DEnepperSurfaceGeodesicDistance::usage="Interesting surface that intersects itself, it is also a minimal surface, that is, its average curvature is zero. We will draw the geodesic lines of this surface from the origin. Geodesic lines are the shortest and most direct paths between two points on a curved surface. In a plane the geodesic lines would be straight. Also calculate the geodesic distance between the points {0, 0, 0} and -
14
3
​,
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​, -3 and compare it with the Euclidean distance. References: Public domain; Personal."
Animation:
In[]:=
Manipulateσ[u_,v_]:=u-
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u
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+u
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v
,v-
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v
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+v
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,
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;geo=ResourceFunction["Geodesic"][σ[u,v],{u,v},t,{-1,2},θ];​​solg=Flatten(NDSolve[#1,{u,v},{t,-5,5}]&)/@Tablegeo,θ,
π
3
,2π,0.1,1;​​ShowParametricPlot3D[σ[u,v],{u,-5,t1},{v,-5,t1},PlotStyleDirective[Orange,Opacity[1],Specularity[White,30]],MeshFalse,MaxRecursion->3],​​ParametricPlot3DEvaluateu[t]-
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u[t]
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+u[t]
2
v[t]
,v[t]-
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v[t]
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+v[t]
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u[t]
,
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u[t]
-
2
v[t]
/.solg,{t,-5,t1},Graphics3D[{Black,Ball[{0,0,0},0.15]}],Graphics3DBlack,Ball-
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,
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,-3,0.2​​,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange5,ViewPoint{-20,-10,-4.},ImageSize900,Style["D'Enneper Surface and Geodesic Lines:",Bold,Large],{{t1,-5+0.00002,"Value (t1)"},-5+0.00001,5,0.00001},ControlPlacementTop
Out[]=
​
D'Enneper Surface and Geodesic Lines:
Value (t1)
The points are in black. The point from which the geodesic lines come out is the point {u = -1, v = 2} or in Cartesian coordinates {-
14
3
,
4
3
,-3
. The other point is the origin. They are on a red geodesic line.
​
​
​
​Geodesic distance:
In[]:=
σ[u_,v_]:=u-
3
u
3
+u
2
v
,v-
3
v
3
+v
2
u
,
2
u
-
2
v
;​​g=ParametricRegion[σ[u,v],{u,v}];​​ShortestCurveDistanceg,{0,0,0},-
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3
,
4
3
,-3
Out[]=
5.96188
Euclidean distance:
In[]:=
NNorm{0,0,0}--
14
3
,
4
3
,-3
Out[]=
5.70575