how to find minimum of discontinuous symbolic function

My function looks something like this

%parameters for a capped cylinder centered on z axis
R = 0.5; %radius of sphere
r = 0.5; %radius of cylinder
L = 3; %length of cylinder 
h = sqrt(R^2-r^2);
shift = (L/2)-h;

A = [-0.4322    0.2514    0.7427] %a point on cylinder surface
B = [-0.0987   -0.2834    1.8999] %a point on sphere surface

a1 = A(1); a2 = A(2); a3 = A(3);
b1 = B(1); b2 = B(2); b3 = B(3);

syms phi
D = @(phi) sqrt((r.*(atan2(a2,a1)-phi)).^2+(a3-L./2).^2) ... %distance between the two points as a function of phi
    + R*acos(((b1.*cos(phi)+b2.*sin(phi)+(b3-shift)*(L./2-shift)))./R.^2); 

And this is its graph over 0,2pi

as shown above, the minimum occurs at one of the asymptote. My question is how can I get the x coordinate of the asymptote?

Thanks for your help!!

[Walter Roberson]

Could we have the other variable values to test with?

[Maggie liu]

@Walter Roberson Sorry for not showing them when I asked the question. I have updated the description with the sample variables. Thanks!

Answer 1

[Walter Roberson]

%parameters for a capped cylinder centered on z axis
R = 0.5; %radius of sphere
r = 0.5; %radius of cylinder
L = 3; %length of cylinder 
h = sqrt(R^2-r^2);
shift = (L/2)-h;

A = [-0.4322    0.2514    0.7427] %a point on cylinder surface

A = 1×3

-0.4322 0.2514 0.7427

B = [-0.0987   -0.2834    1.8999] %a point on sphere surface

B = 1×3

-0.0987 -0.2834 1.8999

a1 = A(1); a2 = A(2); a3 = A(3);
b1 = B(1); b2 = B(2); b3 = B(3);

syms phi real
D = @(phi) sqrt((r.*(atan2(a2,a1)-phi)).^2+(a3-L./2).^2) ... %distance between the two points as a function of phi
    + R*acos(((b1.*cos(phi)+b2.*sin(phi)+(b3-shift)*(L./2-shift)))./R.^2); 

DD = D(phi)

DD = 

DD1c = simplify(cos(2*children(DD,1)))

DD1c = 

limpoint = vpasolve(DD1c == 1, pi)

limpoint = 

3.7910906782081747162135888997947

D(limpoint)

ans = 

0.9588763931357714838022843054125