Plot the Intersection of Two Surfaces

[teochelmus (teo ytriu)]

I would like to know how it can be possible,in Wolfram, to plot an explicit graph of intersection of two surfaces. For instance:

13x^2+40y^2+4z^2+28xy-8xz-8yz-x+y+z+1

and

x-y+z

Thank you very much!

[mroge02 (Michael Rogers)]

If by surfaces you mean where each formula is zero, then it turns out that 13x^2+40y^2+4z^2+28xy-8xz-8yz-x+y+z+1 = 0 has no real solutions. Substituting another formula, here is a way.

ContourPlot3D[{13 x^2 + 8 y^2 + 4 z^2 + 2 x y - 8 x z - 8 y z - x + 
   y + z - 1, x - y + z}, {x, -4, 4}, {y, -2, 2}, {z, -2, 2}, 
 Contours -> {0}, ContourStyle -> None, Mesh -> None, 
 BoundaryStyle -> {1 -> None, 2 -> None, {1, 2} -> {{Thick, Blue}}}]

[fkampas (Frank Kampas)]

Reduce[13 x^2 + 40 y^2 + 4 z^2 + 28 x y - 8 x z - 8 y z - x + y + z + 
   1 == 0, {x, y, z}, Reals]

False

[Vitaliyk (Vitaliy Kaurov)]

Building on Michael Rogers version ( also see this tutorial ) we get:

h = 13 x^2 + 8 y^2 + 4 z^2 + 2 x y - 8 x z - 8 y z - x + y + z - 1;
g = x - y + z;

ContourPlot3D[{h == 0, g == 0}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, 
 MeshFunctions -> {Function[{x, y, z, f}, h - g]}, MeshStyle -> {{Thick, Blue}}, Mesh -> {{0}}, 
 ContourStyle -> Directive[Orange, Opacity[0.5], Specularity[White, 30]], 
 PlotPoints -> 60, SphericalRegion -> True]