masterarbeit/dokumentation/B-Spline-Volumes.md
2017-06-20 20:11:21 +02:00

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# B-Spline Volumes
B-Spline Volumes are a simple extension of B-Splines to 3 Dimensions. This is
a straightforward adaption of the 2-Dimensional version.
## Nomenclature
$x,y,z$ denote space-coordinates,
$u,v,w$ denote spline-coordinates (Between 0-1),
$P_{ijk}$ denote the control-Points on the control-Polygon,
$N_{i,d,\tau}(u)$ denote the value of the underlying Basis-Functions at value
$u$ using the $i$-th Basis-Function of degree $d$ in range $\tau$.
For our case we only care about degree-3 splines, so we omit the d furtheron.
$\tau$ is defined statically (in each direction) with each $P$ as Position on the whole surface/volume and within [0,1].
For a regular Control-Grid this defaults to $\tau_i = i/n$
Given $n,m,o$ control points in $x,y,z$-direction each Point on the curve is
defined by
$$C(u,v,w) = \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} P_{ijk} N_{i}(u) N_j(v) N_k(w)$$
## Calculate $u, v, w$
Given a target-point $\textbf{p}^*$ and an initial guess $\textbf{p}=C(u,v,w)$ we
define the error-function as:
$$Err(u,v,w,\textbf{p}^{*}) = \| \textbf{p}^{*} - C(u,v,w) \|_2^2$$
As the error is just the sum of the components
$$Err(u,v,w,\textbf{p}^{*}) = Err_x(u,v,w,\textbf{p}^{*}) + Err_y(u,v,w,\textbf{p}^{*}) + Err_z(u,v,w,\textbf{p}^{*})$$
we just take one axis into account, as the others are nearly identical. So we yield
$$Err_x(u,v,w,\textbf{p}^{*}) = \left( p^{*}_x - \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} {P_{ijk}}_x N_{i}(u) N_j(v) N_k(w) \right)^2 $$
To solve this we derive:
$$
\begin{array}{rl}
\displaystyle \frac{\partial}{\partial u} & \left( p^{*}_x - \displaystyle \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} {P_{ijk}}_x N_{i}(u) N_j(v) N_k(w) \right)^2 \\
= & \displaystyle - \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} {P_{ijk}}_x N'_{i}(u) N_j(v) N_k(w) \\
& \cdot \left( p^{*}_x - \displaystyle \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} {P_{ijk}}_x N_{i}(u) N_j(v) N_k(w) \right) \\
& \cdot 2
\end{array}
$$
The other partial derivatives follow the same pattern.
## Basis-Splines and Derivatives
The previously mentioned $N_{i,d,\tau}$ are defined recursively:
$$N_{i,0,\tau}(u) = \begin{cases} 1, & u \in [\tau_i, \tau_{i+1}[ \\ 0, & \mbox{otherwise} \end{cases} $$
and
$$N_{i,d,\tau}(u) = \frac{u-\tau_i}{\tau_{i+d}} N_{i,d-1,\tau}(u) + \frac{\tau_{i+d+1} - u}{\tau_{i+d+1}-\tau_{i+1}} N_{i+1,d-1,\tau}(u) $$
This fact can be exploited to get the derivative for an arbitrary $N$:
$$\frac{d}{du} N_{i,d,r}(u) = \frac{d}{\tau_{i+d} - \tau_i} N_{i,d-1,\tau}(u) - \frac{d}{\tau_{i+d+1} - \tau_{i+1}} N_{i+1,d-1,\tau}(u)$$
*Warning:* in the case of $d=1$ the recursion-formula yields a $0$ denominator, but $N$ is also $0$. The right solution for this case is a derivative of $0$