Rational Motion - Rational Bezier and B-spline Motions

Rational Bezier and B-spline Motions

Let denote a unit dual quaternion. A homogeneous dual quaternion may be written as a pair of quaternions, $\hat {\textbf{Q}}= \textbf{Q} + \varepsilon \textbf{Q}^0$ ; where $\textbf{Q} = w\textbf{q}, \textbf{Q}^0 = w\textbf{q}^0 + w^0\textbf{q}$ . This is obtained by expanding using dual number algebra (here, ).

In terms of dual quaternions and the homogeneous coordinates of a point of the object, the transformation equation in terms of quaternions is given by (see for details)

$\tilde {\textbf{P}} = \textbf{Q}\textbf{P}\textbf{Q}^\ast + P_4,$ where and are conjugates of and, respectively and denotes homogeneous coordinates of the point after the displacement.

Given a set of unit dual quaternions and dual weights $\hat {\textbf{q}}_i, \hat {w}_i; i = 0...n$ respectively, the following represents a rational Bezier curve in the space of dual quaternions.

$\hat{\textbf{Q}}(t) = \sum\limits_{i = 0}^n {B_i^n (t)\hat {\textbf{Q}}_i} = \sum\limits_{i = 0}^n {B_i^n (t)\hat {w}_i \hat{\textbf{q}}_i}$

where are the Bernstein polynomials. The Bezier dual quaternion curve given by above equation defines a rational Bezier motion of degree .

Similarly, a B-spline dual quaternion curve, which defines a NURBS motion of degree 2p, is given by,

$\hat {\textbf{Q}}(t) = \sum\limits_{i = 0}^n {N_{i,p}(t) \hat {\textbf{Q}}_i } = \sum\limits_{i = 0}^n {N_{i,p}(t) \hat {w}_i \hat {\textbf{q}}_i }$

where are the pth-degree B-spline basis functions.

A representation for the rational Bezier motion and rational B-spline motion in the Cartesian space can be obtained by substituting either of the above two preceding expressions for in the equation for point transform. In what follows, we deal with the case of rational Bezier motion. The, the trajectory of a point undergoing rational Bezier motion is given by,

$\tilde {\textbf{P}}^{2n}(t) = \textbf{P},$

$H^{2n}(t)] = \sum\limits_{k = 0}^{2n} {B_k^{2n}(t)},$

where is the matrix representation of the rational Bezier motion of degree in Cartesian space. The following matrices (also referred to as Bezier Control Matrices) define the affine control structure of the motion:

$= \frac{1}{C_k^{2n}} \sum\limits_{i+j=k}{C_i^n C_j^n w_i w_j },$

where $= + - + (\alpha_i - \alpha_j )$ .

In the above equations, and are binomial coefficients and $\alpha_i = w_i^0/w_i, \alpha_j = w_j^0/w_j$ are the weight ratios and

$= \left[ \begin{array}{rrrr} q_{j,4} & -q_{j,3} & q_{j,2} & -q_{j,1} \\ q_{j,3} & q_{j,4} & -q_{j,1} & -q_{j,2} \\ -q_{j,2} & q_{j,1} & q_{j,4} & -q_{j,3} \\ q_{j,1} & q_{j,2} & q_{j,3} & q_{j,4} \\ \end{array} \right],$

$= \left[ \begin{array}{rrrr} 0 & 0 & 0 & q_{i,1} \\ 0 & 0 & 0 & q_{i,2} \\ 0 & 0 & 0 & q_{i,3} \\ 0 & 0 & 0 & q_{i,4} \\ \end{array} \right],$

$= \left[ \begin{array}{rrrr} 0 & 0 & 0 & q_{i,1}^0 \\ 0 & 0 & 0 & q_{i,2}^0 \\ 0 & 0 & 0 & q_{i,3}^0 \\ 0 & 0 & 0 & q_{i,4}^0 \\ \end{array} \right],$

$= \left[ \begin{array}{rrrr} 0 & 0 & 0 & -q_{j,1}^0 \\ 0 & 0 & 0 & -q_{j,2}^0 \\ 0 & 0 & 0 & -q_{j,3}^0 \\ 0 & 0 & 0 & q_{j,4}^0 \\ \end{array} \right],$

$= \left[ \begin{array}{rrrr} q_{i,4} & -q_{i,3} & q_{i,2} & q_{i,1} \\ q_{i,3} & q_{i,4} & -q_{i,1} & q_{i,2} \\ -q_{i,2} & q_{i,1} & q_{i,4} & q_{i,3} \\ -q_{i,1} & -q_{i,2} & -q_{i,3} & q_{i,4} \\ \end{array} \right].$

In above matrices, are four components of the real part and are four components of the dual part of the unit dual quaternion .

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