Unwrapping the mesh on 2D plane

How to create a flat pattern of this kind of spherical segment of the mesh shown below?

In CAD there is a function called sheet metal to create a flat pattern of 3D models, but how can I devide a sphere like that in Matlab?

I used

r_x = R*cos(0:pi/100:pi);
r_y = R*sin(0:pi/100:pi);

th = 0:2*pi/NumberOfSegments:2*pi;
[th, r_y] = meshgrid(th, r_y);

[X, Y] = pol2cart(th, r_y);

Z = fliplr(r_x)';
Z = repmat(Z,1,NumberOfSegments + 1);

surf(X,Y,Z)  

to plot that sphere.

Here is an example - image showing a set of 12 gores to make a globe (I want just one segment).

Answer 1

[Meg Noah]

Here's one way - it's actually a bit overcomplicated than it needs to be in the code below to make it more general:

R = 1;
NumberOfSegments = 12;
r_x = R*cos(0:pi/100:pi);
r_y = R*sin(0:pi/100:pi);
th = 0:2*pi/NumberOfSegments:2*pi;
[th, r_y] = meshgrid(th, r_y);
[X, Y] = pol2cart(th, r_y);
Z = fliplr(r_x)';
Z = repmat(Z,1,NumberOfSegments + 1);
surf(X,Y,Z)  

[AZ,EL,radius] = cart2sph(X,Y,Z);
LON_degE = rad2deg(AZ);
LAT_degN = rad2deg(EL);

% last column is wrapping
LON_degE = LON_degE(:,1:NumberOfSegments);
LAT_degN = LAT_degN(:,1:NumberOfSegments);

LON_unique = unique(LON_degE(:));
deltaLon = 360.0/(NumberOfSegments);
% choose a segment
iSegment = 1;
lonC = LON_unique(iSegment);
lon0 = wrapTo180(lonC - deltaLon);
if (lon0 <= -180) 
    lon0 = 180;
end
lon1 = wrapTo180(lonC + deltaLon);
if (lon1 <= -180) 
    lon1 = 180;
end

% get the lat/lon points at the great circle at center longitude
idx = find(abs(LON_degE - lonC) < deltaLon/100);
LATC_degN = LAT_degN(idx);
LONC_degE = LON_degE(idx);

% get the lat/lon points at the great circle at previous longitude
idx = find(abs(LON_degE - lon0) < deltaLon/100);
LAT0_degN = LAT_degN(idx);
LON0_degE = LON_degE(idx);

% get the lat/lon points at the great circle at next longitude
idx = find(abs(LON_degE - lon1) < deltaLon/100);
LAT1_degN = LAT_degN(idx);
LON1_degE = LON_degE(idx);

% find the arclength distances between consequetive points 
% along the center line use the haversine formula for great
% circles on a sphere.
% find the arclength distances from the center to the previous longitude
goreY = R*deg2rad(distance(LATC_degN(1),LONC_degE(1),LATC_degN,LONC_degE));
% arclength distances to the previous longitude
goreX0 = -R*deg2rad(distance(LAT0_degN,LON0_degE,LATC_degN,LONC_degE));
% arclength distances to the next longitude
goreX1 = R*deg2rad(distance(LATC_degN,LONC_degE,LAT1_degN,LON1_degE));
figure()
plot(goreX0,goreY,'--.r');
hold on;
plot(goreX1,goreY,'--.b');
axis equal

[Meg Noah]

OK, here's another attempt:

R = 1;
NumberOfSegments = 24;
NumberOfLatitudes = 100;
r_x = R*cos(0:pi/NumberOfLatitudes:pi);
r_y = R*sin(0:pi/NumberOfLatitudes:pi);
th = 0:2*pi/NumberOfSegments:2*pi;
LON_unique = rad2deg(th);
deltaLon = 360.0/(NumberOfSegments);

[th, r_y] = meshgrid(th, r_y);
[X, Y] = pol2cart(th, r_y);
Z = fliplr(r_x)';
Z = repmat(Z,1,NumberOfSegments + 1);
% surf(X,Y,Z)  

[AZ,EL,radius] = cart2sph(X,Y,Z);
LON_degE = rad2deg(AZ);
LAT_degN = rad2deg(EL);

% last column is wrapping
LON_degE = LON_degE(2:NumberOfLatitudes+1,1:NumberOfSegments);
LAT_degN = LAT_degN(2:NumberOfLatitudes+1,1:NumberOfSegments);

% choose a segment
figure()
clrs = colormap(jet);
deltaClr = floor(size(clrs,1)/NumberOfSegments);

for iSegment = 1:2:NumberOfSegments

lonC = wrapTo180(LON_unique(iSegment));
if (lonC <= -180) 
    lonC = 180;
end
lon0 = wrapTo180(lonC - deltaLon);
if (lon0 <= -180) 
    lon0 = 180;
end
lon1 = wrapTo180(lonC + deltaLon);
if (lon1 <= -180) 
    lon1 = 180;
end

% get the lat/lon points at the great circle at center longitude
idx = find(abs(LON_degE - lonC) < 1e-4);
LATC_degN = LAT_degN(idx);
LONC_degE = LON_degE(idx);

% get the lat/lon points at the great circle at previous longitude
idx = find(abs(LON_degE - lon0) < deltaLon/100);
LAT0_degN = LAT_degN(idx);
LON0_degE = LON_degE(idx);

% get the lat/lon points at the great circle at next longitude
idx = find(abs(LON_degE - lon1) < deltaLon/100);
LAT1_degN = LAT_degN(idx);
LON1_degE = LON_degE(idx);

% find the arclength distances between consequetive points 
% along the center line use the haversine formula for great
% circles on a sphere.
% find the arclength distances from the center to the previous longitude
idxEquator = find(abs(LATC_degN) < 1e-4);
goreY = R*deg2rad(distance(LATC_degN(idxEquator),LONC_degE(idxEquator),LATC_degN,LONC_degE));
goreY(LATC_degN < 0) = -goreY(LATC_degN < 0);
% find the center of the gore in the Longitudes from the first one
goreXC = R*deg2rad(distance(0,lonC,0,0));
if (lonC < 0)
    goreXC = -goreXC;
end
% arclength distances to the previous longitude
goreX0 = goreXC-R*deg2rad(distance(LAT0_degN,LON0_degE,LATC_degN,LONC_degE));
% arclength distances to the next longitude
goreX1 = goreXC+R*deg2rad(distance(LATC_degN,LONC_degE,LAT1_degN,LON1_degE));

iclr = deltaClr*(iSegment-1)+1;
if iclr > size(clrs,1)
    iclr = 1;
end

plot(goreXC*ones(size(goreY)),goreY,':','color',clrs(iclr,:));
hold on;
plot(goreX0,goreY,'.','color',clrs(iclr,:));
plot(goreX1,goreY,'.','color',clrs(iclr,:));
xticks([-pi,-3*pi/4,-pi/2,-pi/4,0,pi/4,pi/2,3*pi/4,pi]);
xticklabels({'-\pi','-3\pi/4','-\pi/2','-\pi/4','0','\pi/4','\pi/2','3\pi/4','\pi'});
xlim([-5*pi/4 5*pi/4]);
yticks([-pi/2,-pi/4,0,pi/4,pi/2]);
yticklabels({'-\pi/2','-\pi/4','0','\pi/4','\pi/2'});
ylim([-3*pi/4,3*pi/4]);
axis equal

end