ClearAll; vmin = 0; vmax = 2; al = Pi/4; xm = 3.6; loxoSph = ParametricPlot3D[{xm + Sech[v Cot[al]] Cos[v + t], Sech[v Cot[al]] Sin[v + t], Tanh[v Cot[al]]}, {t, 0, 2 Pi}, {v, vmin, vmax}, PlotRange -> All]; bt = 0.15; rm = 1.25; rmin = 2 rm; rmax = 3.6 rm; zmin = 0; zmax = 1; plane = ParametricPlot3D[{r, -r Tan[bt], z}, {r, 2 rm, rmax}, {z, zmin, zmax}]; Show[plane, loxoSph] 

One of four (t,v,r,z) is chosen as independent parameter with a uniform increment. Respective $(x,y,z)$ are equated but complicates due to simultaneity.

How is a Table solved for the remaining three coordinates on the line of intersection ? Thanks for help.

ClearAll; vmin = 0; vmax = 2; al = Pi/4; xm = 3.6; loxoSph = ParametricPlot3D[{xm + Sech[v Cot[al]] Cos[v + t], Sech[v Cot[al]] Sin[v + t], Tanh[v Cot[al]]}, {t, 0, 2 Pi}, {v, vmin, vmax}, PlotRange -> All]; bt = 0.15; rm = 1.25; rmin = 2 rm; rmax = 3.6 rm; zmin = 0; zmax = 1; plane = ParametricPlot3D[{r, -r Tan[bt], z}, {r, 2 rm, rmax}, {z, zmin, zmax}]; Show[plane, loxoSph] 

One of four $(t,v,r,z)$ is chosen as independent parameter with a uniform increment. Respective $(x,y,z)$ are equated whle solving but gets bit complicated due to simultaneity.

How is a Table solved for the remaining three coordinates on the line of intersection ? Thanks for help.