Limaçons in Polar Coordinates: Shapes Guide

Based on your image, you are studying the classification of Limaçons defined by the polar equation $r = a(p \pm q \cos \theta)$ or $r = a(p \pm q \sin \theta)$, specifically for the case where $p \ge q$. The notes categorize these curves into "heart" shapes (cardioids), "egg" shapes (convex limaçons), and shapes with a "dimple" (dimpled limaçons).

Answer

The shape of a limaçon is determined by the ratio $\frac{p}{q}$: if $p=q$, it is a cardioid (heart shape with a cusp); if $q < p < 2q$, it is a dimpled limaçon; and if $p \ge 2q$, it is a convex limaçon (egg shape). The orientation depends on whether $\cos \theta$ (horizontal) or $\sin \theta$ (vertical) is used.

Explanation

1. The Cardioid ($p = q$) When $p=q$, the equation simplifies to $r = a(1 \pm \cos \theta)$. At a certain angle, the radius $r$ becomes exactly zero (e.g., at $\theta = \pi$ for $1 + \cos \theta$), creating a "sharp point" known as a cusp at the origin. $r = a(1 + \cos \theta) \implies \text{at } \theta = \pi, r = a(1 - 1) = 0$ This formula shows that the curve touches the pole (origin) exactly once, forming the characteristic heart shape.

2. The Dimpled Limaçon ($q \le p < 2q$) In this range, $r$ is always positive, so the curve never touches the origin. However, because $p$ is less than $2q$, the rate of change of the radius creates a "dent" or dimple on one side. $r = 3 + 2 \cos \theta \implies r_{\text{min}} = 3 - 2 = 1, r_{\text{max}} = 3 + 2 = 5$ The radius transitions from 5 to 1, but the curvature changes sign, creating a non-convex "dimple" rather than a sharp point. ⚠️ This step is required on exams: You must check if $1 < \frac{p}{q} < 2$ to justify the dimpled shape.

3. The Convex Limaçon ($p \ge 2q$) When $p$ is at least twice as large as $q$, the constant term $p$ dominates the fluctuation of the trigonometric term. The resulting shape is "egg-like" and mathematically convex, meaning it has no dimples. $r = 5 + 2 \cos \theta \implies r_{\text{min}} = 5 - 2 = 3$ Because the minimum radius is sufficiently large relative to the variation, the curve stays rounded and flattened rather than indented.

4. Symmetry and Trigonometric Functions The image shows that functions using $\cos \theta$ are symmetric about the initial line ($\theta = 0$), while functions using $\sin \theta$ are symmetric about the vertical line ($\theta = \frac{\pi}{2}$). $\text{Symmetry: } \cos(\theta) = \cos(-\theta) \text{ and } \sin(\theta) = \sin(\pi - \theta)$ This explains why the "heart" points left/right for cosine and up/down for sine.

Final Answer

The behavior of $r = a(p + q \cos \theta)$ is summarized by the ratio:

Ratio Condition	Shape Name	Visual Feature
$\frac{p}{q} = 1$	Cardioid	Cusp (Sharp point) at origin
$1 < \frac{p}{q} < 2$	Dimpled Limaçon	Shallow dent, never hits origin
$\frac{p}{q} \ge 2$	Convex Limaçon	Egg shape, no dent

Numerical Example: For $r = 1 + \cos \theta$, the cusp occurs at: $\boxed{\theta = \pi, r = 0}$

Common Mistakes

• Confusing Cusp vs. Dimple: Students often draw a dimpled limaçon ($p > q$) touching the origin. Remember: the curve ONLY touches the origin if $p = q$.
• Incorrect Symmetry: Aligning a $\cos \theta$ graph along the vertical axis. Remember: $\cos$ corresponds to $x$-axis symmetry, and $\sin$ corresponds to $y$-axis symmetry.