Q:

How to draw a curve line with points between it on the shaded region that situated on the surface of the hemisphere using Matlab?

A hemisphere with shaded region is already formed. Then, planning to form a path or curved line on the shaded region situated on the surface of the hemisphere with points on it. Can't even know what is their latitude or longitude to draw an arc, therefore, any idea on how to obtain the latitude and longitude for the starting and ending point or form a curved line as shown in the figure below?

[x,y,z] = sphere;      % Makes a 21-by-21 point sphere
x = x(11:end,:);       % Keep top 11 x points
y = y(11:end,:);       % Keep top 11 y points
z = z(11:end,:);       % Keep top 11 z points
radius = 0.13;
c = [0.36 0 0];
hs = surf(radius.*x + c(1),radius.*y,radius.*z,'FaceColor','yellow','FaceAlpha',.3);  
axis equal
xlabel('X');ylabel('Y');zlabel('Z');

% Putting the ranges for the elevation and azimuth
minAzimuth = 0;
maxAzimuth = 180;
minElevation = 0;
maxElevation = 80;

% Compute angles 
phi = atan2d(y,x);
theta = acosd (z);

% Highlighting logic (Shading the subset of the hemisphere)
ind = (phi >= minAzimuth & phi <= maxAzimuth) & (theta >= minElevation & theta <= maxElevation); % Find those indices
x2 = x; y2 = y; z2 = z;                                                                          % Make a copy of the hemisphere coordinates
x2(~ind) = NaN; y2(~ind) = NaN; z2(~ind)=NaN;                                                    % Set those out of boundary to NaN

hold on;
surf(radius.*x2+c(1),radius.*y2,radius.*z2,'FaceColor','red');     

A:

Use spherical coordinates:
As you did with your sphere segment, it is easier to define your line in spherical coordinates first, then convert to cartesian when it's time to display.
If you add this after your own code:
% set line parameters
np = 10 ;                % number of points
LineAziStart  =  17 ;    % Starting azimuth
LineAziStop   = 122 ;    % Ending azimuth
LineElevation =  49 ;    % Elevation

% generate spherical coordinates
azp = linspace(deg2rad(LineAziStart),deg2rad(LineAziStop),np).' ;
elp = zeros(np,1) + deg2rad(LineElevation) ;
rp  = zeros(np,1) + radius ;

% convert to cartesian
[xp,yp,zp]=sph2cart(azp,elp,rp) ;

% adjust coordinates for center of the sphere
xp = xp + c(1) ;
yp = yp + c(2) ;
zp = zp + c(3) ;

% display
hp = plot3(xp,yp,zp,'g','LineWidth',2,'Marker','x') ;

You'll obtain a line parallel to the latitude you set:

You can adjust where the line starts, stops, and it elevation by adjusting the first parameters on top of the code (make them match your own values).

Edit:
To explain the spherical coordinate generation:
To get a line in 3D you need a set of 3 coordinates, they can be x/y/z (cartesian referential) or radius/azimuth/elevation (spherical referential).
The constraints for your line are:

Azimuth: (called azp) Has to vary from one value to another. For that I used the statement azp = linspace(deg2rad(LineAziStart),deg2rad(LineAziStop),np).' ;. I recommend you to read the documentation for the linspace function. It will generate a set of np linearly spaced points between LineAziStart and LineAziStop. I also had to use deg2rad because for sperical coordinates you want your angles expressed in radian.
Radius: (called rp) This one is easy. Your line has to be on the surface of the sphere, so all the points of the line will have the same radius value (the radius of the original sphere. We just generate a vector of np points all equal to radius. This is done with rp = zeros(np,1) + radius;.
Elevation: (called elp) Your line has to be parallel to a latitude, so the elevation is also constant for all the points of the line. As with the radius, we generate the set of constant points the same way: elp = zeros(np,1) + deg2rad(LineElevation) ;.

By then you have a set of 3 vectors (rp/azp/elp), which all have the same number of values, and together define a set of point in 3D space, in spherical coordinates. Matlab plotting function requires cartesian coordinates so the last step is just to convert this set of coordinates. This is done with the function sph2cart, which convert rp/azp/elp into xp/yp/zp.
The spherical coordinates we generated were centered on the origin (point 0,0,0,), so the converted cartesian coordinates are also there. The last step is simply to translate these coordinates so they'll be centered on the same point/center than the sphere. After that, your coordinates are ready to be plotted.

