Lecture Notes: Graphing Polar Equations

Introduction

• Topic: Graphing polar equations
• Types of Polar Equations Discussed:
  • Circles
  • Limassons
  • Rose Curves
  • Lamina Gates

Circles

Equation: r = a * cos(θ)

• Positive a: Circle directed towards the right
  • Diameter: a
  • Radius: a/2
• Negative a: Circle directed towards the left
  • Diameter: abs(a)
  • Radius: abs(a)/2

Example 1: r = 4 * cos(θ)

• Graph:
  • Circle on the right side
  • Diameter: 4
  • Connect points at (0, 0), (4, 0), (2, 2), and (2, -2)

Example 2: r = -6 * cos(θ)

• Graph:
  • Circle on the left side
  • Diameter: 6
  • Center at (-3, 0)
  • Radius: 3
  • Connect points at (0, 0), (-6, 0), (-3, 3), and (-3, -3)

Equation: r = a * sin(θ)

• Positive a: Circle centered above x-axis
• Negative a: Circle centered below x-axis

Example 1: r = 2 * sin(θ)

• Graph:
  • Circle centered above x-axis
  • Diameter: 2
  • Radius: 1
  • Connect points at (0, 0), (0, 2), (1, 1), and (-1, 1)

Example 2: r = -8 * sin(θ)

• Graph:
  • Circle centered below x-axis
  • Diameter: 8
  • Radius: 4
  • Connect points at (0, 0), (0, -8), (4, -4), and (-4, -4)

Limassons

Equation: r = a ± b * sin(θ) or r = a ± b * cos(θ)

• Positive Sine: Opens towards positive y-axis
• Negative Sine: Opens towards negative y-axis
• Positive Cosine: Opens towards the right
• Negative Cosine: Opens towards the left

Types Based on a/b Ratio

• a/b < 1: Limasson with inner loop
• a/b = 1: Cardioid (Heart-shaped)
• 1 < a/b < 2: Dimpled Limasson
• a/b >= 2: Limasson without a dimple or inner loop (almost a circle)

Example 1: r = 3 + 5 * cos(θ)

• a/b < 1: Inner loop limasson (3/5 = 0.6)
• Graph:
  • Opens to the right (positive cosine)
  • x-intercepts: 2, 8
  • y-intercepts: 3, -3

Example 2: r = 2 - 5 * sin(θ)

• a/b < 1: Inner loop limasson (2/5 = 0.4)
• Graph:
  • Opens downward (negative sine)
  • x-intercepts: 2, -2
  • y-intercepts: -3, -7

Example 3: r = 3 - 7 * cos(θ)

• a/b < 1: Inner loop limasson (3/7 = 0.43)
• Graph:
  • Opens to the left (negative cosine)
  • x-intercepts: -4, -10
  • y-intercepts: 3, -3

Example 4: r = 3 + 3 * cos(θ)

• a/b = 1: Cardioid
• Graph:
  • Opens to the right (positive cosine)
  • x-intercepts: 0, 6
  • y-intercepts: 3, -3

Note on Cosine vs. Sine

• Cosine: Intercepts mainly on y-axis
• Sine: Intercepts mainly on x-axis

Conclusion

• Covered graphing of circles and limassons
• Important to understand the relationship between a and b to identify the graph type