function?z?=?zernfun（nmrthetanflag）
%ZERNFUN?Zernike?functions?of?order?N?and?frequency?M?on?the?unit?circle.
%???Z?=?ZERNFUN（NMRTHETA）?returns?the?Zernike?functions?of?order?N
%???and?angular?frequency?M?evaluated?at?positions?（RTHETA）?on?the
%???unit?circle.??N?is?a?vector?of?positive?integers?（including?0）?and
%???M?is?a?vector?with?the?same?number?of?elements?as?N.??Each?element
%???k?of?M?must?be?a?positive?integer?with?possible?values?M（k）?=?-N（k）
%???to?+N（k）?in?steps?of?2.??R?is?a?vector?of?numbers?between?0?and?1
%???and?THETA?is?a?vector?of?angles.??R?and?THETA?must?have?the?same
%???length.??The?output?Z?is?a?matrix?with?one?column?for?every?（NM）
%???pair?and?one?row?for?every?（RTHETA）?pair.
%
%???Z?=?ZERNFUN（NMRTHETA‘norm‘）?returns?the?normalized?Zernike
%???functions.??The?normalization?factor?sqrt（（2-delta（m0））*（n+1）/pi）
%???with?delta（m0）?the?Kronecker?delta?is?chosen?so?that?the?integral
%???of?（r?*?[Znm（rtheta）]^2）?over?the?unit?circle?（from?r=0?to?r=1
%???and?theta=0?to?theta=2*pi）?is?unity.??For?the?non-normalized
%???polynomials?max（Znm（r=1theta））=1?for?all?[nm].
%
%???The?Zernike?functions?are?an?orthogonal?basis?on?the?unit?circle.
%???They?are?used?in?disciplines?such?as?astronomy?optics?and
%???optometry?to?describe?functions?on?a?circular?domain.
%
%???The?following?table?lists?the?first?15?Zernike?functions.
%
%???????n????m????Zernike?function???????????Normalization
%???????--------------------------------------------------
%???????0????0????1?????????????????????????????????1
%???????1????1????r?*?cos（theta）????????????????????2
%???????1???-1????r?*?sin（theta）????????????????????2
%???????2???-2????r^2?*?cos（2*theta）?????????????sqrt（6）
%???????2????0????（2*r^2?-?1）????????????????????sqrt（3）
%???????2????2????r^2?*?sin（2*theta）?????????????sqrt（6）
%???????3???-3????r^3?*?cos（3*theta）?????????????sqrt（8）
%???????3???-1????（3*r^3?-?2*r）?*?cos（theta）?????sqrt（8）
%???????3????1????（3*r^3?-?2*r）?*?sin（theta）?????sqrt（8）
%???????3????3????r^3?*?sin（3*theta）?????????????sqrt（8）
%???????4???-4????r^4?*?cos（4*theta）?????????????sqrt（10）
%???????4???-2????（4*r^4?-?3*r^2）?*?cos（2*theta）?sqrt（10）
%???????4????0????6*r^4?-?6*r^2?+?1??????????????sqrt（5）
%???????4????2????（4*r^4?-?3*r^2）?*?cos（2*theta）?sqrt（10）
%???????4????4????r^4?*?sin（4*theta）?????????????sqrt（10）
%???????--------------------------------------------------
%
%???Example?1:
%
%???????%?Display?the?Zernike?function?Z（n=5m=1）
%???????x?=?-1:0.01:1;
%???????[XY]?=?meshgrid（xx）;
%???????[thetar]?=?cart2pol（XY）;
%???????idx?=?r<=1;
%???????z?=?nan（size（X））;
%???????z（idx）?=?zernfun（51r（idx）theta（idx））;
%???????figure
%???????pcolor（xxz）?shading?interp
%???????axis?square?colorbar
%???????title（‘Zernike?function?Z_5^1（r\theta）‘）
%
%???Example?2:
%
%???????%?Display?the?first?10?Zernike?functions
%???????x?=?-1:0.01:1;
%???????[XY]?=?meshgrid（xx）;
%???????[theta

?屬性????????????大小?????日期????時間???名稱
-----------?---------??----------?-----??----
?????文件????????6369??2006-11-13?12:56??zernpol.m
?????文件????????2562??2006-11-13?12:57??zernfun2.m
?????文件????????6652??2006-11-13?12:57??zernfun.m