Parametrizing implicit curves

A proposed parametrization $(x(t), y(t))$ belongs to an implicit curve exactly when substituting both formulas into the equation produces a true statement for every t. For the circle $x^2 + y^2 = 9$, try $x = 3\cos t$, $y = 3\sin t$: substitution gives $9\cos^2 t + 9\sin^2 t = 9(\cos^2 t + \sin^2 t) = 9$. True for all t, so every traced point sits on the circle.

The Pythagorean identity $\cos^2 t + \sin^2 t = 1$ is the engine of every circular and elliptical parametrization, but it only fires when the coefficients cooperate. Check by substituting, never by resemblance: plenty of cos/sin pairs that look right produce $4\cos^2 t + 9\sin^2 t$, which is not constant and belongs to no circle.

1. Circle $x^2 + y^2 = r^2$: use $x = r\cos t$, $y = r\sin t$ on $[0, 2\pi]$. Starts at (r, 0), sweeps counterclockwise.
2. Ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$: use $x = a\cos t$, $y = b\sin t$. Each coefficient is the square root of its own denominator, matched to its own axis; substitution then collapses to $\cos^2 t + \sin^2 t = 1$.
3. A function's graph $y = f(x)$: use $x = t$, $y = f(t)$. Trivial and perfectly legal; every explicit graph is one substitution away from parametric form.
4. A line segment from P to Q: interpolate, $x$ and $y$ each running linearly from their P value to their Q value as t goes 0 to 1.

These are templates, not laws: they are the most convenient covers, chosen among infinitely many.

Substitution proves the journey stays on the curve. It says nothing about how much of the curve gets visited: that is the t interval's job, and it must be audited separately. $x = \cos t, y = \sin t$ on $[0, \pi]$ satisfies $x^2 + y^2 = 1$ at every instant and covers only the top half.

Coverage can also fail through the formulas themselves. $x = t^2, y = t^4$ satisfies $y = x^2$ for every real t, but $t^2 \ge 0$ traps x in the right half: no interval choice ever reaches the parabola's left arm. The audit question is concrete: what set of x values (and y values) do the formulas actually produce over the interval? Match that against the piece you were asked to cover.

An implicit curve never has just one parametrization. The unit circle is covered by $(\cos t, \sin t)$ counterclockwise, by $(\cos t, -\sin t)$ clockwise, by $(\cos 2t, \sin 2t)$ at double speed (a full loop in t-length $\pi$), and by versions starting anywhere on the rim. All substitute correctly; they differ in start, speed, and direction, which is precisely what parametrization adds to an implicit curve.

So treat "find a parametrization" as an engineering task with many right answers, and "is this parametrization correct" as two checks in sequence: substitute (does it stay on the curve?), then audit (does it cover the requested piece, in the requested manner?).