Worked problems: Infinite polar coordinates explained with solutions

📚 Understanding Infinite Polar Coordinates

Infinite polar coordinates might sound intimidating, but they're really just an extension of regular polar coordinates to describe points that are infinitely far away! Let's break it down:

• 🧭 Definition: In standard polar coordinates, a point is defined by $(r, \theta)$, where $r$ is the distance from the origin and $\theta$ is the angle from the positive x-axis. Infinite polar coordinates consider what happens as $r$ approaches infinity. We aren't representing a specific point, but rather a direction.
• 📜 History/Background: The concept evolved from the need to describe asymptotic behavior in mathematics and physics, especially when dealing with fields that extend infinitely. It doesn't have a single inventor, but rather arose organically within the development of calculus and analysis.
• 📐 Key Principles: The key idea is that as $r \rightarrow \infty$, the angle $\theta$ determines the "direction" in which the point is receding. Think of it like shining a laser beam – the angle dictates where the beam points even though it goes on forever.
• 💡 Real-World Examples: Imagine plotting the path of a spacecraft escaping a planet's gravitational pull. As it travels further and further away, its position can be effectively described by an angle, even though its radial distance is constantly increasing. Another example is analyzing the far-field radiation pattern of an antenna.

📝 Worked Problems

Let's solidify this with some examples:

1. 📐 Problem 1: Limit of a Spiral

   Consider the spiral defined by the polar equation $r = \theta$. What happens as $\theta$ approaches infinity?

   Solution: As $\theta$ becomes infinitely large, $r$ also becomes infinitely large. There is no single “infinite polar coordinate” in this case, as the spiral never approaches a fixed direction. However, we can analyze its behavior. As $\theta$ increases, the spiral continuously winds outwards. There's no asymptotic direction; it just keeps going!

2. 🧭 Problem 2: Asymptotic Line

   Consider the curve defined by $r = \frac{1}{\theta}$ as $\theta \rightarrow 0^+$. What is the asymptotic direction?

   Solution: As $\theta$ approaches 0 from the positive side, $r$ approaches infinity. In Cartesian coordinates, $x = r \cos(\theta) = \frac{\cos(\theta)}{\theta}$ and $y = r \sin(\theta) = \frac{\sin(\theta)}{\theta}$. As $\theta \rightarrow 0^+$, $y \rightarrow 1$ and $x \rightarrow \infty$. Therefore, the curve approaches the line $y=1$ as $x$ goes to infinity. The direction is along the positive x-axis, but approaching the line $y=1$.

3. 💡 Problem 3: Hyperbola in Polar Form

   Analyze the behavior of the hyperbola $r^2 \cos(2\theta) = 1$ as $r \rightarrow \infty$.

   Solution: We can rewrite the equation as $r = \pm \frac{1}{\sqrt{\cos(2\theta)}}$. As $r$ goes to infinity, $\cos(2\theta)$ must approach zero. This happens when $2\theta = \frac{\pi}{2} + n\pi$, where $n$ is an integer. Therefore, $\theta = \frac{\pi}{4} + \frac{n\pi}{2}$. The hyperbola has asymptotes at $\theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}$.

✅ Practice Quiz

1. ❓ Question 1

   What happens to the curve $r = e^{\theta}$ as $\theta$ approaches infinity?

2. ❓ Question 2

   Find the asymptotes of the curve $r = \frac{1}{\sin(\theta) - \cos(\theta)}$.

3. ❓ Question 3

   Describe the behavior of the curve $r = \theta^2$ as $\theta$ approaches infinity.

⭐ Conclusion

Understanding infinite polar coordinates helps us analyze the behavior of curves and functions as they extend to infinity. It provides a valuable tool for describing asymptotic behavior and directions in various mathematical and physical contexts. Keep practicing, and you'll master it in no time!