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Given a curve $\mathcal{L}$, define two functions of distance transform $\kappa_{+}$ and $\kappa_{-}$ on \mathbb{R}^{2} as
$\kappa_{+}(x) = d(x,\mathbb{R}^{2}\D), x \in \mathbb{R}^{2}, (1)$
$\kappa_{-}(x) = d(x,\mathbb{R}^{2}\D), x \in \mathbb{R}^{2}, (2)$
I get following error:
Undefined sequence
You need to be careful to use mathmode correctly; in particular, \mathbb needs to be in mathmode.
You didn't specify what the macro \D does, but it looks as if would you like to use the align environment, as follows
\documentclass{article}
\usepackage{amssymb}
\usepackage{amsmath}
\newcommand{\D}{D}
\begin{document}
Given a curve $\mathcal{L}$, define two functions of distance transform $\kappa_{+}$
and $\kappa_{-}$ on $\mathbb{R}^{2}$ as
\begin{align}
\kappa_{+}(x) & = d(x,\mathbb{R}^{2}\D), x \in \mathbb{R}^{2} \\
\kappa_{-}(x) & = d(x,\mathbb{R}^{2}\D), x \in \mathbb{R}^{2}
\end{align}
\end{document}
\D might depend on a certain package, the \mathbb{R} without $ surrounded should not produce the mentioned error, but an missing $. Hence \D should be the cause for the error (though depending on the OPs packages in use).
\mathbb without $? :)
\mathbb R, though the real error message then is of course Error: \mathbb allowed only in math mode. :)
Please always post complete documents so that your problem may be reproduced. However \mathcal is a standard LaTeX command but \mathbb requires a package (such as amssymb)
