-
Notifications
You must be signed in to change notification settings - Fork 31
/
coscos.m
205 lines (188 loc) · 6.21 KB
/
coscos.m
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
function [I,phint,th,pars]=coscos(th,m1,m2,dom,pars)
% [I,phint,th,pars]=coscos(th,m1,m2,dom,pars)
%
% Calculates the longitudinal integral
% \int_{A}{B}\cos(m1\phi)\cos(m2\phi)\,d\phi
% over a specified domain at some (Gauss-Legendre) integration point(s).
%
% INPUT:
%
% th Colatitude(s) that you may need to define the patch
% m1,m2 Angular order of the cosine factors
% dom A matrix with the 'phint' hatchings from PHICURVE, or
% 'patch' spherical patch [default] with the following specs
% 'england' England, Scotland and Wales
% pars [th0,ph0,thR] for 'patch', with
% th0 Colatitude of the cap center, in radians
% ph0 Longitude of the cap center, in radians
% thR Radius of the cap, in radians
% N for 'england' with N the smoothness of the splining
%
% EXAMPLE:
%
% coscos('demo1') % Illustrates the area of a random patch
% coscos('demo2') % Compares result with GL integration
% [a,b]=coscos('demo2',0); % Compares result with GL integration
%
% Last modified by charig-at-princeton.edu, 10/23/2014
% Last modified by fjsimons-at-alum.mit.edu, 10/23/2014
defval('th',[])
defval('m1',10) % Not zero (see demo2)
defval('m2',4)
defval('dom','patch')
if ~isstr(th)
if isstr(dom)
switch dom
case 'patch'
% Get the parameters of the dom
defval('pars',[pi/4 pi/2 pi/9]);
th0=pars(1); ph0=pars(2); thR=pars(3);
defval('th',linspace(th0-thR,th0+thR,100));
phint=dphpatch(th,thR,th0,ph0);
case 'england'
defval('N',10)
% Now we may have multiple pairs
phint=dphregion(th*180/pi,N,'england');
phint=phint*pi/180;
otherwise
error('Specify valid domain')
end
else
phint=dom;
end
I=repmat(0,size(phint,1),1);
for index=1:size(phint,2)/2
A=phint(:,2*index-1); B=phint(:,2*index);
if m1~=m2 & m1~=-m2
I=I+1/2*(sin((m1-m2)*B)*m1+...
sin((m1-m2)*B)*m2+...
sin((m1+m2)*B)*m1-...
sin((m1+m2)*B)*m2-...
sin((m1-m2)*A)*m1-...
sin((m1-m2)*A)*m2-...
sin((m1+m2)*A)*m1+...
sin((m1+m2)*A)*m2)/...
((m1-m2)*(m1+m2));
elseif m1~=0
m=m1;
I=I+1/2*(+cos(m*B).*sin(m*B)+...
m*(B-A)-cos(m*A).*sin(m*A))/m;
else
I=I+(B-A);
end
end
elseif strcmp(th,'demo1')
% Center of the circle, latitude, in degrees
defval('cent',[10+rand*340 90-(rand*180)]);
% Colatitude/Longitude of center, in radians
th0=pi/2-cent(2)/180*pi;
ph0=cent(1)/180*pi;
% Radius of the circle, in degrees
defval('rad',rand*min([90-cent(2) cent(1) abs(360-cent(1))]));
% Radius of the circle, in radians
thR=rad/180*pi;
% Longitude/latitude of the circle, in degrees
[lon,lat]=caploc(cent,rad,101,1);
% North and South latitude, in degrees
lan=cent(2)+rad;
las=cent(2)-rad;
% North and South colatitude, in radians
thN=max(th0-thR,0);
thS=min(th0+thR,pi);
% Some generic sampling
nth=abs(150/pi/2*(thS-thN));
% A set of latitudes from South to North
la=linspace(las,lan,nth)';
% A set of colatitudes from North to South
th=linspace(thN,thS,nth)';
% The corresponding longitudes; all others are wrong
[Id,phintd]=coscos(th,1,1,'patch',[th0 ph0 thR]);
% Make plot
plot(lon,90-lat,'k','LineW',2); hold on
plot(cent(1),90-cent(2),'o','MarkerE','k','MarkerF','g')
a=phintd(:,1)*180/pi;
b=phintd(:,2)*180/pi;
% Don't try to fix the longitude mess-up, it will show up as an exterior
% domain
plot([a b]',[th th]'*180/pi,'k');
hold off
axis ij image
ylim([0 180])
xlim([min(0,min(a)) max(max(b),360)])
grid on
xlabel('Longitude \phi');
ylabel('Colatitude \theta');
longticks(gca)
title(sprintf('(%s,%s,%s) = (%i,%i,%i)',...
'\theta_0','\phi_0','\theta_R',...
round(90-cent(2)),round(cent(1)),round(rad)))
figdisp
elseif strcmp(th,'demo2')
if m1==0; yesplot=0; else yesplot=1; end
defval('cent',[rand*360 90-(rand*180)])
th0=pi/2-cent(2)/180*pi;
ph0=cent(1)/180*pi;
defval('rad',rand*90);
[lon,lat]=caploc(cent,rad,101,1);
thR=rad/180*pi;
mmax=60;
m1=round(rand*mmax-mmax/2);
m2=round(rand*mmax-mmax/2);
thN=max(th0-thR,0);
thS=min(th0+thR,pi);
thd=thN+rand*[thS-thN];
[Id(1),phintd]=coscos(thd,m1,m2,'patch',[th0 ph0 thR]);
Id(2)=sinsin(thd,m1,m2,'patch',[th0 ph0 thR]);
Id(3)=sincos(thd,m1,m2,'patch',[th0 ph0 thR]);
% Remember the dot product!
cc=inline(sprintf('cos(%i*x).*cos(%i*x)',m1,m2));
ss=inline(sprintf('sin(%i*x).*sin(%i*x)',m1,m2));
sc=inline(sprintf('sin(%i*x).*cos(%i*x)',m1,m2));
% This is quite plainly not enough
GL1=abs(m1)+abs(m2);
% This needs to be really, really high to be correct
GL=200;
% It's a good thing we are not doing this
J(1)=gausslegendre(phintd,cc,GL);
J(2)=gausslegendre(phintd,ss,GL);
J(3)=gausslegendre(phintd,sc,GL);
phin=linspace(phintd(1),phintd(2),10000);
K(1)=trapeze(phin,feval(cc,phin));
K(2)=trapeze(phin,feval(ss,phin));
K(3)=trapeze(phin,feval(sc,phin));
if yesplot==1
ph=linspace(min(0,phintd(1)),max(2*pi,phintd(2)),10000);
clf
ah=krijetem(subnum(3,1));
axes(ah(1))
p(1)=plot(ph,feval(cc,ph)); grid on; hold on
yl=ylim; f(1)=fillbox([phintd yl(2) yl(1)]);
yb(1)=ylabel(sprintf('cos(m_1%s)cos(m_2%s)','\phi','\phi'));
axes(ah(2))
p(2)=plot(ph,feval(ss,ph)); grid on; hold on
yl=ylim; f(2)=fillbox([phintd yl(2) yl(1)]);
yb(2)=ylabel(sprintf('sin(m_1%s)sin(m_2%s)','\phi','\phi'));
axes(ah(3))
p(3)=plot(ph,feval(sc,ph)); grid on; hold on
yl=ylim; f(3)=fillbox([phintd yl(2) yl(1)]);
yb(3)=ylabel(sprintf('sin(m_1%s)cos(m_2%s)','\phi','\phi'));
set(p,'LineW',2); longticks(ah,2)
for ind=1:3
axes(ah(ind))
set(ah(ind),'Children',[p(ind) ; f(ind)]);
t(ind)=title(sprintf(...
'I_%i= %8.3f ; J_%i= %8.3f ; K_%i= %8.3f ; |%s_{JI}|= %8.3e',...
ind,Id(ind),ind,J(ind),ind,K(ind),'\Delta',abs(J(ind)-Id(ind))));
end
set(ah,'xlim',[min(0,phintd(1)) max(2*pi,phintd(1))])
nolabels(ah(1:2),1)
axes(ah(3))
xlabel(sprintf('m_1= %i ; m_2= %i ; GL= %i',m1,m2,GL))
end
if any(abs(J(:)-Id(:))>1e-12); error('Something wrong') ; end
% Make decent output
if nargout==1
varargout{1}=abs(J(:)-Id(:));
varargout{2}=GL1;
end
end