Parametric and Polar Representations of Curves

In VCE Specialist Mathematics, curves can be represented in forms other than the standard Cartesian equation $y = f(x)$. Parametric and polar forms allow for the description of complex paths, such as circles, ellipses, and spirals, which may not be functions in the Cartesian plane.

1. Parametric Equations

A curve is defined parametrically if both the $x$ and $y$ coordinates are expressed as functions of a third variable, $t$, known as the parameter.

$$x = f(t), \quad y = g(t) \quad \text{for } t \in [a, b]$$

• Direction of Motion: As $t$ increases, the curve is traced in a specific direction (orientation).
• Domain: The interval of $t$ determines the starting and ending points of the curve.

Eliminating the Parameter

To convert parametric equations to a Cartesian equation, we eliminate $t$. Common methods include:

1. Rearrangement and Substitution: Solve one equation for $t$ and substitute it into the other.
2. Trigonometric Identities: Use identities to link $x$ and $y$ when they involve circular functions.
   • $\sin^2(t) + \cos^2(t) = 1$ (Used for circles and ellipses)
   • $\sec^2(t) - \tan^2(t) = 1$ (Used for hyperbolas)
   • \$1 + \cot^2(t) = \csc^2(t)$

Worked Example: The Ellipse

Given $x = 1 + 3\cos(t)$ and $y = -2 + 2\sin(t)$ for $t \in [0, 2\pi]$:
1. Rearrange for the trig terms: $\cos(t) = \frac{x-1}{3}$ and $\sin(t) = \frac{y+2}{2}$.
2. Substitute into $\cos^2(t) + \sin^2(t) = 1$:
$$\left(\frac{x-1}{3}\right)^2 + \left(\frac{y+2}{2}\right)^2 = 1 \implies \frac{(x-1)^2}{9} + \frac{(y+2)^2}{4} = 1$$
This is the Cartesian equation of an ellipse centered at $(1, -2)$.

EXAM TIP: When converting to Cartesian form, always check if the domain of the parameter $t$ restricts the $x$ or $y$ values. For example, if $x = \cos^2(t)$, then $x$ must be in the interval $[0, 1]$.

2. Polar Coordinates

The polar coordinate system represents a point $P$ in the plane by its distance from the origin ($r$) and the angle ($\theta$) it makes with the positive $x$-axis.

• $r$: The radial distance from the pole (origin). Note that $r$ can be negative, representing a point in the opposite direction of $\theta$.
• $\theta$: The polar angle (usually in radians), measured anticlockwise from the polar axis.

Conversion Formulas

The relationship between Cartesian coordinates $(x, y)$ and polar coordinates $(r, \theta)$ is defined by:

KEY TAKEAWAY: A single point in the plane has infinitely many polar representations because $\theta$ is periodic (e.g., $(r, \theta)$ is the same as $(r, \theta + 2\pi)$). By convention, we often use $r \ge 0$ and $\theta \in (-\pi, \pi]$ or $[0, 2\pi)$.

3. Polar Curves and Graphs

A polar equation is typically given in the form $r = f(\theta)$.

Common Polar Graphs

1. Circles:
   • $r = a$: Circle centered at the origin with radius $a$.
   • $r = 2a\cos(\theta)$: Circle centered at $(a, 0)$ with radius $a$.
   • $r = 2a\sin(\theta)$: Circle centered at $(0, a)$ with radius $a$.
2. Lines:
   • $\theta = \alpha$: A straight line passing through the origin at angle $\alpha$.
   • $r = a\sec(\theta)$: A vertical line $x = a$.
   • $r = a\csc(\theta)$: A horizontal line $y = a$.
3. Spirals:
   • $r = a\theta$ (Archimedean spiral): The distance from the origin increases linearly with the angle.

Conversion of Equations

• To Polar: Replace $x$ with $r\cos(\theta)$ and $y$ with $r\sin(\theta)$.
  • Example: $x^2 + y^2 = 4x \implies r^2 = 4r\cos(\theta) \implies r = 4\cos(\theta)$.
• To Cartesian: Use $r^2 = x^2 + y^2$, $r\cos(\theta) = x$, and $r\sin(\theta) = y$.
  • Example: $r = \frac{2}{1 + \cos(\theta)} \implies r + r\cos(\theta) = 2 \implies \sqrt{x^2+y^2} + x = 2$.
  • Rearrange: $\sqrt{x^2+y^2} = 2 - x \implies x^2 + y^2 = 4 - 4x + x^2 \implies y^2 = 4 - 4x$.

COMMON MISTAKE: When converting $r = f(\theta)$ to Cartesian form, students often forget to square both sides correctly or fail to recognize that $r = \sqrt{x^2+y^2}$ only if $r \ge 0$.

4. Interpretation and Sketching

When sketching parametric or polar curves, consider:

1. Intercepts:
   • $x$-intercepts: Set $y=0$ (in parametric) or find $\theta$ such that $r\sin(\theta)=0$ (in polar).
   • $y$-intercepts: Set $x=0$ (in parametric) or find $\theta$ such that $r\cos(\theta)=0$ (in polar).
2. Symmetry in Polar Graphs:
   • If $f(\theta) = f(-\theta)$, the curve is symmetric about the polar axis ($x$-axis).
   • If $f(\theta) = f(\pi - \theta)$, the curve is symmetric about the line $\theta = \frac{\pi}{2}$ ($y$-axis).
3. Asymptotes: Occur in polar equations when $r \to \infty$ for certain values of $\theta$.

Gradient of a Parametric Curve

To find the gradient $\frac{dy}{dx}$ of a curve defined parametrically:
$$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$$
This is essential for finding stationary points or the equations of tangents to parametric curves.

VCAA FOCUS: VCAA frequently examines the conversion of an ellipse or hyperbola from parametric to Cartesian form and vice versa. Be comfortable with the identity $\sec^2(t) = 1 + \tan^2(t)$ for hyperbolas.