While digging through some code about Perlin noise, I noticed, that a Cubic Hermite Interpolation polynome is used at some point. At this point, I wanted to know, which of the Hermite basis polynomials ($h_{00}$, $h_{10}$, $h_{01}$ or $h_{11}$) it is.
$$ h_{??}(t) = 3 t^2 - 2t^3 $$
The actual problem is, that I took a look to the german and english wikipedia to read about this topic. Unfortunately, the german wikipedia tells me that
$$ h_{01} = t^3 - 2t^2 + t \\ h_{10} = 3t^2 - 2t^3 $$
while the english wikipedia swaps the indices
$$ h_{01} = 3t^2 - 2t^3 \\ h_{10} = t^3 - 2t^2 + t $$
I'd like to know, what the correct indices should be, and if they have a meaning.
The symbols have no universal meaning, and different authors give different names to these four polynomials. The thing that distinguishes one of these polynomials from another is its "cardinality" properties: where the polynomial and its first derivative takes the values $0$ and $1$.
For the particular one you asked about, $h(t) = 3t^2 - 2t^3$, the important cardinality property is: $$ h(0) = 0 \quad ; \quad h(1)=1 \quad ; \quad h'(0)=0 \quad ; \quad h'(1) = 0 $$