Eliminating the Parameter and Converting to Rectangular Form

A parametric curve gives x and y each as separate functions of a third variable t, often standing for time. Eliminating the parameter means removing t so you're left with one equation connecting x and y only — the kind of equation you'd recognize as a line, parabola, circle, or ellipse. The curve was always that shape; t was just the clock telling you where on the shape you are at each moment. Strip away the clock and the shape stays behind.

The mechanical move: pick whichever equation is easier to solve for t, isolate it, then substitute that expression into the other equation. For x = 2cos(t), y = 3sin(t), solve each for the trig function instead: cos(t) = x/2 and sin(t) = y/3. Since cos²(t) + sin²(t) = 1, substituting gives (x/2)² + (y/3)² = 1, an ellipse with horizontal radius 2 and vertical radius 3. Notice that once t is gone, so is every trace of time — the equation can't tell you whether the point moved clockwise or counterclockwise, or where it started at t = 0. That information lived only in the original parametric equations.

This sits right next to parametric differentiation. When you compute dy/dx as (dy/dt)/(dx/dt), you're finding the slope of the same curve without ever eliminating t — useful when elimination is messy or impossible. Eliminating the parameter is the geometric twin of that algebraic tool: one tells you the shape, the other tells you the slope at a point, and both describe the same underlying path.