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How may we find (at least one real) intersection parametrizations $ x(t),y(t),z(t)$ between two implicit surfaces?

p=9; ContourPlot3D[{x^2-y-(z-1)^2==1,(x+1)^2+2Sin[3y]-3z==4},{x,-p,p},{y,-p,p},{z,-p,p},AxesLabel-> {XX,YY,ZZ}]

Thanks in advance.

enter image description here

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    $\begingroup$ Please provide your Mathematica code! $\endgroup$ Commented Nov 1, 2023 at 10:54
  • $\begingroup$ Thanks, given above. Requested intersections parametrizations is for erg as $t,x(t),y(t),z(t) $. $\endgroup$
    – Narasimham
    Commented Nov 1, 2023 at 13:33
  • $\begingroup$ Your "time parametrization t" isn't known! $\endgroup$ Commented Nov 1, 2023 at 13:47
  • $\begingroup$ Is it possible to output a $Table$ of $(x,y,z)$ coordinates from $ erg $ and $NSolve$ without $t$? $\endgroup$
    – Narasimham
    Commented Nov 1, 2023 at 16:40
  • $\begingroup$ Yes it's possible, see my modified answer! $\endgroup$ Commented Nov 1, 2023 at 16:54

1 Answer 1

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Try to solve the two equations for x,z to get a parametrization for the intersection curves {x[y],y,z[y]}:

erg = Solve[{x^2 - y - (z - 1)^2 == 1, (x + 1)^2 + 2 Sin[3 y] - 3 z ==4}, {x, z}];

Show[{p = 9; 
ContourPlot3D[{x^2 - y - (z - 1)^2 ==1, (x + 1)^2 + 2 Sin[3 y] - 3 z == 4}, {x, -p, p}, {y, -p, p}, {z, -p, p} , Mesh -> False, AxesLabel -> {x, y, z}], 
ParametricPlot3D[{x, y, z} /. erg, {y, -9, 9}, PlotStyle -> Red]}]

enter image description here

addendum

The points of intersection follow from

pic=ParametricPlot3D[{x, y, z} /. erg, {y, -9, 9}, PlotStyle -> Red]
points = Cases[pic, Line[p_] :> p, -1];
Graphics3D@Map[{RandomColor[],Thickness[Large], Line[#]} &, points]

enter image description here

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