# Interpolation of points in 3D with smooth space curve

In Mathematica we have ListSurfacePlot3D for reconstructing surface from list of points in 3D. But I can not find something similar if I want to reconstruct a curve instead of a surface. For example if I have this list of points:

{{0.,0.,-1.},{0.06120871905481365,0.2397127693021015,-0.9689124217106447},{0.22984884706593015,0.42073549240394825,-0.8775825618903728},{0.4646313991661485,0.49874749330202717,-0.7316888688738209},{0.7080734182735712,0.4546487134128408,-0.5403023058681397},{0.9005718077734668,0.2992360720519783,-0.3153223623952687},{0.9949962483002227,0.0705600040299336,-0.0707372016677029},{0.9682283436453982,-0.17539161384480992,0.17824605564949209},{0.826821810431806,-0.37840124765396416,0.4161468365471424},{0.6053978997153898,-0.48876505883254856,0.6281736227227391},{0.35816890726838696,-0.4794621373315693,0.8011436155469337},{0.14566511285437003,-0.35277016278519596,0.9243023786324636},{0.01991485667481699,-0.13970774909946293,0.9899924966004454},{0.011706187135988248,0.10755999404390776,0.9941296760805463},{0.12304887282834767,0.32849329935939453,0.9364566872907963},{0.32668234108248706,0.4689999883873694,0.8205593573395608},{0.5727500169043067,0.4946791233116909,0.6536436208636119},{0.8010059513424118,0.39924355631174513,0.4460874899137928},{0.9555651309423384,0.20605924262087827,0.2107957994307797},{0.9985860780981893,-0.03757556023090465,-0.03760215288797655},{0.9195357645382262,-0.2720105554446849,-0.28366218546322625},{0.7377684639979962,-0.4398478799858351,-0.5120854772418407},{0.49778715100597465,-0.49999510327535174,-0.70866977429126},{0.258347620623497,-0.4377260873442142,-0.8611924171615208},{0.07807302063375395,-0.26828645900021747,-0.960170286650366},{0.,0.,-1.}}


Notice that the curve should be a closed loop as the first and last points are the same. I want to find a smooth curve that passes exactly through these points. Something like this:

Any suggestion how to do it?

• Interpolate with option PeriodicInterpolation->True. – Henrik Schumacher Jul 12 '20 at 12:54

 ff = Interpolation[#, PeriodicInterpolation -> True] & /@ Transpose[dat];
{ListPointPlot3D[dat, PlotStyle -> Orange],
ParametricPlot3D[#[t] & /@ ff//Evaluate, {t, 1, Length[dat]}]}//
Show[#,BoxRatios -> {1, 1, 1}, Axes -> None] &


Note that following the OP's request, you can encapsulate this into a function:

ListCurvePlot3D[dat_] :=
Module[{t, ff =
Interpolation[#, PeriodicInterpolation -> True] & /@
Transpose[dat]},
{ListPointPlot3D[dat, PlotStyle -> Orange],
ParametricPlot3D[#[t] & /@ ff // Evaluate, {t, 1, Length[dat]}]} //
Show]

• OP mentions I want to find a smooth curve, but this has corners. Perhaps a BSplineFunction would be more appropriate? – flinty Jul 12 '20 at 12:56
• There is no interpolation. The curve is not smooth. – azerbajdzan Jul 12 '20 at 12:57
• What is the purpose of PeriodicInterpolation -> True when it works exactly same without it? – azerbajdzan Jul 12 '20 at 13:15
• @azerbajdzan i/ I tend to listen to what Henrik says because he is very smart. ii/ I guess you requested the path to be periodic: the documentation states that if this option is used you can evaluate t at values higher than Length[dat] and remain on the curve. I assume that it means internally it is using periodic functions to do the interpolation. For instance {t, -5, Length[dat] + 5} – chris Jul 12 '20 at 13:21
• I see... Now I wonder why they did not make one command like they did for surface. If we have ListSurfacePlot3D we could also have ListCurvePlot3D. Thanks. – azerbajdzan Jul 12 '20 at 13:28