🟢 Parabola

In this tutorial, we show how to cover a parabola.

using CairoMakie, MultiParamContinuationconst MPC = MultiParamContinuationF(u,p) = [u[1]^12 + u[2]^12 - u[3]]prob = ManifoldProblem(F, [0,0,0.], nothing;    finalize_solution = (u,p) -> (u[3] < 2) * (u[1]>-0.1) * (u[2]>-0.1))

2-d Manifold Problem
    ├─ n = 3
    └─ m = 1

show(prob)

2-d Manifold Problem
    ├─ n = 3
    └─ m = 1

We now compute a covering of the manifold

S = MPC.continuation(prob,            Henderson(                      use_curvature = true,                      ),            CoveringPar(max_charts = 10000,                    max_steps = 2000,                    verbose = 0,                    newton_options = NonLinearSolveSpec(;maxiters = 5, abstol = 1e-12, reltol = 1e-10),                    Rmax = .2, # maximal radius of validity                    R0 = .05, # initial radius of validity                    ϵ = 0.005, # used to estimated current radius                    )            )show(S)

Surface 2-d
   ├─ # charts = 2001
   └─ problem =
          2-d Manifold Problem
              ├─ n = 3
              └─ m = 1

You plot the result as follows

f = MPC.plotd(S; draw_edges = true,)