How can we define piecewise sinusoidal or triangular functions along a path on a hemisphere?

Here are paths on the cylinder and hemisphere structure.

if true

    clear all
N1=6; %determines number of segments on top 
N2=10; %determines length as L=delta*N2 
a=1; % diameter of top 
aa=linspace(0, pi, N1);
z=a*sin(aa);
y=a*cos(aa);
delta=sqrt((y(2)-y(1))^2+(z(2)-z(1))^2); %Width of each segment
L=delta*N2;   %height of cylinder
z=z+L;
z1=0:delta:L-delta;
z2=fliplr(z1);
z=[z1 z z2];
y1=ones(1,length(z1)).*y(1);
y2=ones(1,length(z2)).*y(end);
y=[y1 y y2];
x=zeros(1,length(z));
rho=sqrt(x.^2+y.^2);
L=delta*90; %whole length 
N=length(y)
figure(1)
plot3(x,y,z,'*-r'); 
axis equal;
grid on
xlabel('x');ylabel('y');zlabel('z')
%Rotate 
hold on
a=36
xRot     = x.*cosd(a) - y.*sind(a);
yRot     = x.*sind(a) + y.*cosd(a);
zRot=z;
plot3(xRot,yRot,zRot,'-k*');
  end

Defining piecewise functions on the cylinder path is easy because its the function of z only. For example, on the red path, we can take f(z')=(sin((z(8)-z'))+sin((z'-z(7))))./sin(delta) to get a sine piece between 7th and 8th element and integrate it from 7th to 8th element of z. How can we define it on the hemisphere part of the path?

Thanks,

Dave