I want to find the shortest function between the point $(a_1, a_2)$ and the point $(c_1,c_2)$ that passes through the point $(b_1,b_2)$. This is what I have done:
First, I parametrize the coordinates as: $(a_1, u(a_1)), (b_1, u(b_1)), (c_1, u(c_1))$
My variational problem is:
$\min_{u()} \int_{a_1}^{c_1} \sqrt{1+(u'(x))^2}dx$
With boundary conditions $u(a_1)=a_2, u(c_1)=c_2$, however I still have the constraint in only one point $u(b_1)=b_2$ that I am not sure how to include. The solution of the second order differential equation $u()$ will have two constants that I can find using the boundary conditions, but I do not know how to include the constraint in the picture.
Thank you