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{SECT 0 {SECT 0 {PARA 3 "" 0 "" {HYPERLNK 17 "AESOP" 2 "AESOP" "" }
{TEXT -1 25 " routine beam_intersect()" }}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 6 "Syntax" }}{PARA 0 "" 0 "" {TEXT 272 201 "beam_intersect( i
nputbeam, surface, coordframe, \n                small_parms1, order1,
 small_params2, order2, \n                varlist, nodenom_list, subli
st, siderels, \n                siderels_list )" }{TEXT -1 1 " " }}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "Purpose" }}{PARA 0 "" 0 "" {TEXT 
271 16 "beam_intersect()" }{TEXT -1 192 " finds the intersection point
 of a beam and a quadratic surface up to and including order 'order' i
n the small parameters list 'params'.  If order < 0, then no series ex
pansions are performed." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Argume
nts" }}{PARA 15 "" 0 "" {TEXT -1 13 "The argument " }{TEXT 257 10 "inp
utbeam " }{TEXT -1 228 "must be a table of data type 'beam':\n      be
am[pos]     anchor position of beam (type 'point')\n      beam[dir]   \+
   direction of propogation (type 'vector')\n      beam[path]    accum
ulated optical path\n      beam[type] = 'beam'" }}{PARA 15 "" 0 "" 
{TEXT -1 13 "The argument " }{TEXT 258 8 "surface " }{TEXT -1 164 "mus
t be indexed as follows:\n      surface[eqn]    := f(xname,yname,zname
);\n      surface[coord] := [xname,yname,zname];\n      surface[pos]\n
      surface[dir]\nwhere " }{XPPEDIT 18 0 "f(x,y,z)" "-%\"fG6%%\"xG%
\"yG%\"zG" }{TEXT -1 40 " is the equation for the surface,\n      " }
{XPPEDIT 18 0 "f(x,y,z) = 0" "/-%\"fG6%%\"xG%\"yG%\"zG\"\"!" }{TEXT 
-1 330 "\nin the local coordinates, and xname, yname, and zname are th
e labels used for the x, y, and z coordinates in the equation describi
ng the surface.  surface[pos] is the position of the surface vertex in
 the global coordinate system, and surface[dir] is the direction of th
e surface axis of symmetry in the global coordinate system." }}{PARA 
15 "" 0 "" {TEXT -1 13 "The argument " }{TEXT 259 11 "coordframe " }
{TEXT -1 376 "signals whether the coordinate frame in which the input \+
beam is expressed is the surface LOCAL frame or the GLOBAL reference f
rame.  If coordframe=LOCAL, no frame conversion is done.  If coordfram
e=GLOBAL, the input beam is transformed to the surface LOCAL frame, th
e intersection is found, then the intersection point is transformed to
 the GLOBAL frame before being returned." }}{PARA 15 "" 0 "" {TEXT 
260 12 "small_parms1" }{TEXT 263 5 " and " }{TEXT 264 12 "small_parms2
" }{TEXT 265 4 " are" }{TEXT -1 244 " Maple lists of small parameters \+
to expand the expressions in at the intersection point.  The lists wil
l be treated as independent.  Expansions will first be performed on th
e variables in small_params1, then on the variables in small_params2. \+
 " }{TEXT 266 6 "order1" }{TEXT -1 5 " and " }{TEXT 267 6 "order2" }
{TEXT -1 262 " are the corresponding desired expansion orders.  This a
llows independent expansions to different orders for different variabl
es.  For example, one may want to expand in radius to 12th order at th
e same time as expanding to 4th order in a perturbation parameter." }}
{PARA 15 "" 0 "" {TEXT 261 8 "varlist " }{TEXT -1 137 "is a list of \"
important\" variables by which the expressions may be organized and th
en recursively subexpression-simplified via collect()." }}{PARA 15 "" 
0 "" {TEXT 262 12 "nodenom_list" }{TEXT -1 113 " is a list of variable
s which are not allowed in an expression denominator when finding beam
 intersection points." }}{PARA 15 "" 0 "" {TEXT 268 7 "sublist" }
{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT 269 8 "siderels" }}{PARA 15 "
" 0 "" {TEXT 270 13 "siderels_list" }{TEXT -1 1 " " }}}{SECT 1 {PARA 
4 "" 0 "" {TEXT -1 12 "Return Value" }}{PARA 15 "" 0 "" {TEXT -1 298 "
intersct() returns a table containing the first solution for the inter
section of the beam with the surface, along with the distance from the
 beam anchor to the intersection point, indexed as follows:\n     retu
rn_value[pos]       an AESOP 'point'\n     return_value[path]      the
 optical path length" }}{PARA 15 "" 0 "" {TEXT -1 146 "Coordinates are
 wrt the surface local reference frame if intersct() is called with co
ordframe=LOCAL, or wrt the global frame if coordframe=GLOBAL." }}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "Additional Requirements" }}{PARA 
15 "" 0 "" {TEXT -1 103 "The global variables\n     GLOBAL := 1;\n    \+
 LOCAL  := 0;\nmust be defined before calling this procedure." }}
{PARA 15 "" 0 "" {TEXT -1 86 "The floating point variable time0 must b
e initialized before calling this procedure:  " }{TEXT 19 14 "time0:=t
ime();" }}{PARA 15 "" 0 "" {TEXT -1 93 "The files startup.p and opsys.
p must have been read by Maple before this routine can be used." }}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Method" }}{PARA 0 "" 0 "" {TEXT 
-1 148 "The input beam must be in the local coordinate frame of the su
rface.  The parameterized equation for a line coincident with the inpu
t beam is\n      " }{XPPEDIT 18 0 "r(t)=r[0] + t*n" "/-%\"rG6#%\"tG,&&
F$6#\"\"!\"\"\"*&F&F+%\"nGF+F+" }{TEXT -1 7 "\nwhere " }{XPPEDIT 18 0 
"r[0]=[x[0],y[0],z[0]]" "/&%\"rG6#\"\"!7%&%\"xG6#F&&%\"yG6#F&&%\"zG6#F
&" }{TEXT -1 129 " is a point on the line (the beam anchor point), t i
s the parameterization of the position on the line (ie, the path lengt
h from " }{XPPEDIT 18 0 "r[0]" "&%\"rG6#\"\"!" }{TEXT -1 44 "), and n \+
is the unit direction vector\n      " }{XPPEDIT 18 0 "n=[ sin psi * co
s lambda, sin psi * sin lambda, cos psi ]" "/%\"nG7%**%$sinG\"\"\"%$ps
iGF'%$cosGF'%'lambdaGF'**F&F'F(F'F&F'F*F'*&F)F'F(F'" }{TEXT -1 3 " = \+
" }{XPPEDIT 18 0 "[ n[x], n[y], n[z] ]" "7%&%\"nG6#%\"xG&F$6#%\"yG&F$6
#%\"zG" }{TEXT -1 8 "\nwhere (" }{XPPEDIT 18 0 "psi,lambda" "6$%$psiG%
'lambdaG" }{TEXT -1 192 ") are the polar and azimuthal angls wrt the +
z axis and counterclockwise from the +x axis in the local coordinate f
rame.\n\nThe equation of the surface (in the local frame) is of the fo
rm\n      " }{XPPEDIT 18 0 "f(x,y,z) = 0" "/-%\"fG6%%\"xG%\"yG%\"zG\"
\"!" }{TEXT -1 94 "\nWe can find the value of t which corresponds to t
he intersection point by substituting\n      " }{XPPEDIT 18 0 "x = x[0
] + t*sin psi * cos lambda" "/%\"xG,&&F#6#\"\"!\"\"\"*,%\"tGF(%$sinGF(
%$psiGF(%$cosGF(%'lambdaGF(F(" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "x[0] \+
+ t*n[x]" ",&&%\"xG6#\"\"!\"\"\"*&%\"tGF'&%\"nG6#F$F'F'" }{TEXT -1 7 "
\n      " }{XPPEDIT 18 0 "y = y[0] + t*sin psi * sin lambda" "/%\"yG,&
&F#6#\"\"!\"\"\"*,%\"tGF(%$sinGF(%$psiGF(F+F(%'lambdaGF(F(" }{TEXT -1 
5 "   = " }{XPPEDIT 18 0 "y[0] + t*n[y]" ",&&%\"yG6#\"\"!\"\"\"*&%\"tG
F'&%\"nG6#F$F'F'" }{TEXT -1 7 "\n      " }{XPPEDIT 18 0 "z = z[0] + t*
sin psi * sin lambda" "/%\"zG,&&F#6#\"\"!\"\"\"*,%\"tGF(%$sinGF(%$psiG
F(F+F(%'lambdaGF(F(" }{TEXT -1 6 "    = " }{XPPEDIT 18 0 "z[0] + t*n[z
]" ",&&%\"zG6#\"\"!\"\"\"*&%\"tGF'&%\"nG6#F$F'F'" }{TEXT -1 6 "\ninto \+
" }{XPPEDIT 18 0 "f(x,y,z)" "-%\"fG6%%\"xG%\"yG%\"zG" }{TEXT -1 171 " \+
and solving for t.  For a quadratic surface there will in general be t
wo solutions.  Then we substitute the solution(s) for t back into the \+
equations for the line,\n      " }{XPPEDIT 18 0 "r(t) = r[0] + t*n" "/
-%\"rG6#%\"tG,&&F$6#\"\"!\"\"\"*&F&F+%\"nGF+F+" }{TEXT -1 126 "\nto de
termine the x, y, and z values of the intersection point.\n\nAll calul
ations and results are in the local reference frame." }}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 18 "Contact the Author" }}{PARA 15 "" 0 "" 
{TEXT -1 123 "Dr. Marc A. Murison\nAstronomical Applications Dept.\nU.
S. Naval Observatory\n3450 Massachusetts Ave., NW\nWashignton, DC 2039
2" }}{PARA 15 "" 0 "" {TEXT 256 29 "murison@riemann.usno.navy.mil" }}
{PARA 15 "" 0 "" {TEXT 256 35 "http://riemann.usno.navy.mil/AESOP/" }}
}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "See Also" }}{PARA 0 "" 0 "" 
{HYPERLNK 17 "AESOP" 2 "AESOP" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "plan
e" 2 "AESOP,quadric" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "quadric" 2 "AE
SOP,quadric" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "reflect" 2 "AESOP,refl
ect" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "surfnorm" 2 "AESOP,surface_nor
mal" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Zernike" 2 "AESOP,Zernike" "" 
}}}}}{MARK "0 1 1 0" 27 }{VIEWOPTS 1 1 0 1 1 1803 }