A parametric curve defines $x$ and $y$ as functions of a parameter $t$:
$$x = f(t), \quad y = g(t), \quad t \in [a, b]$$
The parameter $t$ often represents time or angle. The curve is traced as $t$ increases.
Eliminating the parameter: Solve one equation for $t$ and substitute, or use an identity.
Example 1: $x = 3\cos t$, $y = 2\sin t$, $t \in [0, 2\pi]$.
Use $\cos^2 t + \sin^2 t = 1$: $\left(\dfrac{x}{3}\right)^2 + \left(\dfrac{y}{2}\right)^2 = 1$.
This is an ellipse with semi-axes $a = 3$ (horizontal) and $b = 2$ (vertical).
Example 2: $x = t^2$, $y = t^3 - t$.
Here $t = \pm\sqrt{x}$, giving $y = \pm\sqrt{x}(x-1)$. The curve has a self-intersection at $t = \pm 1$.
Gradient of a parametric curve:
$$\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{g’(t)}{f’(t)}$$
Arc length:
$$L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\,dt$$
A point is represented as $(r, \theta)$ where $r \geq 0$ is the distance from the origin and $\theta$ is the angle from the positive $x$-axis.
Conversion between polar and Cartesian:
$$x = r\cos\theta, \quad y = r\sin\theta, \quad r = \sqrt{x^2+y^2}, \quad \tan\theta = \frac{y}{x}$$
Common polar curves:
| Polar equation | Cartesian form | Shape |
|---|---|---|
| $r = a$ | $x^2+y^2 = a^2$ | Circle, radius $a$ |
| $\theta = \alpha$ | $y = x\tan\alpha$ | Line through origin |
| $r = 2a\cos\theta$ | $(x-a)^2 + y^2 = a^2$ | Circle, centre $(a,0)$ |
| $r = 2a\sin\theta$ | $x^2 + (y-a)^2 = a^2$ | Circle, centre $(0,a)$ |
| $r = a(1 + \cos\theta)$ | Cardioid | Heart-shaped |
Example 3: Convert $r = 4\cos\theta$ to Cartesian.
Multiply both sides by $r$: $r^2 = 4r\cos\theta \Rightarrow x^2+y^2 = 4x$.
Complete the square: $(x-2)^2 + y^2 = 4$. Circle, centre $(2,0)$, radius $2$.
Example 4: Convert $x^2 + y^2 = 9$ to polar.
$r^2 = 9 \Rightarrow r = 3$.
To sketch $r = f(\theta)$:
1. Make a table of $(\theta, r)$ for key angles: \$0, \pi/6, \pi/4, \pi/3, \pi/2, \ldots, 2\pi$.
2. Note symmetry: $f(-\theta) = f(\theta)$ (symmetric about polar axis); $f(\pi - \theta) = f(\theta)$ (symmetric about $y$-axis).
3. Plot each $(r, \theta)$ point in polar coordinates.
Example 5: Sketch $r = 1 + 2\cos\theta$.
| $\theta$ | $0$ | $\pi/3$ | $\pi/2$ | $2\pi/3$ | $\pi$ |
|---|---|---|---|---|---|
| $r$ | $3$ | $2$ | $1$ | $0$ | $-1$ |
When $r < 0$, the point is plotted in the opposite direction. This produces a limaon with inner loop.
KEY TAKEAWAY: Parametric equations describe curves that may not be functions of $x$; polar equations describe curves whose natural symmetry is rotational. Both require conversion skills.
EXAM TIP: When converting parametric to Cartesian, look for Pythagorean identities ($\sin^2 t + \cos^2 t = 1$, \$1 + \tan^2 t = \sec^2 t$) before solving for $t$ algebraically.
COMMON MISTAKE: Forgetting to multiply both sides by $r$ when converting polar equations — e.g., $r = 2\cos\theta$ must become $r^2 = 2r\cos\theta$ before substituting.
