Two parametrizations of a curve. Are both valid?

Question

I want to find the curve of intersection of $z=4x^2+y^2$ and $y=x^2$.

To do this, I can assign a parameter $t$ to the variable $y$, so that $$y=t, \quad x=\sqrt t, \quad z=4t+t^2$$ and the parametric form of the curve is $$( \sqrt t, t, 4t+t^2).$$ But I can also choose to assign the parameter $x=t$, so that $$x=t, \quad y=t^2, \quad z=4t^2+t^4$$ and the parametric form of the curve is $$( t, t^2, 4t^2+t^4).$$

These expressions don't seem to be equivalent to each other, which make me wonder: Is there any mistake on the previous calculations? Or are these two parametric representations valid for the curve I am looking for?

Answer 1

You say “These expressions don't seem to be equivalent to each other.” How so? Can you find a point $(x,y,z)$ that only one of the two parameterizations hits? The interpretation of $t$ is different in the parameterizations (it is the $x$ coordinate in one and the $y$ coordinate in the other), but the curve they describe is the same. Similarly, the equations $y=x^3$ and $\sqrt[3]y=x$ don’t look the same, but they describe the same curve.

Different parameterizations can describe the same curve.

Answer 2

you can write $$z=4x^2+x^4,y=x^2$$

Answer 3

There is no need to expect a unique parametrisation should exist, i.e. many different parametrisation could represent the same curve.

If the parameter were physically interpreted as time, the parametrisation could be interpreted as depending on the speed an imaginary $1$D creature crawls along the curve at.

Nevertheless, quantities such the length should be parametrisation-independent.

Answer 4

One thing, never assign a domain restricting function to the coordinate variables i.e choose $x = t, \cos(t), \sin(t)$, etc. Basically anything which is defined for all $t$. This way you don't have to worry about the domain in which you need to get the whole curve.

Also, for two parametrizations $\gamma_1(t)$ and $\gamma_1(s)$ which trace the same underlying curve, then the parameters $t,s$ are related by a diffeomorphism $\phi$ i.e $\gamma_1( \phi(t)) = \gamma_1(s)$. I'll give you an example. Let $s = t^2$ then we have,

$$ \gamma(t) = (t,t^2) \ \ \ \ \ \gamma(s) = (t^2, t^4) = (s,s^2)$$

The underlying curve in which these functions parametrize is the parabola $y = x^2$.