= Moment curve
{wiki}
The degree-$d$ moment curve is $\gamma(t)=(t,t^2,\ldots,t^d)$ in $\mathbb R^d$. Its tangent vectors at distinct parameters are linearly independent:
$$
\det[\gamma'(t_1)\ \cdots\ \gamma'(t_d)]=d!\prod_{i<j}(t_j-t_i).
$$
Factor $1,2,\ldots,d$ from the rows and apply the <Vandermonde determinant> formula. Parameter separation bounded below therefore gives quantitative <transversality> of the tangent directions.
For radial position vectors, the determinant instead equals $(\prod_jt_j)\prod_{i<j}(t_j-t_i)$. They are independent only when all parameters are nonzero as well as distinct. In particular $\gamma(0)=0$ supplies no radial direction. For $d=2$, the curve is the <parabola>, with tangent determinant $2(t_2-t_1)$. Its nondegeneracy is a basic source of curvature in <Fourier restriction theory> and <decoupling inequalities>.
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