Graphing in Polar Coordinates & Geometric Applications

1. Coordinate Transformation Equations

In the plane $\mathbb{R}^2$, a point $P$ is determined by its directed distance $r$ from the origin (pole) and counterclockwise angle $\theta$ from the positive horizontal axis (polar axis):

2. Symmetry Criteria for Polar Curves $r = f(\theta)$

1. Symmetry about Polar Axis (Horizontal Axis): The equation is unchanged when replacing $(r, \theta)$ with $(r, -\theta)$ or $(-r, \pi - \theta)$.
2. Symmetry about Normal Axis $\theta = \pi/2$ (Vertical Axis): The equation is unchanged when replacing $(r, \theta)$ with $(r, \pi - \theta)$ or $(-r, -\theta)$.
3. Symmetry about the Pole (Origin): The equation is unchanged when replacing $(r, \theta)$ with $(-r, \theta)$ or $(r, \theta + \pi)$.

3. Zoology of Classical Polar Curves

• Cardioids: $r = a(1 \pm \cos\theta)$ or $r = a(1 \pm \sin\theta)$ ($a > 0$). Heart-shaped curves passing through the pole with a cusp at $r = 0$.
• Limaçons: $r = a + b\cos\theta$. If $a < b$, it features an inner loop; if $a = b$, it is a cardioid; if $a > b$, it is dimpled or convex.
• Rose Curves: $r = a\cos(n\theta)$ or $r = a\sin(n\theta)$. If $n \in \mathbb{N}$ is odd, the curve has $n$ petals; if $n$ is even, the curve has $2n$ petals.
• Lemniscates: $r^2 = a^2\cos(2\theta)$. Figure-eight curves with nodal tangents at $\theta = \pm \pi/4$.

1. Slope of the Cartesian Tangent $\frac{dy}{dx}$

Treating $\theta$ as a parametric variable with $x(\theta) = r(\theta)\cos\theta$ and $y(\theta) = r(\theta)\sin\theta$:

2. The Polar Angle $\psi$ (Angle Between Radius Vector and Tangent)

Let $\phi$ be the inclination angle of the tangent line with the polar axis ($\tan\phi = \frac{dy}{dx}$) and $\theta$ be the vectorial angle of $P$. The angle $\psi = \phi - \theta$ between the radius vector $OP$ and the tangent line satisfies:

Substituting $\tan\phi = \frac{r' \sin\theta + r \cos\theta}{r' \cos\theta - r \sin\theta}$ and $\tan\theta = \frac{\sin\theta}{\cos\theta}$:

This remarkably elegant identity proves that the geometry of polar tangents depends solely on $r$ and its angular rate of change.

1. Derivation of the Polar Area Element

Consider an infinitesimal sector between rays $\theta$ and $\theta + d\theta$ bounded by $r = f(\theta)$. Approximating this sector by a circular sector of radius $r$ and central angle $d\theta$:

Integrating from $\theta = \alpha$ to $\theta = \beta$ yields the total swept area:

2. Area Between Two Polar Curves

If $r_{\text{outer}}(\theta) \ge r_{\text{inner}}(\theta) \ge 0$ for $\theta \in [\alpha, \beta]$:

3. Critical Precaution: Multi-Loop & Overlapping Limits

For rose curves $r = a\cos(n\theta)$, a single petal is bounded between adjacent roots where $r = 0$. For $r = a\cos(2\theta)$, setting $2\theta = \pm \frac{\pi}{2} \implies \theta \in [-\frac{\pi}{4}, \frac{\pi}{4}]$ bounds exactly one petal. Integrating over $[0, 2\pi]$ directly without symmetry can count overlapping loops multiple times.

1. Analytical Derivation of $ds$

Recall $dx = (r'\cos\theta - r\sin\theta)d\theta$ and $dy = (r'\sin\theta + r\cos\theta)d\theta$. Squaring and summing:

2. Total Arc Length Integral

1. Revolution About the Polar Axis ($x$-axis)

When revolving about the polar axis ($y = 0$), the perpendicular distance to the rotation axis is $y = r\sin\theta$.

• Surface Area of Revolution: $$\mathbf{S = 2\pi \int_\alpha^\beta r\sin\theta \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} \, d\theta}$$
• Volume of Revolution: Revolving differential triangular sectors $dA$ gives conical rings: $$\mathbf{V = \frac{2}{3}\pi \int_\alpha^\beta r^3 \sin\theta \, d\theta}$$

2. Revolution About the Normal Axis $\theta = \pi/2$ ($y$-axis)

The perpendicular distance is $x = r\cos\theta$: