How do you find the inscribed rectangles in an arbitrary Jordan curve? In this video, a proof is presented that there always exists at least one inscribed rectangle in any enclosed curve. Is there an algorithmic method for finding the coordinates of the inscribed rectangles for an arbitrary curve?
There are many articles about that prove such a rectangle must exist, but I have found no methods for actually finding those rectangles.
Who cares about topology? (Inscribed rectangle problem)
 A: Suppose the curve is given implicitly by $F(x) = 0$.  The vertices $v_1, v_2, v_3, v_4$ of your rectangle are then solutions of the system
$$ \eqalign{F(v_1) &= 0 \cr
            F(v_2) &= 0 \cr
            F(v_3) &= 0 \cr
            F(v_4) &= 0 \cr
            (v_1 - v_2) \cdot (v_2 - v_3) &= 0\cr
            (v_2 - v_3) \cdot (v_3 - v_4) &= 0\cr
            (v_3 - v_4) \cdot (v_4 - v_1) &= 0\cr
            (v_4 - v_1) \cdot (v_1 - v_2) &= 0\cr} $$
We can try solving this system, numerically or symbolically (maybe using Groebner bases if $F$ is a polynomial), avoiding the trivial solutions  $v_1 = v_2 = v_3 = v_4$.  Hopefully there will be only finitely many such solutions, and some of them give your inscribed rectangles.
EDIT: For an example such as yours (produced using splines), an implicit equation isn't going to be easy to find.  However, a parametric equation can be obtained.  If the curve is given parametrically as $x = X(t)$, $0 \le t \le L$, and the vertices are $X(t_1)$, $X(t_2)$, $X(t_3)$, $X(t_4)$, then the equations are
$$ \eqalign{(X(t_1) - X(t_2)) \cdot (X(t_2) - X(t_3)) &= 0\cr
 (X(t_2) - X(t_3)) \cdot (X(t_3) - X(t_4)) &= 0\cr
 (X(t_3) - X(t_4)) \cdot (X(t_4) - X(t_1)) &= 0\cr
 (X(t_4) - X(t_1)) \cdot (X(t_1) - X(t_2)) &= 0\cr}$$
