Question

Consider the family of limaçons \(r=1+b \cos \theta .\) Describe how the curves change as \(b \rightarrow \infty\)

Short answer

Answer: As \(b \rightarrow \infty\), the limaçon curve approaches a vertical line at \(r=0\) on the polar plane, with angles \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\).

Step-by-step solution

Understanding limaçons in polar coordinates

Limaçons are curves represented by the equation \(r = a + b \cos \theta\) or \(r = a + b \sin \theta\) in polar coordinates, where \(a\) and \(b\) are constants, and \(\theta\) is the angle. In this exercise, we are considering the family of limaçons given by \(r = 1 + b \cos \theta\), where \(b\) is the variable parameter.

Cases for the value of b

These curves have different shapes depending on the value of \(b\): 1. If \(b < 1\), the limaçon is a "diminishing" curve with a loop. 2. If \(b = 1\), the limaçon is a cardioid. 3. If \(b > 1\), the limaçon does not have a loop but has a "dent" or "cusp" on the inner part. 4. If \(b \rightarrow \infty\), we need to analyze the behavior of the limaçon.

Analyzing the behavior of the limaçon as b approaches infinity

To study the behavior of the limaçon as \(b \rightarrow \infty\), we can observe the equation \(r = 1 + b \cos \theta\). As \(b\) becomes larger, the value of \(b \cos \theta\) will dominate the equation, and the constant term \(1\) becomes negligible. The equation looks like: \(r \approx b\cos\theta\). By dividing both sides of the equation by \(b\), we get \(\frac{r}{b} \approx \cos \theta\). As \(b \rightarrow \infty\), the fraction \(\frac{r}{b}\) approaches \(0\), so we have \(0 \approx \cos\theta\). The solution of this equation is \(\theta = \frac{\pi}{2}\) or \(\theta = \frac{3\pi}{2}\)

Concluding the behavior of the limaçon as b approaches infinity

As \(b\) becomes larger, the limaçon curve becomes closer and closer to a vertical line passing through \(r=0\), and with angles \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\). So, as \(b \rightarrow \infty\), the limaçon curve approaches a vertical line at \(r=0\) on the polar plane.

Key concepts

Limaçons

Limaçons are an intriguing family of curves in polar coordinates identified by equations of the form \(r = a + b \cos \theta\) or \(r = a + b \sin \theta\). Here, \(a\) and \(b\) are constants, and \(\theta\) is the angle in radians. These curves are known for their diverse shapes, potentially exhibiting loops, dimples, or resembling a circle depending on the respective values of \(a\) and \(b\).
The exercise in question focuses on a specific type of limaçon where the equation is \(r = 1 + b \cos \theta\). This reveals how the parameter \(b\) drastically alters the curve's appearance:

• If \(b < 1\), the curve manifests as a dimpling or looping structure, hinting at a smaller enclosed area.
• If \(b = 1\), it transitions into a unique heart-shaped curve termed a "cardioid".
• When \(b > 1\), the limaçon loses its loop but retains a cusp-like appearance, forming a defined dimple.

The beauty of limaçons lies in their geometric versatility, showcasing a spectrum of forms with minimal adjustments to the constants involved.

Cardioid

Among the fascinating shapes limaçons take is the cardioid, which occurs specifically when \(b = 1\) in the equation \(r = 1 + b \cos \theta\). A cardioid, deriving its name from the Greek word for "heart," resembles the shape of a heart and is a particularly symmetrical and aesthetically pleasing curve.
This charm arises from its balanced properties:

• All points on a cardioid are equidistant from the radial line, which contributes to its symmetry.
• The curve lacks the internal loop seen in some other limaçons, maintaining a continuous and smooth outline.

The cardioid is often observed in optical and acoustical applications due to its focused properties, serving as a key concept in understanding polar plots and their symmetry.

Infinity Behavior

Understanding the behavior of limaçons as \(b\) approaches infinity provides insight into the fundamental dynamics of polar coordinates. As we analyze the equation \(r = 1 + b \cos \theta\), it becomes evident that as \(b\) increases significantly, the influence of the term \(b \cos \theta\) overshadows the constant term 1.
The behavior can be described as follows:

• In the limit where \(b\) is extremely large, the equation simplifies to \(r \approx b \cos \theta\).
• Analyzing this form, by dividing through by \(b\), we find \(\frac{r}{b} \approx \cos \theta\), which converges to 0 as \(b\) goes to infinity.
• Thus, \(\cos \theta\) centers around 0, meaning \(\theta\) approaches \(\frac{\pi}{2}\) or \(\frac{3\pi}{2}\), indicating the curve approximates the vertical line at \(r = 0\).

Therefore, the limaçon effectively transforms into a straight vertical line, emphasizing the dramatic changes in geometric structure as parameters evolve towards extreme values.