By pinching a flexible cylinder at intervals its cross section gradually transitions from a circle to a straight line resulting in a shape like:
L = 8; zm = 0.6 L; b[t_] = Sqrt[(L/(2 Pi))^2 - t^2];
ParametricPlot3D[{t Cos[u], b[t] Sin[u], zm Sqrt[1 - (b[t]/t)^2]}, {u,
0, 2 Pi}, {t, 0, 3}, PlotLabel -> " ToothPasteSurface",
Axes -> True, Boxed -> False]
but this shape has only its perimeter held approximately constant. Given here to show its probable form.
To find it numerically a ( Monge-Ampère equation ?) pde is to be solved to obtain in Monge form $z=f(x,y)$ when its $K= \dfrac{rt-s^2}{(1+p^2+q^2)^2}=0 $ is retained zero due to isometric mapping invariance.
{Derivative[2, 0][z[x, y]] Derivative[0, 2][z[x, y]] - Derivative[1, 1][z[x, y]]^2 == 0}
BoundaryConditions {(z = 0, x^2 + y^2 = 1), (z = 5, y = 0)}
Please help to form pde to solve and plot 3D shape.
