Implicit form and parametric form

A curve in the plane can be represented in two complementary ways.

Observation — Implicit form vs parametric form

A curve in the plane is represented in two ways:

• implicit form: $F(x,y)=0$ (e.g. $x^2+y^2-r^2=0$, $\;x^2/a^2 + y^2/b^2 - 1 = 0$).
• parametric form: $x=x(t)$, $y=y(t)$ (circle, ellipse, Lissajous, helices).

Merits of the parametric form: we know how, not just where, the curve exists (velocity, direction); it is indispensable for line integrals and for physics. Merits of the implicit form: a single equation, direct intersections with other curves. One passes from one to the other: implicit $\to$ parametric by inspection (with sines/cosines or roots); parametric $\to$ implicit by eliminating $t$.

Example — Eliminating the parameter

From $x=2\cos t$, $y=3\sin t$ ($t\in[0,2\pi)$) one obtains the implicit form: $\cos t = x/2, \quad \sin t = y/3 \implies (x/2)^2 + (y/3)^2 = 1 \implies \frac{x^2}{4} + \frac{y^2}{9} = 1.$ It is an ellipse with horizontal semi-axis $a=2$ and vertical semi-axis $b=3$.

Links

Topics: Complex numbers
Concepts: Implicit form · Parametric form · Parametrisation
Skills: Model