3 cos θ is a circle. Find its center and radius in Cartesian form.
In Octave, plot r = 2 + 3 sin(2θ) and compute its area numerically.
Find the slope dy/dx for r = e^{θ} at θ = π/2.