[求助]ansys分析后面型数据如何进行zernike多项式拟合？ [复制链接]

小弟不是学光学的，所以想请各位大侠指点啊！谢谢啦 VV54$a  
就是我用ansys计算出了镜面的面型的数据，怎样可以得到zernike多项式的系数，然后用zemax各阶得到像差！谢谢啦！ j0Cj&x%qF}  

可以用matlab编程，用zernike多项式进行波面拟合，求出zernike多项式的系数，拟合的算法有很多种，最简单的是最小二乘法，你可以查下相关资料，挺简单的

泽尼克多项式的前9项对应象差的

非常感谢啊，我手上也有zernike多项式的拟合的源程序，也不知道对不对，不怎么会有 f#7=N
{wm  
function z = zernfun(n,m,r,theta,nflag) bR:hu}YS  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. K:Z(jF!j  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N IGlyx'\_  
%   and angular frequency M, evaluated at positions (R,THETA) on the B[#n,ay  
%   unit circle.  N is a vector of positive integers (including 0), and ;kR=vv  
%   M is a vector with the same number of elements as N.  Each element tGbx/$Y   
%   k of M must be a positive integer, with possible values M(k) = -N(k) BJ'pe[Xa5  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, zKaj<Og  
%   and THETA is a vector of angles.  R and THETA must have the same 5j0 Ib>\  
%   length.  The output Z is a matrix with one column for every (N,M) ?4aW^l6/  
%   pair, and one row for every (R,THETA) pair. t
TubW=H  
% OQKc_z'"  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ^|hVFM2  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), >LH}A6dUC  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral f|F=)tJO  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, =*zde0T?l  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized JR&yaOws  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. -XK;B--c  
% p&)d]oV>  
%   The Zernike functions are an orthogonal basis on the unit circle. R?tjobk!  
%   They are used in disciplines such as astronomy, optics, and yx*<c#Uf  
%   optometry to describe functions on a circular domain. Of$R+n.  
% \IudS{ .?;  
%   The following table lists the first 15 Zernike functions. \j BA4?(S  
% >El]5M7h7  
%       n    m    Zernike function           Normalization gSj0+|  
%       -------------------------------------------------- &@BAV
c z  
%       0    0    1                                 1  ylS
6D  
%       1    1    r * cos(theta)                    2 Q"c/]Sk)  
%       1   -1    r * sin(theta)                    2 UWK|_RT6SA  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) +9pock  
%       2    0    (2*r^2 - 1)                    sqrt(3) 0M&~;`W}  
%       2    2    r^2 * sin(2*theta)             sqrt(6) W2zG"Q  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) ^Oeixi@f  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) kUT^o  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) C@zG(?X  
%       3    3    r^3 * sin(3*theta)             sqrt(8) ._<, Eodv  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) sX3qrRY  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ;_|4c7
  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Uq{$j5p8  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) :xbj&
l  
%       4    4    r^4 * sin(4*theta)             sqrt(10) |-S+x]9  
%       -------------------------------------------------- :*DWL!a  
% lFSvHs5  
%   Example 1:
_'X  
% b?lRada{I  
%       % Display the Zernike function Z(n=5,m=1) E`hR(UL ?  
%       x = -1:0.01:1; XZ3fWcw[  
%       [X,Y] = meshgrid(x,x); jAv3qMQA  
%       [theta,r] = cart2pol(X,Y); bhbTloCR  
%       idx = r<=1; 2mMi=pv9  
%       z = nan(size(X)); ?~.:C'  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); 0E,QOF{o  
%       figure {.[EXMX  
%       pcolor(x,x,z), shading interp JRZp'Ln  
%       axis square, colorbar gu~R4@3  
%       title('Zernike function Z_5^1(r,\theta)') zxH<~2
 
% 4sRg+mMI  
%   Example 2: "USzk7=&.  
% R$A%Zh6  
%       % Display the first 10 Zernike functions 4<)*a]\c5M  
%       x = -1:0.01:1; z 0zB&}  
%       [X,Y] = meshgrid(x,x); suW|hh1/Ya  
%       [theta,r] = cart2pol(X,Y); .X"&kO>G  
%       idx = r<=1; v6[VdWOx5  
%       z = nan(size(X)); 8O60pB;4  
%       n = [0  1  1  2  2  2  3  3  3  3]; h(J$-SUs  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; e>.^RtDF  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; ],~[^0  
%       y = zernfun(n,m,r(idx),theta(idx)); J=(i0A  
%       figure('Units','normalized') zxD=q5in  
%       for k = 1:10 2Ub-uf
kU  
%           z(idx) = y(:,k); 5} ur,0{  
%           subplot(4,7,Nplot(k)) #CAZ}];Qx  
%           pcolor(x,x,z), shading interp j6$@vA)  
%           set(gca,'XTick',[],'YTick',[]) }$qrNbLJ  
%           axis square JKO*bbj  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) yJO Jw o^  
%       end *O@Zn  
% j!oX\Y-:&  
%   See also ZERNPOL, ZERNFUN2. S')
DAx  
D^P0X:T]  
%   Paul Fricker 11/13/2006 YWDgRb  
5L~lF8  
(: kn)  
% Check and prepare the inputs: 0dS(g&ZR  
% ----------------------------- N#)Klq87z  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) S1@r.z2L  
    error('zernfun:NMvectors','N and M must be vectors.') Nq\)o{<1  
end <SOG?Lh~  
IR:{{ (  
if length(n)~=length(m) 2@pEiq3  
    error('zernfun:NMlength','N and M must be the same length.') P$N5j~*  
end Mqk|H~l5c  
* a1q M?  
n = n(:); "lC>_A  
m = m(:); F2_'U' a  
if any(mod(n-m,2)) ~)>.%`v&  
    error('zernfun:NMmultiplesof2', ... s|c}9/Xe)  
          'All N and M must differ by multiples of 2 (including 0).') cXf/  
end tlg}"lY  
nhC8Tq[m  
if any(m>n) MZcvr9y  
    error('zernfun:MlessthanN', ... i O?f&u  
          'Each M must be less than or equal to its corresponding N.') PNo:vRtsq  
end [q_62[-X  
qdKqc,R1{  
if any( r>1 | r<0 ) Ie=gI+2  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') x%goyXK  
end %hZX XpuO  
v
dB2T2F  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) (JnEso-V  
    error('zernfun:RTHvector','R and THETA must be vectors.') }Y!s:w#  
end .m>Qlh  
O'#;Ge/,  
r = r(:); w'$>E4\  
theta = theta(:); n+Conp/  
length_r = length(r); "$K]+0ryG<  
if length_r~=length(theta) $FX$nY  
    error('zernfun:RTHlength', ... a_{'I6a*,  
          'The number of R- and THETA-values must be equal.') *b0z/6  
end v,ni9DIu  
@|">j#0  
% Check normalization: 5rCJIl.  
% -------------------- &(Hw:W9  
if nargin==5 && ischar(nflag) |wQ3+WN|  
    isnorm = strcmpi(nflag,'norm'); ~]?EV?T  
    if ~isnorm u8|CeA  
        error('zernfun:normalization','Unrecognized normalization flag.') !Y7$cU &  
    end Cc`-34/%  
else r2i]9>w  
    isnorm = false; \+Y=}P>  
end *&_cp]3-WF  
cq gCcO,  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4oryTckS  
% Compute the Zernike Polynomials
ePv`R'#  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% T\6,@7  
1
{d;Ngx  
% Determine the required powers of r: N=T}  
% ----------------------------------- q*Hg-J}  
m_abs = abs(m); 5[)#
3vY  
rpowers = []; fz|_c*&64  
for j = 1:length(n) $dK430_B  
    rpowers = [rpowers m_abs(j):2:n(j)]; T3SFG]H  
end GVn'p Wg  
rpowers = unique(rpowers); #8M^;4N>[  
%{:pBt:Z  
% Pre-compute the values of r raised to the required powers, 7 H:y=?X6  
% and compile them in a matrix: 0YfmAF$/B  
% ----------------------------- 0o6o<ggi  
if rpowers(1)==0 +\&6Zbn  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); rLmc(-q  
    rpowern = cat(2,rpowern{:}); ~Jsu"kr  
    rpowern = [ones(length_r,1) rpowern]; 'o0o.&/=  
else 6|3 X*Orn  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); '|5o(6u'  
    rpowern = cat(2,rpowern{:}); `ZM$\Q=:  
end 6w m-uu  
$""kZ  
% Compute the values of the polynomials: ;XjXv'  
% -------------------------------------- #;@I.  
y = zeros(length_r,length(n)); bXXX-Xc  
for j = 1:length(n) 'X6Y!VDd  
    s = 0:(n(j)-m_abs(j))/2;
S'ms>ZENC  
    pows = n(j):-2:m_abs(j); \{~CO{II  
    for k = length(s):-1:1 d=uGB"  
        p = (1-2*mod(s(k),2))* ... H`URJ8k$Q  
                   prod(2:(n(j)-s(k)))/              ... VGxab;#,:3  
                   prod(2:s(k))/                     ... :~srl)|)  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... whP5u/857  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); W_Hoa*~  
        idx = (pows(k)==rpowers); 1x\k:2U  
        y(:,j) = y(:,j) + p*rpowern(:,idx); DS7L}]  
    end 1MnC5[Q  
     =Bm|9A1  
    if isnorm \*b  .f  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 9b,0_IMHH  
    end 59W~bWHCP  
end ~$j;@4  
% END: Compute the Zernike Polynomials l`:u5\ rM  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $G }9iV7  
Y{#*;p*I  
% Compute the Zernike functions: R)*l)bpZ#  
% ------------------------------ M3F1O6=4j  
idx_pos = m>0; dw5"}-D  
idx_neg = m<0; zF{~Md1  
/]-yZ0hX0O  
z = y; ~!g2+^G7+P  
if any(idx_pos) f/IQ2yT-:D  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); +Ig%h[1a  
end z#P`m,~t0  
if any(idx_neg) .7LQ l?  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); c|aX4=Z  
end WQiRbbX  
L+ XAbL)  
% EOF zernfun

function z = zernfun2(p,r,theta,nflag) cS
:O|R#%t  
%ZERNFUN2 Single-index Zernike functions on the unit circle. >G%oWRk  
%   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated v'Ehr**]+  
%   at positions (R,THETA) on the unit circle.  P is a vector of positive 2QAP$f0Ln  
%   integers between 0 and 35, R is a vector of numbers between 0 and 1, CnZEBAU  
%   and THETA is a vector of angles.  R and THETA must have the same FkuD Gg~a  
%   length.  The output Z is a matrix with one column for every P-value, CxJ3u  
%   and one row for every (R,THETA) pair. lA1  
% d6JW"  
%   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike / j%~#@  
%   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ,c?( |tF  
%   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) aBG^Xhx  
%   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 w%X@os}E  
%   for all p. tK/,U =+  
% Ml{ ]{n  
%   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 oaPWeM+  
%   Zernike functions (order N<=7).  In some disciplines it is 4KR`  
%   traditional to label the first 36 functions using a single mode ISK 8t  
%   number P instead of separate numbers for the order N and azimuthal l:JVt`A4?  
%   frequency M. v7KBYN  
% +WMXd.iN,  
%   Example: \f(zMP  
% -LUZ7,!/>o  
%       % Display the first 16 Zernike functions 8Yf*vp>T/x  
%       x = -1:0.01:1; oA7DhU5n  
%       [X,Y] = meshgrid(x,x); ZS(%!+M  
%       [theta,r] = cart2pol(X,Y); /d:hW4}<}.  
%       idx = r<=1; c(2?
./\|  
%       p = 0:15; <rK[&JlJ  
%       z = nan(size(X)); }+pwSjsno  
%       y = zernfun2(p,r(idx),theta(idx)); %*npLDi  
%       figure('Units','normalized') K?!W9lUq  
%       for k = 1:length(p) GK1nGdT]  
%           z(idx) = y(:,k); 8IxIW0  
%           subplot(4,4,k) 0!ZaR6  
%           pcolor(x,x,z), shading interp %Y=r5'6l  
%           set(gca,'XTick',[],'YTick',[]) w{xa@Q]t-  
%           axis square 8M,@Mbn  
%           title(['Z_{' num2str(p(k)) '}']) 0,0Z!-Y  
%       end T|D^kL%m!  
% JA9NTu(  
%   See also ZERNPOL, ZERNFUN. PlS
)Zv3  
00dY?d{[D  
%   Paul Fricker 11/13/2006 3F!)7  
h%W,O,K/  
D]}~`SO  
% Check and prepare the inputs: \<T7EV.  
% ----------------------------- )8#-IXxp  
if min(size(p))~=1 _a& Z$2O  
    error('zernfun2:Pvector','Input P must be vector.') ]a&riPh"  
end
c*E7nc)u  
>4kQ9lXL  
if any(p)>35 Wex2Fd?DO  
    error('zernfun2:P36', ... U\qbr.<  
          ['ZERNFUN2 only computes the first 36 Zernike functions ' ... $|J+  
           '(P = 0 to 35).']) oi,KA  
end u pUJF`3  
0uW)&>W  
% Get the order and frequency corresonding to the function number: '/ Hoq  
% ---------------------------------------------------------------- Fv %@k{  
p = p(:); =>3,]hnep  
n = ceil((-3+sqrt(9+8*p))/2); I(7iD. ^:  
m = 2*p - n.*(n+2); >]gB@tn[  
er-0i L@  
% Pass the inputs to the function ZERNFUN: @%L  
% ---------------------------------------- *aWh]x9TlU  
switch nargin HpIi-Es7C  
    case 3 9
e>2kd  
        z = zernfun(n,m,r,theta); /
<$|tp\Rc  
    case 4 w42{)S"  
        z = zernfun(n,m,r,theta,nflag);  +@f  
    otherwise U; <{P  
        error('zernfun2:nargin','Incorrect number of inputs.') 7_qsVhh]$E  
end oPaoQbR(A  
XP}5i!}}7=  
% EOF zernfun2

function z = zernpol(n,m,r,nflag) W1xPK*  
%ZERNPOL Radial Zernike polynomials of order N and frequency M. bDh4p]lm  
%   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of )zz{~Cf  
%   order N and frequency M, evaluated at R.  N is a vector of v*JKLA  
%   positive integers (including 0), and M is a vector with the Rc4=zimr+  
%   same number of elements as N.  Each element k of M must be a d#bg(y\G|  
%   positive integer, with possible values M(k) = 0,2,4,...,N(k) A=|a!N/  
%   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is G"u4]!$/  
%   a vector of numbers between 0 and 1.  The output Z is a matrix mSu$1m8  
%   with one column for every (N,M) pair, and one row for every *)^ZUk  
%   element in R. g +gcH  
% 3PRU  
%   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- <im}R9eJ1  
%   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 7zE1>.  
%   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to !J<0.nO/:  
%   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 =B;qy7?  
%   for all [n,m]. :KG=3un]  
% $J)`Ru6.  
%   The radial Zernike polynomials are the radial portion of the Xg_l4!T_l  
%   Zernike functions, which are an orthogonal basis on the unit IV':sNV  
%   circle.  The series representation of the radial Zernike &{a#8sbf#c  
%   polynomials is G@jx&#v  
% 06.8m;{N  
%          (n-m)/2 OT|0_d?bD  
%            __ pMViq0  
%    m      \       s                                          n-2s [U^Cz{G  
%   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r _G<Wq`0w)  
%    n      s=0 l"X,[  
% knV*,   
%   The following table shows the first 12 polynomials. Ic!x y  
% \?8q&o1=]  
%       n    m    Zernike polynomial    Normalization ax;<idC}  
%       --------------------------------------------- 8JR&s  
%       0    0    1                        sqrt(2) 5{1=BZftZ  
%       1    1    r                           2 DyZ6&*s$  
%       2    0    2*r^2 - 1                sqrt(6) \21Gg%W5AE  
%       2    2    r^2                      sqrt(6) .Sa=VC?EZ  
%       3    1    3*r^3 - 2*r              sqrt(8) 7AGUi+!ICl  
%       3    3    r^3                      sqrt(8) =c&.I}^1L  
%       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) &%/T4$'+Y+  
%       4    2    4*r^4 - 3*r^2            sqrt(10) e F}KOOfC  
%       4    4    r^4                      sqrt(10) DXO'MZon3  
%       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) eUR+j?5I  
%       5    3    5*r^5 - 4*r^3            sqrt(12) :2vuc!Pu  
%       5    5    r^5                      sqrt(12) wCv9VvF`  
%       --------------------------------------------- /#eS3`48  
% E|_8#xvb  
%   Example: gy|o#&e]%  
% /6y{?0S  
%       % Display three example Zernike radial polynomials !a!4^zqp  
%       r = 0:0.01:1; 3N2d@R  
%       n = [3 2 5]; xy&*s\=:  
%       m = [1 2 1]; 7aPA+gA/  
%       z = zernpol(n,m,r); Z
zR0k  
%       figure L|-|DOgw  
%       plot(r,z) `}$bJCSF.n  
%       grid on aAn p7\7  
%       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') Z:x`][vg  
% mC0Dj O  
%   See also ZERNFUN, ZERNFUN2. 368 g>/#'  
hV) `e"r\s  
% A note on the algorithm. tBl(E  
% ------------------------ z?|bs?HKS  
% The radial Zernike polynomials are computed using the series 4,Uqcw?!F'  
% representation shown in the Help section above. For many special #a
tL2(wJ  
% functions, direct evaluation using the series representation can -*J!Ws(9  
% produce poor numerical results (floating point errors), because /"~UGn]R  
% the summation often involves computing small differences between z" ?WT$  
% large successive terms in the series. (In such cases, the functions j:2F97  
% are often evaluated using alternative methods such as recurrence m&(qr5>b  
% relations: see the Legendre functions, for example). For the Zernike 8.IenU9  
% polynomials, however, this problem does not arise, because the a gL@A  
% polynomials are evaluated over the finite domain r = (0,1), and mC(YO y  
% because the coefficients for a given polynomial are generally all EaL>~:j  
% of similar magnitude. {/aHZ<I&^h  
% N Mx:Jh-YN  
% ZERNPOL has been written using a vectorized implementation: multiple {9XNh[NbP  
% Zernike polynomials can be computed (i.e., multiple sets of [N,M] I};*O6D`  
% values can be passed as inputs) for a vector of points R.  To achieve +/M%%:>mY  
% this vectorization most efficiently, the algorithm in ZERNPOL #@1(  
% involves pre-determining all the powers p of R that are required to em- <V5fb  
% compute the outputs, and then compiling the {R^p} into a single :LdPqFXj  
% matrix.  This avoids any redundant computation of the R^p, and S>j.i  
% minimizes the sizes of certain intermediate variables. 0XqxW\8_l  
% N:+d=G`x  
%   Paul Fricker 11/13/2006 Wfc~"GQq4  
?FR-aXx  
H(M{hfa|  
% Check and prepare the inputs: j}(m$j'  
% ----------------------------- AA=eWg  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) $ye>;
Ek  
    error('zernpol:NMvectors','N and M must be vectors.') ub`zS-vb  
end WNKP';(a@G  
P6'Se'f8  
if length(n)~=length(m) ;mwnAO  
    error('zernpol:NMlength','N and M must be the same length.') ga S}>?qk  
end _.BT%4  
KU]o=\ak%  
n = n(:); Unb3 Gv#O  
m = m(:); "Y- WY,H  
length_n = length(n); *8)va  
M#m;jJqON  
if any(mod(n-m,2)) )1HWD]>4  
    error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') L*vKIP<EMM  
end Qj(ppep\U"  
`
c-omNu  
if any(m<0) n"Bc2}{  
    error('zernpol:Mpositive','All M must be positive.') ]bpgsW:Xu  
end Q[|*P ] w  
HTvUt*U1  
if any(m>n) +PKsiUJ|  
    error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') m&'!^{av  
end 1Ax;|.KQH  
$7i[7S4  
if any( r>1 | r<0 ) /m 7~-~$V  
    error('zernpol:Rlessthan1','All R must be between 0 and 1.') be5N{lPT@;  
end Vry_X2  
;_E|I=%'E  
if ~any(size(r)==1) e6jA4X+a  
    error('zernpol:Rvector','R must be a vector.') 6CW5ay_,  
end ~vM99hW  
d~{$,"!-f  
r = r(:); urCTP.F  
length_r = length(r); 4i+%~X@p  
d2-oy5cEB  
if nargin==4 n5^57[(  
    isnorm = ischar(nflag) & strcmpi(nflag,'norm'); #h4FLF_w  
    if ~isnorm BpYxH#4  
        error('zernpol:normalization','Unrecognized normalization flag.') TOS'|xQ  
    end ;YW@ 3F-h  
else 6DgdS5GhT_  
    isnorm = false; (+/d*4  
end V{A`?Jl6{  
SO[ u4b_"h  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% RgQs`aI  
% Compute the Zernike Polynomials mdEl CC0  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ^w<aS w  
uLPBl~Y  
% Determine the required powers of r: Fkq^2o ]  
% ----------------------------------- cF8X  
rpowers = []; ,u)jZ7  
for j = 1:length(n) $AE5n>ZD$  
    rpowers = [rpowers m(j):2:n(j)]; 1+XM1(|c`  
end 0_MtmmL.  
rpowers = unique(rpowers); 5$cjCjY  
'\xE56v)F  
% Pre-compute the values of r raised to the required powers, RwOOe7mv  
% and compile them in a matrix: VO=Ibu&X  
% ----------------------------- 5$N#=i`V  
if rpowers(1)==0 h,]VWG  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); jC{KI!kPt  
    rpowern = cat(2,rpowern{:}); 7C,giCYU  
    rpowern = [ones(length_r,1) rpowern]; 6yMZ2%  
else GdVF;  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); >8=lX`9f{  
    rpowern = cat(2,rpowern{:}); g=@d!]Z~[  
end 4nh=Dq[  
/sT?p=[.  
% Compute the values of the polynomials: b.@4yW  
% -------------------------------------- ms&1P  
z = zeros(length_r,length_n); 5
\#I4\  
for j = 1:length_n : MjDcI~  
    s = 0:(n(j)-m(j))/2; ci$J?a  
    pows = n(j):-2:m(j); K%x]:|,>M  
    for k = length(s):-1:1 Ro"'f7(v.  
        p = (1-2*mod(s(k),2))* ... m|c[C\)By  
                   prod(2:(n(j)-s(k)))/          ... 6l;2kztGp  
                   prod(2:s(k))/                 ... =W:=}ODD  
                   prod(2:((n(j)-m(j))/2-s(k)))/ ... $[,4Ib_|  
                   prod(2:((n(j)+m(j))/2-s(k))); 6L$KMYHE  
        idx = (pows(k)==rpowers); s8>y&b.  
        z(:,j) = z(:,j) + p*rpowern(:,idx); (=Kv1 HaD  
    end -eyF9++`  
     *qk7e[IP  
    if isnorm T]-MrnO  
        z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 9 i/ (  
    end <<A#4!f  
end !U m9ceK  
6uFw+Ya#  
% EOF zernpol

这三个文件，我不知道该怎样把我的面型节点的坐标及轴向位移用起来，还烦请指点一下啊，谢谢啦！

我也正在找啊

我也一直想了解这个多项式的应用，还没用过呢

guapiqlh:我也一直想了解这个多项式的应用，还没用过呢 (2014-03-04 11:35)  @\y{q;  

_,L_H[FN  
数值分析方法看一下就行了。其实就是正交多项式的应用。zernike也只不过是正交多项式的一种。 @FN|=
?8%  
(2#Xa,pb  
07年就写过这方面的计算程序了。