<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">%============================================================
% CPA Math Definitions
%============================================================
\newcommand{\mcrs}{\theta_{os}}
\newcommand{\mspd}{v_{os}}

%\newcommand{\crs}{\theta}
%\newcommand{\spd}{v}
\newcommand{\mtol}{t}
\newcommand{\dpt}{$\langle \mcrs, \mspd, \mtol \rangle$}
\newcommand{\ddpt}{\langle \mcrs, \mspd, \mtol \rangle}
\newcommand{\tol}{{$t$}}
\newcommand{\ttol}{{$t_{min}$}}
\newcommand{\mttol}{{t_{min}}}

\newcommand{\oslat}{{y}_{os}}
\newcommand{\oslon}{{x}_{os}}
\newcommand{\cnlat}{{y}_{cn}}
\newcommand{\cnlon}{{x}_{cn}}
\newcommand{\cnspd}{{v}_{cn}}
\newcommand{\cncrs}{{\theta}_{cn}}

%============================================================
% K TWO
%============================================================
\newcommand{\ktwo}{$ 
  \cos^2(\mcrs) \mspd^2 -                    % (1,1)a
2 \cos(\mcrs) \mspd \cos(\cncrs) \cnspd +    % (1,3)(3,1)a 
  \cos^2(\cncrs) \cnspd^2 +          % (3,3)a
  \sin^2(\mcrs) \mspd^2 - \newline           % (1,1)b
2 \sin(\mcrs) \mspd \sin(\cncrs) \cnspd +    % (1,3)(3,1)b
  \sin^2(\cncrs) \cnspd^2  $                 % (3,3)b
}

%---------------------------------------------------------
% A narrower version of k_two, to fit on one page
%---------------------------------------------------------
\newcommand{\ktwoa}{
  \cos^2(\mcrs) \mspd^2 -                    % (1,1)a
2 \cos(\mcrs) \mspd \cos(\cncrs) \cnspd +    % (1,3)(3,1)a 
  \cos^2(\cncrs) \cnspd^2 +                  % (3,3)a
  \sin^2(\mcrs) \mspd^2 -                   % (1,1)b
}
\newcommand{\ktwob}{
2 \sin(\mcrs) \mspd \sin(\cncrs) \cnspd +    % (1,3)(3,1)b
  \sin^2(\cncrs) \cnspd^2                   % (3,3)b
}

%---------------------------------------------------------
% A even narrower version of k_two, to fit on one page
%---------------------------------------------------------
\newcommand{\ktwoA}{
  \cos^2(\mcrs) \mspd^2 -                    % (1,1)a
2 \cos(\mcrs) \mspd \cos(\cncrs) \cnspd +    % (1,3)(3,1)a 
}

\newcommand{\ktwoB}{
  \cos^2(\cncrs) \cnspd^2 +                  % (3,3)a
  \sin^2(\mcrs) \mspd^2 -                   % (1,1)b
}

\newcommand{\ktwoC}{
2 \sin(\mcrs) \mspd \sin(\cncrs) \cnspd +    % (1,3)(3,1)b
  \sin^2(\cncrs) \cnspd^2                   % (3,3)b
}


%============================================================
% K ONE
%============================================================
\newcommand{\kone}{$
2 \cos(\mcrs) \mspd \oslat -                 % (1,2)(2,1)a
2 \cos(\mcrs) \mspd \cnlat -                 % (1,4)(4,1)a
2 \oslat \cos(\cncrs) \cnspd  +      % (2,3)(3,2)a
2 \cos(\cncrs) \cnspd \cnlat + \newline              % (3,4)(4,3)a
2 \sin(\mcrs) \mspd \oslon -                 % (1,2)(2,1)b
2 \sin(\mcrs) \mspd \cnlon -         % (1,4)(4,1)b
2 \oslon \sin(\cncrs) \cnspd +               % (2,3)(3,2)b
2 \sin(\cncrs) \cnspd \cnlon $               % (3,4)(4,3)b
}
%---------------------------------------------------------
% A narrower version of k_one, to fit on one page
%---------------------------------------------------------
\newcommand{\konea}{
2 \cos(\mcrs) \mspd \oslat -                 % (1,2)(2,1)a
2 \cos(\mcrs) \mspd \cnlat -                 % (1,4)(4,1)a
2 \oslat \cos(\cncrs) \cnspd  +              % (2,3)(3,2)a
2 \cos(\cncrs) \cnspd \cnlat +              % (3,4)(4,3)a
}
\newcommand{\koneb}{
2 \sin(\mcrs) \mspd \oslon -                 % (1,2)(2,1)b
2 \sin(\mcrs) \mspd \cnlon -                 % (1,4)(4,1)b
2 \oslon \sin(\cncrs) \cnspd +               % (2,3)(3,2)b
2 \sin(\cncrs) \cnspd \cnlon                % (3,4)(4,3)b
}
%---------------------------------------------------------
% A even narrower version of k_one, to fit on one page
%---------------------------------------------------------
\newcommand{\koneA}{
2 \cos(\mcrs) \mspd \oslat -                 % (1,2)(2,1)a
2 \cos(\mcrs) \mspd \cnlat -                 % (1,4)(4,1)a
}

\newcommand{\koneB}{
2 \cos(\cncrs) \cnspd \oslat +              % (2,3)(3,2)a
2 \cos(\cncrs) \cnspd \cnlat +              % (3,4)(4,3)a
}

\newcommand{\koneC}{
2 \sin(\mcrs) \mspd \oslon -                 % (1,2)(2,1)b
2 \sin(\mcrs) \mspd \cnlon -                 % (1,4)(4,1)b
}

\newcommand{\koneD}{
2 \sin(\cncrs) \cnspd \oslon +               % (2,3)(3,2)b
2 \sin(\cncrs) \cnspd \cnlon                % (3,4)(4,3)b
}


%============================================================
\newcommand{\kzero}{
  \oslat^2 -                 % (2,2)a
2 \oslat \cnlat +            % (2,4)(4,2)a
  \cnlat^2  +                % (4,4)a      
  \oslon^2 -                 % (2,2)b
2 \oslon \cnlon +            % (2,4)(4,2)b
  \cnlon^2                  % (4,4)b      
}




%---------------------------------------------------------
% A version of k_two, with static and dynamic broken out
%---------------------------------------------------------
\newcommand{\ktwoAS}{
  \cos^2(\cncrs) \cnspd^2 +                  % (3,3)a
  \sin^2(\cncrs) \cnspd^2                   % (3,3)b
}

\newcommand{\ktwoAD}{
  \cos^2(\mcrs) \mspd^2 -                    
2 \cos(\mcrs) \mspd \cos(\cncrs) \cnspd +   
}

\newcommand{\ktwoBD}{
  \sin^2(\mcrs) \mspd^2 -                   % (1,1)b
2 \sin(\mcrs) \mspd \sin(\cncrs) \cnspd     % (1,3)(3,1)b
}




%---------------------------------------------------------
% A version of k_one, with static and dynamic broken out
%---------------------------------------------------------


\newcommand{\koneAS}{
2 \cos(\cncrs) \cnspd \cnlat  -            % (3,4)(4,3)a
2 \cos(\cncrs) \cnspd \oslat -            % (2,3)(3,2)a
}

\newcommand{\koneBS}{
2 \sin(\cncrs) \cnspd \oslon +             % (2,3)(3,2)b
2 \sin(\cncrs) \cnspd \cnlon               % (3,4)(4,3)b
}

\newcommand{\koneAD}{
2 \cos(\mcrs) \mspd \oslat -               % (1,2)(2,1)a
2 \cos(\mcrs) \mspd \cnlat                 % (1,4)(4,1)a
}

\newcommand{\koneBD}{
2 \sin(\mcrs) \mspd \oslon -               % (1,2)(2,1)b
2 \sin(\mcrs) \mspd \cnlon                 % (1,4)(4,1)b
}

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