Spline Specifications
There are three equivalent methods for specifying a particular spline representation: (1) We van state the set of boundary conditions that are imposed on the spline; or (2) we can state the matrix that characterizes the spline; or (3) we can state the set of blending functions (or basis functions) that determine how specified geometric constraints on the curve are calculate positions along the curve path.
To illustrate these three equivalent specifications, suppose we have the following parametric cubic polynomial representation for the x coordinate along the path of a cubic spline section:
Boundary conditions for this curve might be set, for example, on the endpoint coordinates x(0)and x(1) and on the parametric first derivatives at the endpoints x'(0) and x'(1). These four boundary conditions are sufficient to determine the values of the four coefficients ax,bx,cx and dx.
From the boundary conditions, we can obtain the matrix that characterizes this spline curve by first rewriting Eq. above as the matrix product.
Where U is the row matrix of powers of parameter u, and C is the coefficient column matrix. If x(0), x(1), x'(0) and x'(1) are known using the equation above we can right the boundary conditions in matrix form and solve for the coefficient matrix C as
Where
is a four-element column matrix containing the geometric constraint values (boundary conditions) on the spline
and C is the 4-by-4 matrix of the polynomial coefficients given by
and M is the matrix of the coefficients in the equation.
the equation x=UC can now be rewritten as follows:
or as
Finally, we can expand equation above to obtain a polynomial representation for coordinate x in terms of the geometric constraint parameters
where gk are the constraint parameters, such as the control-point coordinates and slope of the curve at the control points, and BFk(u) are the polynomial blending functions.
These blending functions can be written in a matrix form as
where Mblend is the set of coefficients of these blending functions. The curve equation can then be expressed as
where B is the matrix of the input points.
In the following sections, we discuss some commonly used splines and their matrix and blending-function specifications.