By using similar arguments to the ones from my answer to this question, I can prove that the homogeneous coordinate ring of the rational quartic curve in $\mathbb P^3$, that is, $$R = K[x_1, x_2, x_3, x_4]/\left< x_1^2x_3-x_2^3,x_1x_3^2-x_2^2x_4,x_2x_4^2-x_3^3,x_1x_4-x_2 x_3\right>,$$ is isomorphic to $K[s^4,s^3t,st^3,t^4]$. (This is also Exercise 18.8 in Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry.)
I'm interested in geometric arguments which I expect to be simpler than the algebraic ones. References are also welcome.
Edit. There is some connection with this topic where it is proved that $R$ is an integral domain.