Random Gaussian cubic surface Seed: 67 (Gaussian coefficient and line-tracking seed)
Cubic: f(x,y,z) = -0.138 x*z^2 +0.00464 y*z^2 -0.0271 z^2 -0.0908 x^2*z +0.347 x*y*z +0.203 x*z +0.124 y^2*z +0.157 y*z -0.267 z +0.226 x^3 -0.153 x^2*y -0.0503 x^2 +0.297 x*y^2 -0.315 x*y -0.385 x -0.255 y^3 +0.432 y^2 -0.0072 y +0.184 1
Affine cubic: f(x,y,z) = -0.138 x*z^2 +0.00464 y*z^2 -0.0271 z^2 -0.0908 x^2*z +0.347 x*y*z +0.203 x*z +0.124 y^2*z +0.157 y*z -0.267 z +0.226 x^3 -0.153 x^2*y -0.0503 x^2 +0.297 x*y^2 -0.315 x*y -0.385 x -0.255 y^3 +0.432 y^2 -0.0072 y +0.184 1
Shadow: the quartic B(x,y)^2 - 4 A(x,y) C(x,y) = 0 is drawn on the z=-4 xy-plane in the same 3D scene. The translucent ruled curtain rises upward from the shadow to the ramification curve where the lifted point has z >= -2 and lies above the displayed projection plane. The 15 real projected lines on the xy-plane are bitangents found from the projective lines on the cubic surface. Same-color transparent line projection cones are included but hidden by default; use the page toggle to show them.
Viewing box: axis ranges are [-4, 4].