Multivariate parameters appear to present a jagged appearance of integrands (using default Runge-Kutta ODE integration intervals?) in ParametricPlot3D plotting on a single argument.
Higher Mesh.. ing to 200 improves sectors' jagging (large step secants appearing instead of tangent) somewhat, but still color from PlotStyle for lines does not come through, e.g., like in case of the straight computed helical lines shown. How to get PlotStyle line colors with smoother lines?
thmax=6Pi ;
EQU={SI'[th]==Sin[PH[th]],SI[0]==.123,
PH'[th]==Cos[th],PH[0]==-1.234,
Z'[th]==2 Cos[PH[th]]Cos[SI[th]],
R'[th]==3 Sin[PH[th]] Cos[SI[th]],
R[0]==1.321,Z[0]==0.};
NDSolve[EQU,{SI,PH,R,Z},
{th,0,thmax}];
{ph[u_],si[u_],
r[u_],z[u_]}={PH[u],
SI[u],R[u],Z[u]}/.First[%];
ParametricPlot[{z[th],r[th]},{th,0,thmax},
AspectRatio->Automatic,PlotLabel->MERIDIAN_,PlotStyle->{Thick,Magenta}]
ParametricPlot3D[{r[th] Cos[th+t],z[th],r[th] Sin[th+t]},{t,0,2 Pi},
{th,0,thmax},PlotLabel->SURFACE_]
SpC1=ParametricPlot3D[{r[th] Cos[th],z[th],r[th] Sin[th]},{t,0,2 Pi},
{th,0,thmax},PlotStyle->{Thick,Magenta},PlotLabel->SPACE_CURVE,Mesh->20];
SpC2=ParametricPlot3D[{2 Sin[th],th,1.8 Cos[th]},{th,0,thmax/3},
PlotLabel->HELIX_,PlotStyle->{Thick,Magenta}];
Show[SpC1,SpC2]
{}
) in the toolbar. Right now it is difficult to read. $\endgroup$