Graphs of Polar Equations

Introduction

• Graphing polar equations is similar to graphing rectangular equations.
• Polar coordinate system consists of circles extending from the pole and lines for specific angles.

Graphing Polar Equations

• Accomplished through point-plotting, trigonometric functions, and symmetry.
• Pick angle measurements for ( \theta ) to determine corresponding ( r ) values.

Symmetry Tests for Polar Coordinates

1. Replace ( \theta ) with ( -\theta ); symmetry with respect to the polar axis.
2. Replace ( \theta ) with ( -\theta ) and ( r ) with ( -r ); symmetry with respect to ( \theta = \pi/2 ).
3. Replace ( r ) with ( -r ); symmetry with respect to the pole.

• Note: A polar equation might fail a test but still exhibit symmetry.

General Types of Polar Equations

Circles in Polar Form

1. ( r = a \cos \theta ): circle with diameter ( a ), left-most edge at the pole.
2. ( r = a \sin \theta ): circle with diameter ( a ), bottom-most edge at the pole.

Limacons (Snails)

• Equations: ( r = a \pm b \sin \theta ) or ( r = a \pm b \cos \theta ) where ( a > 0 ), ( b > 0 ).
• Shape Determination:
  • Ratio ( \frac{a}{b} ) determines shape:
    • ( \frac{a}{b} < 1 ): inner and outer loop.
    • ( \frac{a}{b} = 1 ): cardoid shape.
    • ( 1 < \frac{a}{b} < 2 ): dimpled limacon.
    • ( \frac{a}{b} \geq 2 ): convex shape.

Rose Curves

• Equations: ( r = a \sin n\theta ) or ( r = a \cos n\theta ), with ( n > 1 ) and integer.
• Petal Count:
  • Even ( n ): ( 2n ) petals.
  • Odd ( n ): ( n ) petals.

Lemniscates

• Equations: ( r^2 = a^2 \sin 2\theta ) or ( r^2 = a^2 \cos 2\theta ).
• Shape resembles figure-8 or propeller.

Example 1: Graph ( r = 1 - 2 \cos \theta )

• Type: Limacon.
• Shape: Inner and outer loop (since ( \frac{1}{2} < 1 )).
• Symmetry: Passes polar axis symmetry test.
• Plot points for ( \theta ) values ranging from ( 0 ) to ( \pi ).

Example 2: Graph ( r = 3 \cos 2\theta )

• Type: Rose curve with 4 petals (since ( n = 2 )).
• Symmetry: Passes polar axis symmetry test.
• Plot points for various ( \theta ) values to complete the graph.

This document provides a comprehensive explanation of graphing polar equations through examples and symmetry tests, offering a foundation for understanding the shapes and symmetries involved in these graphs.