By using uniform cubic B-splines we can produce long continuous curves without having to ensure continuity between joins of segments. These curves have \(C^2\) continuity, and do not necessarily pass through all of their control points. For a uniform cubic B-spline, values of \(t\) between \(3\) and the total number of control points are valid.
At any point along the curve (i.e. for a given value of \(t\)), the location of the curve is determined by a weighted sum of four control points. In a uniform cubic B-spline the range of possible values for \(t\) is divided so that \(t_i=i\). We can then write the location on the curve (looking at the x coordinate only) as
$$x(t)=\begin{pmatrix}(t-t_i)^3&(t-t_i)^2&t-t_i&1\end{pmatrix}M\begin{pmatrix}x_{i-3}\\x_{i-2}\\x_{i-1}\\x_i\end{pmatrix}$$
when \(t_i\leq t\leq t_{i+1}\). See the lecture slides for the form of \(M\).