# LaTeX2HTML 2023 (Released January 1, 2023)
# Associate images original text with physical files.


$key = q//;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 12.36ex; " SRC="|."$dir".q|img69.svg"
 ALT="\begin{displaymath}\alpha_p =
\begin{cases}
\alpha_0 + \phi_p - 180^\circ &amp; \tex...
...a_0}{\cos\delta_p\cos\delta_0}) &amp; \text{otherwise.}
\end{cases}\end{displaymath}">|; 

$key = q/(-200,0);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.53ex; vertical-align: -0.69ex; " SRC="|."$dir".q|img110.svg"
 ALT="$(-200,0)$">|; 

$key = q/(-y,-x)=;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.53ex; vertical-align: -0.69ex; " SRC="|."$dir".q|img89.svg"
 ALT="$(-y,-x) =$">|; 

$key = q/(0,0);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.53ex; vertical-align: -0.69ex; " SRC="|."$dir".q|img5.svg"
 ALT="$(0,0)$">|; 

$key = q/(0,pm90^circ);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.53ex; vertical-align: -0.69ex; " SRC="|."$dir".q|img15.svg"
 ALT="$(0,\pm90^\circ)$">|; 

$key = q/(3600,-1800),(3600,+1800);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.53ex; vertical-align: -0.69ex; " SRC="|."$dir".q|img109.svg"
 ALT="$(3600,-1800), (3600,+1800)$">|; 

$key = q/(alpha,delta);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.53ex; vertical-align: -0.69ex; " SRC="|."$dir".q|img6.svg"
 ALT="$(\alpha ,\delta )$">|; 

$key = q/(alpha,delta)=(0,0);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.53ex; vertical-align: -0.69ex; " SRC="|."$dir".q|img25.svg"
 ALT="$(\alpha,\delta) = (0,0)$">|; 

$key = q/(alpha_0,delta_0);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.53ex; vertical-align: -0.69ex; " SRC="|."$dir".q|img16.svg"
 ALT="$(\alpha_0,\delta_0)$">|; 

$key = q/(alpha_p,delta_p);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.62ex; vertical-align: -0.78ex; " SRC="|."$dir".q|img18.svg"
 ALT="$(\alpha_p,\delta_p)$">|; 

$key = q/(phi,theta);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.53ex; vertical-align: -0.69ex; " SRC="|."$dir".q|img3.svg"
 ALT="$(\phi ,\theta )$">|; 

$key = q/(phi_0,theta_0);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.53ex; vertical-align: -0.69ex; " SRC="|."$dir".q|img4.svg"
 ALT="$(\phi _0,\theta _0)$">|; 

$key = q/(phi_p,theta_p);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.62ex; vertical-align: -0.78ex; " SRC="|."$dir".q|img17.svg"
 ALT="$(\phi_p,\theta_p)$">|; 

$key = q/(x,y);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.53ex; vertical-align: -0.69ex; " SRC="|."$dir".q|img2.svg"
 ALT="$(x,y)$">|; 

$key = q/(x^prime_;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.57ex; vertical-align: -0.69ex; " SRC="|."$dir".q|img102.svg"
 ALT="$(x^\prime_$">|; 

$key = q/(x_0,y_0);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.53ex; vertical-align: -0.69ex; " SRC="|."$dir".q|img14.svg"
 ALT="$(x_0,y_0)$">|; 

$key = q/(y,x)+180^circ;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.53ex; vertical-align: -0.69ex; " SRC="|."$dir".q|img90.svg"
 ALT="$(y,x) + 180^\circ$">|; 

$key = q/(y,x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.53ex; vertical-align: -0.69ex; " SRC="|."$dir".q|img61.svg"
 ALT="$(y,x)$">|; 

$key = q/)=(x_{text{SFL},0},y_{text{SFL},0});MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.63ex; vertical-align: -0.80ex; " SRC="|."$dir".q|img104.svg"
 ALT="$) =
(x_{\text{SFL},0},y_{\text{SFL},0})$">|; 

$key = q/,y^prime_;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.56ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img103.svg"
 ALT="$,y^\prime_$">|; 

$key = q/-22000;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.90ex; vertical-align: -0.30ex; " SRC="|."$dir".q|img47.svg"
 ALT="$-22000$">|; 

$key = q/20times12;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.90ex; vertical-align: -0.30ex; " SRC="|."$dir".q|img1.svg"
 ALT="$20 \times 12$">|; 

$key = q/30^{circ};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.82ex; vertical-align: -0.11ex; " SRC="|."$dir".q|img8.svg"
 ALT="$30^{\circ }$">|; 

$key = q/50^{circ};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.82ex; vertical-align: -0.11ex; " SRC="|."$dir".q|img11.svg"
 ALT="$50^{\circ }$">|; 

$key = q/=x_{text{SFL},0};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.90ex; vertical-align: -0.80ex; " SRC="|."$dir".q|img106.svg"
 ALT="$= x_{\text{SFL},0}$">|; 

$key = q/=y_{text{SFL},0}-delta_0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.51ex; vertical-align: -0.80ex; " SRC="|."$dir".q|img108.svg"
 ALT="$= y_{\text{SFL},0} - \delta_0$">|; 

$key = q/^{1,2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.14ex; vertical-align: -0.11ex; " SRC="|."$dir".q|img13.svg"
 ALT="$^{1,2}$">|; 

$key = q/^{1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.14ex; vertical-align: -0.11ex; " SRC="|."$dir".q|img12.svg"
 ALT="$^{1}$">|; 

$key = q/alpha_0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.56ex; vertical-align: -0.45ex; " SRC="|."$dir".q|img10.svg"
 ALT="$\alpha _0$">|; 

$key = q/alpha_p;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.88ex; vertical-align: -0.78ex; " SRC="|."$dir".q|img68.svg"
 ALT="$\alpha_p$">|; 

$key = q/backslash;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.53ex; vertical-align: -0.69ex; " SRC="|."$dir".q|img46.svg"
 ALT="$\backslash$">|; 

$key = q/delta-delta_0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.17ex; vertical-align: -0.45ex; " SRC="|."$dir".q|img51.svg"
 ALT="$\delta-\delta_0$">|; 

$key = q/delta;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.82ex; vertical-align: -0.11ex; " SRC="|."$dir".q|img52.svg"
 ALT="$\delta$">|; 

$key = q/delta_0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.17ex; vertical-align: -0.45ex; " SRC="|."$dir".q|img7.svg"
 ALT="$\delta _0$">|; 

$key = q/delta_0=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.17ex; vertical-align: -0.45ex; " SRC="|."$dir".q|img9.svg"
 ALT="$\delta _0 = 0$">|; 

$key = q/delta_0ne0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.27ex; vertical-align: -0.56ex; " SRC="|."$dir".q|img111.svg"
 ALT="$\delta_0 \ne 0$">|; 

$key = q/delta_p;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.49ex; vertical-align: -0.78ex; " SRC="|."$dir".q|img64.svg"
 ALT="$\delta_p$">|; 

$key = q/delta_p=pm90^circ;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.50ex; vertical-align: -0.78ex; " SRC="|."$dir".q|img67.svg"
 ALT="$\delta_p = \pm 90^\circ$">|; 

$key = q/delta_{p,1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.49ex; vertical-align: -0.78ex; " SRC="|."$dir".q|img62.svg"
 ALT="$\delta_{p,1}$">|; 

$key = q/delta_{p,2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.49ex; vertical-align: -0.78ex; " SRC="|."$dir".q|img63.svg"
 ALT="$\delta_{p,2}$">|; 

$key = q/displaystyle(-cosdeltasin(alpha-alpha_p),sindeltacosdelta_p-cosdeltasindelta_pcos(alpha-alpha_p));MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.62ex; vertical-align: -0.78ex; " SRC="|."$dir".q|img78.svg"
 ALT="$\displaystyle (-\cos\delta\sin(\alpha-\alpha_p),
\sin\delta\cos\delta_p-\cos\delta\sin\delta_p\cos(\alpha-\alpha_p))$">|; 

$key = q/displaystyle(-costhetasin(phi-phi_p),sinthetacosdelta_p-costhetasindelta_pcos(phi-phi_p));MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.62ex; vertical-align: -0.78ex; " SRC="|."$dir".q|img74.svg"
 ALT="$\displaystyle (-\cos\theta\sin(\phi-\phi_p),
\sin\theta\cos\delta_p-\cos\theta\sin\delta_p\cos(\phi-\phi_p))$">|; 

$key = q/displaystyle(0,cosphi_p)+cos^{-1}(sindelta_0);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.93ex; vertical-align: -0.78ex; " SRC="|."$dir".q|img58.svg"
 ALT="$\displaystyle (0,\cos\phi_p) + \cos^{-1}(\sin\delta_0)$">|; 

$key = q/displaystyle(0,cosphi_p)-cos^{-1}(sindelta_0);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.93ex; vertical-align: -0.78ex; " SRC="|."$dir".q|img60.svg"
 ALT="$\displaystyle (0,\cos\phi_p) - \cos^{-1}(\sin\delta_0)$">|; 

$key = q/displaystyle(alpha-alpha_0)cosdelta,;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.53ex; vertical-align: -0.69ex; " SRC="|."$dir".q|img22.svg"
 ALT="$\displaystyle (\alpha-\alpha_0) \cos\delta,$">|; 

$key = q/displaystyle(alpha-alpha_0)cosdelta;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.53ex; vertical-align: -0.69ex; " SRC="|."$dir".q|img94.svg"
 ALT="$\displaystyle (\alpha-\alpha_0) \cos \delta$">|; 

$key = q/displaystyle(alpha-alpha_0)timescos(delta-delta_0);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.53ex; vertical-align: -0.69ex; " SRC="|."$dir".q|img48.svg"
 ALT="$\displaystyle (\alpha-\alpha_0) \times cos(\delta-\delta_0)$">|; 

$key = q/displaystyle(phi_0,theta_0)=(0,0);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.53ex; vertical-align: -0.69ex; " SRC="|."$dir".q|img53.svg"
 ALT="$\displaystyle (\phi_0,\theta_0) = (0,0)$">|; 

$key = q/displaystyle+90^circ;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.13ex; vertical-align: -0.30ex; " SRC="|."$dir".q|img84.svg"
 ALT="$\displaystyle +90^\circ$">|; 

$key = q/displaystyle+delta_0=delta;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.17ex; vertical-align: -0.45ex; " SRC="|."$dir".q|img101.svg"
 ALT="$\displaystyle + \delta_0 = \delta$">|; 

$key = q/displaystyle0^circ;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.94ex; vertical-align: -0.11ex; " SRC="|."$dir".q|img82.svg"
 ALT="$\displaystyle 0^\circ$">|; 

$key = q/displaystyle=(alpha-alpha_0)cosdelta;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.53ex; vertical-align: -0.69ex; " SRC="|."$dir".q|img98.svg"
 ALT="$\displaystyle = (\alpha-\alpha_0) \cos \delta$">|; 

$key = q/displaystyle=;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.06ex; vertical-align: -0.11ex; " SRC="|."$dir".q|img21.svg"
 ALT="$\displaystyle =$">|; 

$key = q/displaystylealpha;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.21ex; vertical-align: -0.11ex; " SRC="|."$dir".q|img72.svg"
 ALT="$\displaystyle \alpha$">|; 

$key = q/displaystylealpha_0-180^circ;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.29ex; vertical-align: -0.45ex; " SRC="|."$dir".q|img86.svg"
 ALT="$\displaystyle \alpha_0 - 180^\circ$">|; 

$key = q/displaystylealpha_p+;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.24ex; vertical-align: -0.78ex; " SRC="|."$dir".q|img73.svg"
 ALT="$\displaystyle \alpha_p +$">|; 

$key = q/displaystylealpha_p;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.88ex; vertical-align: -0.78ex; " SRC="|."$dir".q|img85.svg"
 ALT="$\displaystyle \alpha_p$">|; 

$key = q/displaystylecos^{-1}((sindelta-zcosdelta_0)slashsindelta_0);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.84ex; vertical-align: -0.69ex; " SRC="|."$dir".q|img30.svg"
 ALT="$\displaystyle \cos^{-1} ( (\sin\delta - z \cos\delta_0)/\sin\delta_0 )$">|; 

$key = q/displaystylecostheta=sindeltasindelta_0+cosdeltacosdelta_0cosDeltaalpha;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.17ex; vertical-align: -0.45ex; " SRC="|."$dir".q|img39.svg"
 ALT="$\displaystyle \cos\theta = \sin\delta\sin\delta_0 + \cos\delta\cos\delta_0\cos\Delta\alpha$">|; 

$key = q/displaystyledelta-delta_0.;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.17ex; vertical-align: -0.45ex; " SRC="|."$dir".q|img24.svg"
 ALT="$\displaystyle \delta-\delta_0.$">|; 

$key = q/displaystyledelta-delta_0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.17ex; vertical-align: -0.45ex; " SRC="|."$dir".q|img49.svg"
 ALT="$\displaystyle \delta-\delta_0$">|; 

$key = q/displaystyledelta;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.82ex; vertical-align: -0.11ex; " SRC="|."$dir".q|img75.svg"
 ALT="$\displaystyle \delta$">|; 

$key = q/displaystyledelta_p;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.49ex; vertical-align: -0.78ex; " SRC="|."$dir".q|img83.svg"
 ALT="$\displaystyle \delta_p$">|; 

$key = q/displaystyledelta_{p,1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.49ex; vertical-align: -0.78ex; " SRC="|."$dir".q|img57.svg"
 ALT="$\displaystyle \delta_{p,1}$">|; 

$key = q/displaystyledelta_{p,2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.49ex; vertical-align: -0.78ex; " SRC="|."$dir".q|img59.svg"
 ALT="$\displaystyle \delta_{p,2}$">|; 

$key = q/displaystylefrac{x}{cosy};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 4.81ex; vertical-align: -2.15ex; " SRC="|."$dir".q|img71.svg"
 ALT="$\displaystyle \frac{x}{\cos y}$">|; 

$key = q/displaystylephi;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.27ex; vertical-align: -0.56ex; " SRC="|."$dir".q|img70.svg"
 ALT="$\displaystyle \phi$">|; 

$key = q/displaystylephi_p+;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.49ex; vertical-align: -0.78ex; " SRC="|."$dir".q|img77.svg"
 ALT="$\displaystyle \phi_p +$">|; 

$key = q/displaystylephi_p;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.49ex; vertical-align: -0.78ex; " SRC="|."$dir".q|img81.svg"
 ALT="$\displaystyle \phi_p$">|; 

$key = q/displaystylephi_{text{SFL},0}=text{atan2}(-cosdeltasin(alpha-alpha_0-180^circ),-cosdeltacos(alpha-alpha_0-180^circ))=alpha-alpha_0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.63ex; vertical-align: -0.80ex; " SRC="|."$dir".q|img88.svg"
 ALT="$\displaystyle \phi_{\text{SFL},0} = \text{atan2}(-\cos\delta\sin(\alpha-\alpha_0-180^\circ),
-\cos\delta\cos(\alpha-\alpha_0-180^\circ))
= \alpha-\alpha_0$">|; 

$key = q/displaystylephicostheta;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.27ex; vertical-align: -0.56ex; " SRC="|."$dir".q|img80.svg"
 ALT="$\displaystyle \phi \cos \theta$">|; 

$key = q/displaystyler;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.21ex; vertical-align: -0.11ex; " SRC="|."$dir".q|img31.svg"
 ALT="$\displaystyle r$">|; 

$key = q/displaystyler_mathrm{AIPS}=2frac{sintheta}{1+costheta};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 5.16ex; vertical-align: -1.89ex; " SRC="|."$dir".q|img40.svg"
 ALT="$\displaystyle r_\mathrm{AIPS} = 2\frac{\sin\theta}{1+\cos\theta}$">|; 

$key = q/displaystyler_mathrm{AIPS}=2tanfrac{theta}{2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 4.97ex; vertical-align: -1.70ex; " SRC="|."$dir".q|img41.svg"
 ALT="$\displaystyle r_\mathrm{AIPS} = 2 \tan\frac{\theta}{2}$">|; 

$key = q/displaystylercosp;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.66ex; vertical-align: -0.56ex; " SRC="|."$dir".q|img34.svg"
 ALT="$\displaystyle r \cos p$">|; 

$key = q/displaystylersinp;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.21ex; vertical-align: -0.56ex; " SRC="|."$dir".q|img33.svg"
 ALT="$\displaystyle r \sin p$">|; 

$key = q/displaystylesin^{-1}(sindeltasindelta_p+cosdeltacosdelta_pcos(alpha-alpha_p));MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.93ex; vertical-align: -0.78ex; " SRC="|."$dir".q|img79.svg"
 ALT="$\displaystyle \sin^{-1}(\sin\delta\sin\delta_p+\cos\delta\cos\delta_p\cos(\alpha-\alpha_p))$">|; 

$key = q/displaystylesin^{-1}(sinthetasindelta_p+costhetacosdelta_pcos(phi-phi_p));MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.93ex; vertical-align: -0.78ex; " SRC="|."$dir".q|img76.svg"
 ALT="$\displaystyle \sin^{-1}(\sin\theta\sin\delta_p+\cos\theta\cos\delta_p\cos(\phi-\phi_p))$">|; 

$key = q/displaystylesindeltacosdelta_0-cosdeltasindelta_0cos(alpha-alpha_0);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.53ex; vertical-align: -0.69ex; " SRC="|."$dir".q|img28.svg"
 ALT="$\displaystyle \sin\delta\cos\delta_0 - \cos\delta\sin\delta_0\cos(\alpha-\alpha_0)$">|; 

$key = q/displaystyletanfrac{theta}{2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 4.97ex; vertical-align: -1.70ex; " SRC="|."$dir".q|img32.svg"
 ALT="$\displaystyle \tan \frac{\theta}{2}$">|; 

$key = q/displaystyletheta;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.82ex; vertical-align: -0.11ex; " SRC="|."$dir".q|img29.svg"
 ALT="$\displaystyle \theta$">|; 

$key = q/displaystyletheta_{text{SFL},0}=sin^{-1}(sindelta)=delta;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.94ex; vertical-align: -0.80ex; " SRC="|."$dir".q|img92.svg"
 ALT="$\displaystyle \theta_{\text{SFL},0} = \sin^{-1}(\sin\delta)
= \delta$">|; 

$key = q/displaystylex;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.21ex; vertical-align: -0.11ex; " SRC="|."$dir".q|img20.svg"
 ALT="$\displaystyle x$">|; 

$key = q/displaystylex^prime_;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.68ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img96.svg"
 ALT="$\displaystyle x^\prime_$">|; 

$key = q/displaystylex_;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.56ex; vertical-align: -0.45ex; " SRC="|."$dir".q|img97.svg"
 ALT="$\displaystyle x_$">|; 

$key = q/displaystylex_{text{SFL},0};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.90ex; vertical-align: -0.80ex; " SRC="|."$dir".q|img93.svg"
 ALT="$\displaystyle x_{\text{SFL},0}$">|; 

$key = q/displaystyley;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.66ex; vertical-align: -0.56ex; " SRC="|."$dir".q|img23.svg"
 ALT="$\displaystyle y$">|; 

$key = q/displaystyley^prime_;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.68ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img99.svg"
 ALT="$\displaystyle y^\prime_$">|; 

$key = q/displaystyley_;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.66ex; vertical-align: -0.56ex; " SRC="|."$dir".q|img100.svg"
 ALT="$\displaystyle y_$">|; 

$key = q/displaystyley_{text{SFL},0};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.90ex; vertical-align: -0.80ex; " SRC="|."$dir".q|img95.svg"
 ALT="$\displaystyle y_{\text{SFL},0}$">|; 

$key = q/displaystylez;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.21ex; vertical-align: -0.11ex; " SRC="|."$dir".q|img27.svg"
 ALT="$\displaystyle z$">|; 

$key = q/includegraphics[width=0.9textwidth]{greg-projections-formalism};LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 89.87ex; vertical-align: -0.11ex; " SRC="|."$dir".q|img19.svg"
 ALT="\includegraphics[width=0.9\textwidth]{greg-projections-formalism}">|; 

$key = q/includegraphics[width=0.9textwidth]{greg-projections-gls2sfl};LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 85.46ex; vertical-align: -0.11ex; " SRC="|."$dir".q|img26.svg"
 ALT="\includegraphics[width=0.9\textwidth]{greg-projections-gls2sfl}">|; 

$key = q/p;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.66ex; vertical-align: -0.56ex; " SRC="|."$dir".q|img37.svg"
 ALT="$p$">|; 

$key = q/phi_p;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.49ex; vertical-align: -0.78ex; " SRC="|."$dir".q|img54.svg"
 ALT="$\phi_p$">|; 

$key = q/phi_{text{SFL},0};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.51ex; vertical-align: -0.80ex; " SRC="|."$dir".q|img87.svg"
 ALT="$\phi_{\text{SFL},0}$">|; 

$key = q/r;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.21ex; vertical-align: -0.11ex; " SRC="|."$dir".q|img36.svg"
 ALT="$r$">|; 

$key = q/rsimeqtheta;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.82ex; vertical-align: -0.11ex; " SRC="|."$dir".q|img45.svg"
 ALT="$r
\simeq \theta$">|; 

$key = q/rsimeqthetaslash2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.53ex; vertical-align: -0.69ex; " SRC="|."$dir".q|img44.svg"
 ALT="$r
\simeq \theta/2$">|; 

$key = q/sinphi_p=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.49ex; vertical-align: -0.78ex; " SRC="|."$dir".q|img56.svg"
 ALT="$\sin\phi_p = 0$">|; 

$key = q/tan(thetaslash2)simeqthetaslash2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.53ex; vertical-align: -0.69ex; " SRC="|."$dir".q|img43.svg"
 ALT="$\tan (\theta/2) \simeq \theta/2$">|; 

$key = q/tanx=frac{sinx}{cosx}=frac{2sinxcosx}{2cos^2x}=frac{sin2x}{1+cos2x};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 3.23ex; vertical-align: -1.08ex; " SRC="|."$dir".q|img42.svg"
 ALT="$\tan x =
\frac{\sin x}{\cos x} = \frac{2 \sin x \cos x}{2 \cos^2 x} =
\frac{\sin 2x}{1 + \cos 2x}$">|; 

$key = q/theta;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.82ex; vertical-align: -0.11ex; " SRC="|."$dir".q|img38.svg"
 ALT="$\theta$">|; 

$key = q/theta_p;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.49ex; vertical-align: -0.78ex; " SRC="|."$dir".q|img65.svg"
 ALT="$\theta_p$">|; 

$key = q/theta_{text{SFL},0};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.51ex; vertical-align: -0.80ex; " SRC="|."$dir".q|img91.svg"
 ALT="$\theta_{\text{SFL},0}$">|; 

$key = q/x;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.21ex; vertical-align: -0.11ex; " SRC="|."$dir".q|img50.svg"
 ALT="$x$">|; 

$key = q/x_;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.56ex; vertical-align: -0.45ex; " SRC="|."$dir".q|img105.svg"
 ALT="$x_$">|; 

$key = q/y_;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.66ex; vertical-align: -0.56ex; " SRC="|."$dir".q|img107.svg"
 ALT="$y_$">|; 

$key = q/z;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.21ex; vertical-align: -0.11ex; " SRC="|."$dir".q|img35.svg"
 ALT="$z$">|; 

$key = q/{displaymath}delta_p={cases}delta_{p,1}&text{if}delta_{p,1}>0text{and}delta_{p,1}leq90^circdelta_{p,2}&text{otherwise.}{cases}{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 8.74ex; " SRC="|."$dir".q|img66.svg"
 ALT="\begin{displaymath}\delta_p =
\begin{cases}
\delta_{p,1} &amp; \text{if } \delta_{p,...
... \leq 90^\circ \\\\
\delta_{p,2} &amp; \text{otherwise.}
\end{cases}\end{displaymath}">|; 

$key = q/{displaymath}phi_p={cases}0^circ&text{fordelta_0geq0}180^circ&text{fordelta_0<0}{cases}{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 8.74ex; " SRC="|."$dir".q|img55.svg"
 ALT="\begin{displaymath}\phi_p =
\begin{cases}
0^\circ &amp; \text{for $\delta_0 \geq 0$} \\\\
180^\circ &amp; \text{for $\delta_0 &lt; 0$}
\end{cases}\end{displaymath}">|; 

1;

