I am trying to use Catmull spline curve in my program , I am trying to understand it but why we only use below given Matrix , because the examples I saw I only found the below one
In Catmull spline curve , why its speciality ? and if we can use other matrix then how and what change it effects ?
A Catmull-Rom spline is just a sequence of cubic segments, strung together so that their tangents match at the junctions.
You can think of each cubic segment in "Hermite" form. This means the segment is defined in terms of its end-points $\mathbf{p}_0$ and $\mathbf{p}_1$ and its end tangents $\mathbf{m}_0$ and $\mathbf{m}_1$. Because two adjacent cubic segments share the same point and tangent, they join smoothly.
Of course, you do this same construction with any set of points $\mathbf{p}_i$ and tangent vectors $\mathbf{m}_i$. Usually the points $\mathbf{p}_i$ are given, and the splining function somehow makes up a set of tangent vectors $\mathbf{m}_i$. Different splining techniques make up the tangent vectors in different ways. The Catmull-Rom spline calculates the tangent vector $\mathbf{m}_i$ just by scaling the vector $\mathbf{p}_{i+1} - \mathbf{p}_{i-1}$. That's where the matrix comes from. This page shows the derivation.
You can invent some other way to fabricate the tangents $\mathbf{m}_i$, and this would give rise to a different matrix. There is nothing really special about the Catmull-Rom approach or its associated matrix.