Mathematical definition

In the toolbox fold curves are computed by minimally extended defining systems cf. [22], §4.1.2. The fold curve is defined by the following system
\begin{displaymath}
\left\{
\begin{array}{ccl}
f(u,\alpha) & = & 0, \\
g(u,\alpha) & = & 0,
\end{array}
\right.
\end{displaymath} (55)

where $(u,\alpha) \in {\bf R}^{n+2}$, while $g$ is obtained by solving
\begin{displaymath}
\left(
\begin{array}{cc}
f_u(u,\alpha) & w_{bor} \\
v_{...
...
=\left(
\begin{array}{c}
0_n \\ 1
\end{array}
\right),
\end{displaymath} (56)

and $w_{bor},v_{bor} \in \ensuremath{\mathbf{R}}^n$ are chosen such that the matrix in (56) is nonsingular. An advantage of this method is that the derivatives of $g$ can be obtained easily from the derivatives of $f_u(u,\alpha)$:

\begin{displaymath}
g_z = -w^T(f_u)_z v
\end{displaymath}

where $z$ is a state variable or an active parameter and w is obtained by solving
\begin{displaymath}
\left(
\begin{array}{cc}
f_u^T(u,\alpha) & v_{bor} \\
w...
...\left(
\begin{array}{c}
0_n \\
1
\end{array}
\right),
\end{displaymath} (57)

This method is implemented in the curve definition file limitpoint.m.