Rational B-Spline Curves

A rational B-spline curve is the projection of a nonrational B-spline curve defined in four-dimensional homogeneous coordinate space back to the three-dimensional physical space.

A rational B-spline curve is defined as

\[P(t) = \sum_{i=1}^{n+1} B_i^hN_{i,k}(t)\]

where $B_i^h$ are the four-dimensional homogeneous control polygon vertices for the nonrational four-dimensional B-spline curve and $N_{i,k}(t)$ is the nonrational B-spline basis function.

Projecting into the three-dimensional space by dividing through by thr homogeneous coordinate yelds

\[P(t) = \dfrac{\sum_{i=1}^{n+1} B_ih_iN_{i,k}(t)}{\sum_{i=1}^{n+1}h_iN_{i,k}(t)} = \sum_{i=1}^{n+1}B_iR_{i,k}(t)\]

here the $B_i$ are the three-dimensional control polygon vertices and $R_{i,k}(t) = \dfrac{h_iN_{i,k}(t)}{\sum_{i=1}^{n+1}h_iN_{i,k}(t)}$ are the rational B-spline basis function where $h_i>0$ for every value of $i$.

RB-Spline Properties

RB-spline are a generalization of nonrational B-spline, thus they carry forward nearly all the analytic and geometric characteristics:

Every rational basis function is positive or zero for all parameter values.
The sum of the rational B-spline basis functions for any value of $t$ is one.
Each basis function has one maximum, except for the first order basis.
The maximum order of the rational B-spline is equal to the number of control polygon vertices.
A RB-spline generally follows the shape of the control polygon.

RB-Spline Basis Functions

To generate RB-spline basis functions and curves are used open uniform, periodic uniform and nonuniform knot vectors.

The homogeneous coordinates $h_i$, also called homogeneous weight factors, provide additional blendig capability, i.e. as a certain weight $h_j$ increases the curve is pulled closer to the polygon vertex $B_j$.

RB-Spline Derivatives

The derivatives are obtained by formal differantiation of the curve's function

\[P'(t) = \sum_{i=1}^{n+1}B_iR'_{i,k}(t)\]

where

\[R'_{i,k}(t)=\dfrac{h_iN'_{i,k}(t)}{\sum_{i=1}^{n+1}h_iN_{i,k}}-\dfrac{h_iN_{i,k}\sum_{i=1}^{n+1}h_iN'_{i,k}}{(\sum_{i=1}^{n+1}h_iN_{i.k})^2}\]

For example, evaluating this result at $t=0$ and $t=n-k+2$

\[P'(0)=(k-1)\dfrac{h_2}{h_1}(B_2-B_1)\]

and

\[P'(n-k+2)=(k-1)\dfrac{h_n}{h_{n+1}}(B_{n+1}-B_n)\]

Higher order dervatives are obtained in a similar manner.