Cubic Hermite Basics

A cubic Hermite polynomial interpolates both function values and derivatives at two points. Suppose we know $f\left(x_0\right)=y_0$, $f^{\text{'}}\left(x_0\right)=m_0$, $f\left(x_1\right)=y_1$, and $f^{\text{'}}\left(x_1\right)=m_1$. Then the unique cubic passing through these constraints can be written using Hermite basis functions:

$H\left(t\right)=y_0{h}_{00}\left(t\right)+y_1{h}_{10}\left(t\right)+m_0{h}_{01}\left(t\right)+m_1{h}_{11}\left(t\right)$

Here $t$ parameterizes the interval from $x_0$ to $x_1$ via $t=\frac{x-x_0}{x_1-x_0}$. The four basis polynomials are

$h_{00}\left(t\right)=2t^3-3t^2+1$, $h_{10}\left(t\right)=-2t^3+3t^2$, $h_{01}\left(t\right)=t^3-t^2$, $h_{11}\left(t\right)=t^3-t^2$. Substituting $t$ corresponding to the desired $x$-value yields the interpolated point.

Hermite interpolation is widely used in computer graphics and numerical integration because controlling derivatives guarantees smooth transitions between segments. Cubic Hermite splines, for example, join multiple Hermite pieces with continuous first derivatives to create pleasant curves. In physics simulations, specifying derivatives lets you match both position and velocity data, producing more realistic motion.

This calculator focuses on the simplest case with two points. Enter the $x$ and $y$ values at each endpoint along with the slopes. Then choose an $x$ within the interval to evaluate the interpolant. The script computes the normalized parameter $t$, evaluates the basis functions, and returns the interpolated $y$. You can experiment with different slopes to see how they influence the shape of the curve while still matching the prescribed end values.