# Parametrization of intersections

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}]

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

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]}]


pic=ParametricPlot3D[{x, y, z} /. erg, {y, -9, 9}, PlotStyle -> Red]