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.

• 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.