Parametric Equations

The central idea

Parametric equations $x=x(t)$, $y=y(t)$ assign both coordinates as functions of a parameter over a stated interval. The same geometric curve may have different parametrizations, speeds, and orientations. Eliminating $t$ identifies a Cartesian relation but can lose direction or parameter restrictions.

Definitions, hypotheses, and notation

A parametrization is regular when its velocity vector does not vanish. Regularity helps guarantee a well-defined tangent direction, but a curve may still be meaningful at a singular parameter value. Reparametrizing by an increasing function preserves orientation; a decreasing change reverses it.

For $x=a+h t,y=b+k t$, the path is a line through $(a,b)$ with direction $\langle h,k\rangle$. For periodic trigonometric coordinates, compare the parameter interval length with the fundamental period to determine whether the path is partial, complete, or repeatedly traced.

Conceptual meaning

Rather than requiring one output $y$ for each $x$, a parameter traces a path. This represents circles, loops, and motion naturally. The ordered pair at each parameter value is the position of the moving point.

A dependable method and decision rule

1. Record the permitted parameter interval.
2. Make a small table of parameter values and corresponding points.
3. Eliminate the parameter when useful, retaining coordinate restrictions.
4. Determine orientation from increasing $t$.
5. Check whether different parameter values repeat points or the entire curve.

Fully worked example

Common mistakes and why they fail