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On the cubic hermite spline Wikipedia page, the formula for interpolating between $x_k$ and $x_{k+1}$ is given by

$$h_{00}(t)p_k+h_{10}(t)(x_{k+1}-x_k)m_k+h_{01}(t)p_{k+1}+h_{11}(t)(x_{k+1}-x_k)m_{k+1}$$

Where $t = (x-x_k)/(x_{k+1}-x_k)$ and $h$ refers to the basis functions.

I have a slight problem in understanding this: When given $h(t)$, is it $h \times t$, or $h$ of $t$?

This of course makes a big difference in the equation. Judging from the 0 to 1 interval, I'd say it's $h$ of $t$, but I'd rather make sure.

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Yes, it's the function $h$ applied to $t$. It's a linear combination of basis functions, stretched to fit the interval $[x_k, x_{k+1}]$.

If they had meant $h$ times $t$, they would probably just have written $ht$.

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