Cartesian and parametric forms

Two ways to describe the same curve

• A Cartesian equation relates $x$ and $y$ directly: $x^2 + y^2 = 25$. It answers "is this point on the curve?"
• Parametric equations give $x$ and $y$ each in terms of a third variable, the parameter: $x = 5\cos\theta$, $y = 5\sin\theta$. They answer "where is the point when the parameter is $\theta$?"
• The parameter usually has a meaning: an angle for a circle or ellipse, time for a moving object, or a position along the curve.
• Both describe exactly the same set of points. Choosing between them is a matter of which question you are answering.

The standard parametric forms

Conic	Cartesian	Parametric
Circle radius $r$, centre $(h,k)$	$(x-h)^2 + (y-k)^2 = r^2$	$x = h + r\cos\theta$, $y = k + r\sin\theta$
Ellipse	$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$	$x = a\cos\theta$, $y = b\sin\theta$
Parabola $y^2 = 4ax$	$y^2 = 4ax$	$x = at^2$, $y = 2at$
Hyperbola	$\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$	$x = a\sec\theta$, $y = b\tan\theta$
Rectangular hyperbola	$xy = c^2$	$x = ct$, $y = \dfrac{c}{t}$

• Notice the pattern. The trigonometric forms rely on an identity: $\cos^2\theta + \sin^2\theta = 1$ for the ellipse, and $\sec^2\theta - \tan^2\theta = 1$ for the hyperbola. The identity is what makes the substitution work.

• Sketch the curve before you start manipulating it. Knowing which of these three shapes you are dealing with tells you immediately what a sensible answer looks like — an ellipse is bounded, a parabola and a hyperbola are not, and a hyperbola has two separate branches that a single point cannot bridge.

Converting parametric to Cartesian — eliminate the parameter

• The goal is to remove $t$ or $\theta$ entirely, leaving a relationship between $x$ and $y$ alone.

• Three standard techniques:

  • Rearrange and substitute — make the parameter the subject of one equation and put it into the other. Works whenever the parameter appears algebraically.
  • Use a trigonometric identity — isolate $\cos\theta$ and $\sin\theta$, then use $\cos^2 + \sin^2 = 1$.
  • Multiply or divide the two equations — especially effective for $x = ct$, $y = c/t$, where multiplying gives $xy = c^2$ in one step.

• Worked through — $x = 3\cos\theta$, $y = 5\sin\theta$:

  • Isolate: $\cos\theta = \dfrac{x}{3}$ and $\sin\theta = \dfrac{y}{5}$
  • Apply the identity: $\left(\dfrac{x}{3}\right)^2 + \left(\dfrac{y}{5}\right)^2 = 1$
  • $\dfrac{x^2}{9} + \dfrac{y^2}{25} = 1$ — an ellipse with a vertical major axis

• Worked through — $x = 2t^2$, $y = 4t$:

  • From the second, $t = \dfrac{y}{4}$
  • Substitute: $x = 2\left(\dfrac{y}{4}\right)^2 = \dfrac{2y^2}{16} = \dfrac{y^2}{8}$
  • $y^2 = 8x$ — a parabola with $4a = 8$, so $a = 2$ ✓ matching the form $x = at^2$, $y = 2at$

Converting Cartesian to parametric

• Match the equation to a standard form and read off the parameters.
• The choice is not unique. $y = x^2$ can be written $x = t$, $y = t^2$, or $x = 2t$, $y = 4t^2$ — both trace the same parabola. Any valid parametrisation is acceptable unless the question asks for a particular one.
• State the range of the parameter when it matters. $x = 3\cos\theta$, $y = 3\sin\theta$ traces the whole circle for $0 \leq \theta < 2\pi$; restricting to $0 \leq \theta \leq \pi$ gives only the upper semicircle.

Restrictions the parametric form imposes

• Parametric forms can miss points, and saying so is often the point of the question:
  • $x = a\sec\theta$, $y = b\tan\theta$ is undefined at $\theta = \frac{\pi}{2}$, where $\sec$ and $\tan$ are undefined.
  • $x = ct$, $y = \frac{c}{t}$ excludes $t = 0$, so the curve never touches either axis — correctly, since $xy = c^2 \neq 0$.
  • $x = at^2$ is never negative, so this parametrisation covers only the right-opening parabola.
• Check the domain and range your parametrisation produces, and compare it with the Cartesian curve.

Why parametric forms are worth having

• They handle vertical tangents naturally. At the top of a circle, $\frac{dy}{dx}$ is undefined in Cartesian terms, but the parametric description is perfectly well behaved.
• They give each point a label. "The point with parameter $t$" is a single symbol that stands for a general point, which is exactly what a proof about every point needs.
• They describe motion. With $t$ as time, the parametric equations give position, and their derivatives give velocity.
• Tangent equations become formulas in the parameter, which is how the standard's general tangent results are derived.