\def\reals{\hbox{\bf R\/}} % set of real numbers
\def\integers{\hbox{\bf Z\/}} % set of all integers
\def\sphere{\hbox{\bf S\/}} % needs dimensionality subscript
\def\ball{\hbox{\bf B\/}} % needs dimensionality subscript

\def\dist{\mathop{\rm dist}} % distance.
\def\transl#1{\underline #1} % translation by a vector
\def\basis#1{{\hbox{\bi e}
{#1}}} % i-th standard basis vector
\def\hull{\mathop{\rm hull}}
\def\boundary#1{{\partial #1}} % boundary
\def\closure#1{{\bar{#1}}} % closure

\def\hom{\Vert}
\def\hop#1{[\,#1\,]}
\def\hol#1{\langle\, #1\, \rangle}
\def\sdist{\mathop{\rm sdist}} % signed dist.
\def\hdist{\mathop{\rm hdist}} % homogeneous dist.
\def\adist{\mathop{\rm adist}} % angular dist.
\def\CW{\mathop{\rm CW}}
\def\CCW{\mathop{\rm CCW}}
\def\POS{\mathop{\rm POS}}
\def\NEG{\mathop{\rm NEG}}

\def\plane{\mathop{\rm plane}}
\def\circle{\mathop{\rm circle}}

\def\leftof{\mathrel{\rm left\ of}}
\def\rightof{\mathrel{\rm right\ of}}
\def\online{\mathrel{\rm on}}

\def\incircle{\mathop{ \rm InCircle}}
\def\outcircle{\mathop{\rm OutCircle}}

\def\la{\langle} % left angle bracket
\def\ra{\rangle} % right angle bracket

\def\lbb{\lbrack\!\lbrack} % double "["
\def\rbb{\dbrack\!\dbrack} % double "]"

\def\lpp{(\!(} % double "("
\def\rpp{)\!)} % double ")"

\def\laa{\langle\!\langle} % double "<"
\def\raa{\rangle\!\rangle} % double ">"

\def\dpr{{\raise1pt \hbox{$\scriptstyle \odot$}}} % dual product
\def\N{N} % 180 degree rotation

\def\detmat#1{\left|\;\;
\vcenter{\vskip2pt \hbox{$ \dispstyle #1 $}\vskip2pt}
\;\;\right|}