Polar Coordinates

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1. What Are Polar Coordinates?

Unlike Cartesian coordinates (x, y), polar coordinates represent a point using:

• r = the distance from the origin (radius)

• θ = the angle (in radians or degrees) from the positive x-axis

So a point P is written as:

$$P=(r,θ)$$

2. Converting Between Cartesian and Polar Coordinates

A. Polar ➡️ Cartesian

$$\begin{matrix} x=r⋅cos(θ) \\ \\ y=r⋅sin(θ) \end{matrix}$$

B. Cartesian ➡️ Polar

$$\begin{matrix} r=\sqrt{x^{2}+y^{2}} \\ \\ \theta=tan^{-1}\left( \frac{y}{x} \right) \end{matrix}$$

Note: Adjust 𝜃 based on quadrant.

Python Example 1: Convert Polar to Cartesian

import math

# Polar coordinates
r = 5
theta = math.radians(45)  # Convert degrees to radians

# Convert to Cartesian
x = r * math.cos(theta)
y = r * math.sin(theta)

print(f"Cartesian coordinates: x = {x:.2f}, y = {y:.2f}")

Output:

Cartesian coordinates: x = 3.54, y = 3.54

Python Example 2: Convert Cartesian to Polar

import math

# Cartesian coordinates
x, y = 3, 4

# Convert to Polar
r = math.hypot(x, y)
theta = math.atan2(y, x)

print(f"Polar coordinates: r = {r:.2f}, theta = {math.degrees(theta):.2f}°")

Output:

Polar coordinates: r = 5.00, theta = 53.13°

3. Graphing Polar Equations

Common Polar Graphs:

Equation	Shape
r=a	Circle
r=a⋅cos(θ)	Circle shifted
r=a⋅sin(nθ)	Rose curve
r=a+b⋅cos(θ)	Limaçon
r=a⋅ebθ	Spiral (logarithmic)

Python Example 3: Plot Polar Equation

Let’s plot:

$$r=2+2⋅cos(θ)$$

import numpy as np
import matplotlib.pyplot as plt

theta = np.linspace(0, 2 * np.pi, 1000)
r = 2 + 2 * np.cos(theta)

plt.figure(figsize=(6,6))
ax = plt.subplot(111, projection='polar')
ax.plot(theta, r)

plt.title(r'$r = 2 + 2\cos(\theta)$')
plt.show()

Output:

Python Example 4: Plot Polar Equation

Let’s plot:

$$r=3⋅cos(5θ)$$

import numpy as np
import matplotlib.pyplot as plt

# Plot a Rose Curve: r = a * cos(n * theta)
def plot_polar_equation(a, n):
    theta = np.linspace(0, 2 * np.pi, 1000)
    r = a * np.cos(n * theta)
    plt.polar(theta, r)
    plt.title(f"Rose Curve: r = {a} * cos({n} * θ)")
    plt.show()

plot_polar_equation(3, 5)

Output:

Python Example 5: Plot Polar Equation

Let’s plot:

$$r=0.5\theta$$

def plot_spiral(a):
    theta = np.linspace(0, 4 * np.pi, 1000)  # Cover multiple rotations
    r = a * theta
    plt.polar(theta, r)
    plt.title(f"Spiral: r = {a} * θ")
    plt.show()

plot_spiral(0.5)

Output:

4. Symmetry in Polar Graphs

Symmetry	Test
Polar Axis (x-axis)	Replace 𝜃 with −𝜃
Line 𝜃=𝜋/2	Replace 𝜃 with 𝜋−𝜃
Origin	Replace 𝑟 with −𝑟

5. Area in Polar Coordinates

To find the area under a polar curve:

$$A=\frac{1}{2}\int_{\alpha}^{\beta}r^{2}d\theta$$

Python Example 6: Area of a Petal of r=sin(2𝜃)

import sympy as sp

theta = sp.Symbol('theta')
r = sp.sin(2 * theta)

# Area of one petal (0 to pi/2)
area_expr = 0.5 * r**2
area = sp.integrate(area_expr, (theta, 0, sp.pi/2))

print("Area of one petal:", area.evalf())

Output:

Area of one petal: 0.392699081698724

6. Direction of Rotation

In polar plots:

• Increasing θ: rotates counterclockwise

• Negative θ: rotates clockwise

• Negative r: reflects across origin

Keywords

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