This portion is from 'Advanced Engineering Mathematics'-Erwin Kreyzig[9 th edition]
Doubt from the Red box:-
How can I bravely write the explicit form of the of a curve $C$ in terms of its projection into the $xy$-plane, that is $y=f(x)$. For which equation should I apply implicit function theorem. Please help me. This would help me to give justification for the second one.
Also how parameterization yields advantage over the projection representation as said in the marked box of the text?
I would use the inverse function theorem rather than the implicit function theorem. The projection of $r(t)$ onto the $xy$-plane is just $s(t) = (x(t), y(t))$. Choose an interval $U$ on which $x(t)$ and $y(t)$ are smooth and $x'(t)$ is nonzero; by the inverse function theorem $x$ has a smooth inverse $g \colon V \to U$ defined on an interval $V$. Reparametrizing $s$ over $V$ via $g$ gives:
$$s(g(t)) = (x(g(t)), y(g(t))) = (t, y(g(t)))$$
so we see that the curve $s$ traverses the graph of the function $f = y \circ g$.
You also ask about the advantages of parametric representations. The text below your highlighted box gives two: parametric representations place $x$, $y$, and $z$ all on the same footing rather than arbitrarily singling out an independent variable, and a parametric representation naturally induces an orientation on the curve.
If neither of those reasons are persuasive, I'll add a third: projection representations can't describe all curves. Even with just two variables there is no way to describe a circle as the graph of a single function, but it has many straightforward parametric representations, such as $t \mapsto (\cos(t), \sin(t))$. Actually, here's a fourth: the projection representation encodes only the shape of the curve (meaning the collection of points that it traverses) while a parametric representation can encode how quickly it is traversed. For instance, the projection representations can't distinguish between a spaceship that starts at rest, accelerates for awhile, and then comes to a stop versus a spaceship that travels at constant velocity for the whole trip.