Mathematical definition

In the toolbox branch point curves are computed by minimally extended defining systems, cf. [22], §4.1.2. The branch point curve is defined by the following system
\begin{displaymath}
\left\{
\begin{array}{ccl}
f(u,\alpha) & = & 0, \\
g_1(...
...) & = & 0,\\
g_2(u,\alpha) & = & 0,
\end{array}
\right.
\end{displaymath} (81)

where $(u,\alpha) \in {\bf R}^{n+2}$, while $g_1$ and $g_2$ are obtained by solving
\begin{displaymath}
N^4\left(
\begin{array}{cc}
v_{11}&v_{21}\\
v_{12}&v_{2...
...y}{cc}
0_n& 0_n\\
1 & 0\\
0 & 1
\end{array}
\right).
\end{displaymath} (82)

Here $v_{11}$ and $v_{21}$ are functions and $v_{12},v_{22},g_1$ and $g_2$ are scalars and

\begin{displaymath}N^4=
\left[\begin{array}{ccc}
f_u(u,\alpha)&f_{\beta}(u,\al...
...}&v_0^{12T}&0\\
v_0^{21T}&v_0^{22T}&0
\end{array}
\right]
\end{displaymath}

where the bordering functions $v_0^{11},v_0^{21},w_{01}$ and scalars $v_0^{12},v_0^{22}$ are chosen so that $N^4$ is nonsingular. This method is implemented in the curve definition file branchpoint.m.