Witch of Agnesi

Witch of Agnesi is a surface of revolution.

γ = {2 a Tan[x_1], 2 a Cos[x_1]^2} ;

ParametricPlot[Evaluate[γ/.a→1], {x_1, -1.2, 1.2}] ;

Clear :: ssym : x  is not a symbol or a string. More…                  1

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f = [γ[[1]], γ[[2]] {Cos[x_2], Sin[x_2]}]

p1 = ParametricPlot3D[Evaluate[.9 f/.a→1], {x_1, -1.2, 1.2}, {x_2, -3.5, 2}] ;

{2 a Tan[x_1], 2 a Cos[x_1]^2 Cos[x_2], 2 a Cos[x_1]^2 Sin[x_2]}

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The Euclidean space ^3 induces the following metric

df = Μd[f, 1, 2] ;

MF[ℊ = [df] . df//]

( {{-2 a^2 (-1 + Cos[4 x_1] - 2 Sec[x_1]^4), 0}, {0, 4 a^2 Cos[x_1]^4}} )

The coefficient a, does not appear in the Riemann (1,3)-tensor field R.

Μℛ[ℊ]

Geodesics on the witch of agnesi have the following intriguing property...

sol = Χ★[x, 2]/.NDSolve[[ΧGeodesic[ℊ], {x_1[0] - .1, x_2[0] - .5}, {x_1^′[0] + 3., x_2^′[0] - .5}], Χ★[x, 2], {t, 0, T = 10}][[1]] ;

Show[p1, PlotLine3D[f◦sol/.a→1, {t, 0, T, .01}, {.7, .7, 0}, .005], ImageSize→400] ;

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sol = Χ★[x, 2]/.NDSolve[[ΧGeodesic[ℊ], {x_1[0] - .1, x_2[0] - .5}, {x_1^′[0] + 3., x_2^′[0] - 2.5}], Χ★[x, 2], {t, 0, T = 10}][[1]] ;

Show[p1, PlotLine3D[f◦sol/.a→1, {t, 0, T, .01}, {.7, .7, 0}, .005], ImageSize→400] ;

[Graphics:../HTMLFiles/index_310.gif]


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