Drawing geodesic lines on the membrane (Plot3D)

Question

The most common method of creating patterns on a membrane involves drawing geodesic lines on the membrane. Justification of the layout of the geodesic lines is beyond the scope of this text .

Although the geodesic lines can be drawn freely to form patterns of any shape and position, it is common that the route of the geodesic lines follow some models determined:

how to draw these roughly parallel geodesic lines is something that every user usually does in a particular way depending on different objectives I need repeat the same exercise with a geodesic line located on the Plot3D of f, like generatrix line.

For example, let the function

Plot3D[x + y - x y, {x, -5, 5}, {y, -5, 5}, PlotTheme -> "Scientific"]

Can I draw their lines of geodesic as shown below

A method of multi-geodesic uses end points of the new geodesic lines found at the edge of the membrane. If we are with a membrane formed by different parts, we can not perform the function of multi-geodesic directly.

What we must do is break down the membrane in two simple membranes and find the pattern in each. I would like to create images based on above method as shown below: Is this possible in Mathematica? I do not know how this can be implemented. Please Help.

Answer 1

Plot the surface.

surface = Plot3D[x + y - x y, {x, -5, 5}, {y, -5, 5}, AxesLabel -> {"x", "y", "z"}]

The aim is to draw geodesics - i.e. minimum distance paths - on this surface.

Define the geodesic as a parametric curve.

geo[t_] = {x[t], y[t], z[t]};

Define z[t] in terms of x[t] and y[t].

z[t_] = x[t] + y[t] - x[t] y[t];

A minimum distance path can be computed as the solution of the variational problem of finding a path on the surface whose length is stationary with respect to perturbations of the path.

Get the variational methods package. At this point it is useful to review the Variational Methods documentation.

<< VariationalMethods`

Construct the function that computes the rate of movement along the geodesic.

f[t] = Sqrt[#.#] &[D[{x[t], y[t], z[t]}, t]]

Construct the Euler equations which the geodesic solution must satisfy.

diffeqns = EulerEquations[f[t], {x[t], y[t]}, t]

<omitted messy Euler equations>

Define some initial conditions at t=0 to fix the position and direction at the start of the geodesic.

initconds = {x[0] == -5, y[0] == 5, x'[0] == 1, y'[0] == -1};

Define the maximum value of the geodesic parameter to be used. This is used to determine how far to compute the numerical solution of the Euler equations.

tmax = 9;

Numerically solve the Euler equations subject to the initial conditions.

This throws the warning "NDSolve::ntdvdae: Cannot solve to find an explicit formula for the derivatives. NDSolve will try solving the system as differential-algebraic equations.", but this appears to be a benign problem.

EDIT: I wrapped diffeqns in FullSimplify to fix an error that occurred in version 12.0.

solution = NDSolve[Join[FullSimplify[diffeqns], initconds], {x, y}, {t, 0, tmax}];

Plot the geodesic, and superimpose it on the plot of the surface.

geodesic = ParametricPlot3D[
  {x[t], y[t], z[t]} /. solution[[1]] // Evaluate, {t, 0, tmax}];
Show[surface, geodesic]

This geodesic is parameterised by the initial conditions and tmax, and these parameters can be varied to generate other geodesic solutions.

WARNING: I found that the numerical solution of the Euler equations is not in general as easy to obtain as in the specific example above, which is therefore merely a guide to how variational methods can be used to compute geodesics subject to initial conditions. More generally, you will have to tweak the NDSolve options, and so on.