1
$\begingroup$ 
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]

ParaPlaneCutsParaSph

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.

Improve this question
$\endgroup$

1 Answer 1

Reset to default
2
$\begingroup$

You can do find and visualize intersection in an ugly way, e.g.:

f[t_, v_] := {3.6` + Cos[t + v] Sech[v], Sech[v] Sin[t + v], Tanh[v]} g[r_, z_] := {r, -r Tan[bt], z} sol = Quiet@Solve[f[t, v] == g[r, z], {v, t, r, z}]; int1 = ParametricPlot3D[g @@ ({r, z} /. sol[[10]]), {v, 0, 2}, PlotStyle -> Red]; int2 = ParametricPlot3D[g @@ ({r, z} /. sol[[11]]), {v, 0, 2}, PlotStyle -> Red]; surf1 = ParametricPlot3D[f[t, v], {v, 0, 2}, {t, 0, 2 Pi}, Mesh -> None, PlotStyle -> {LightBlue, Opacity[0.5]}]; surf2 = ParametricPlot3D[{r, -r Tan[bt], z}, {r, 2.5, 3.6 1.25}, {z, 0, 1}, Mesh -> None]; Show[surf1, surf2, int1, int2]

enter image description here

where solution 10 and 11 give the desired octants (it doesn't handle adding constraint).

However, the loxoSph is a unit sphere with centre (3.6,0,0).

Solving the equivalent problem centred on (0,0,0) with translated plane yields circle parametrization:

h[u_, v_] := g[u, v] - {3.6, 0, 0}; par = v /. Solve[h[u, v].h[u, v] == 1, {u, v}]; func1[p_] := h[p, par[[1]] /. u -> p] + {3.6, 0, 0} func2[p_] := h[p, par[[2]] /. u -> p] + {3.6, 0, 0} pp = ParametricPlot3D[func2[t], {t, 2.5, 4.5}, PlotStyle -> Red]; Show[surf1, surf2, pp]

enter image description here

Improve this answer
$\endgroup$
2
  • $\begingroup$ I wish to only see a Table of numerical data of points of intersection with uniform increment of one parameter. $\endgroup$ Commented Nov 7, 2016 at 11:16
  • $\begingroup$ @Narasimham then simply use Table[func2[j],{j, 2.5,4.5,inc} above for whatever increment you desire. $\endgroup$ Commented Nov 7, 2016 at 11:22

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.