Function Repository Resource:

AsymptoticCurves

Compute the asymptotic curves of a parameterized surface

Contributed by: Wolfram Staff (original content by Alfred Gray)

ResourceFunction["AsymptoticCurves"][s,{u,v}]

compute the equations for asymptotic curves of a surface s parameterized by variables u and v.

Details and Options

For a regular surface s, an asymptotic curve is a curve c(t) for which the normal curvature vanishes in the direction c'(t).

The differential equation for the asymptotic curves on a surface is e (u')² + 2f u'v' +g (v')²=0, where e, f and g are coefficients of the second fundamental form.

Examples

Basic Examples (4)

Define a hyperbolic paraboloid:

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$hyperbolicparaboloid[u_, v_] := {u, v, u v}$

The asymptotic curves are just straight lines:

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The mesh itself represents the asymptotic curves:

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Define a funnel:

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$funnel[a_, b_, c_][u_, v_] := {a v Cos[u], b v Sin[u], c Log[v]}$

Plot it:

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$ParametricPlot3D[funnel[1, 1, 1][u, v], {u, 0, 3 \[Pi]/2}, {v, .1, 1}]$

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Get the equations for asymptotic curves:

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Solve the equations:

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$Equal @@@ Flatten[{DSolve[eqs[[1]], u[v], v] /. C[1] -> p, DSolve[eqs[[2]], u[v], v] /. C[1] -> q} /. u[v] -> \[Theta]] // PowerExpand$

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Solve for variables:

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$Solve[%, {\[Theta], v}, Reals] // PowerExpand$

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Redefine the funnel parameterization in terms of r=e^p-q and θ=p+q:

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$funnelasym[c_][p_, q_] := {c Exp[p - q] Cos[p + q], -c Exp[p - q] Sin[p + q], c (p - q)}$

Plot the asymptotic curves computed in terms of the new coordinates:

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$ParametricPlot3D[funnelasym[1][u, v], {u, 0, \[Pi]}, {v, 0, \[Pi]}, PlotPoints -> 50, MaxRecursion -> 3, Mesh -> {30, 30}, MeshStyle -> Red, PlotRange -> {{-4, 2}, {-4, 4}, {-3, 1.3}}]$

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Define the exponentially twisted helicoid:

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$exptwist[a_, c_][u_, v_] := {a v Cos[u], a v Sin[u], a Exp[c u]}$

Plot the surface:

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$ParametricPlot3D[ exptwist[1, .3][u, v], {u, -\[Pi], 2 \[Pi]}, {v, -1, 1}, PlotPoints -> {24, 12}, Mesh -> {40, 0}, MeshStyle -> Red]$

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The asymptotic curve equations:

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The asymptotic curve solutions:

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Solve for parameters:

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$Solve[{u[v] == p, u[v] == q + (2 Log[v])/c} /. u[v] -> \[Theta], {\[Theta], v}, Reals]$

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Redefine the parameterization in terms of r =e^c(p-q) and θ=p:

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$exptwistasym[a_, c_][p_, q_] := a {Exp[1/2 c (p - q)] Cos[p], Exp[1/2 c (p - q)] Sin[p], Exp[c p]}$

Plot the asymptotic curves on the exponentially twisted helicoid:

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$ParametricPlot3D[ exptwistasym[1, .3][p, q], {p, -\[Pi], 2 \[Pi]}, {q, -\[Pi], 2 \[Pi]}, PlotPoints -> {36, 12}, PlotRange -> All, Mesh -> {40, 0}, MeshStyle -> Red]$

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Superimpose both the surface and its asymptotic curves:

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$ParametricPlot3D[ Evaluate[{exptwist[1, .3][u, v], exptwistasym[1, .3][u, v]}], {u, -\[Pi], 2 \[Pi]}, {v, -1, 1}, PlotPoints -> {24, 12}, Mesh -> {40, 0}, MeshStyle -> Red]$

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Define a wrinkled surface:

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$wrinkles[a_][u_, v_] := {u, v, a (u/v + v/u)}$

Plot the surface:

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Get the equations for asymptotic curves:

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Solve the equations:

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$eqs = Equal @@@ Flatten[{DSolve[Derivative[1][u][v] == u[v]^3/v^3, u[v], v] /. C[1] -> q, DSolve[Derivative[1][u][v] == u[v]/v, u[v], v] /. C[1] -> p} /. u[v] -> u] // PowerExpand$