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Question: How to prove or disprove flatness of surface?

Question:How to prove or disprove flatness of surface?

Markiyan Hirnyk 7784 Maple 2016
surface differential-geometry
September 06 2016
0

How to prove or disprove the flatness of the surface x = (u-v)^2, y =  u^2-3*v^2, z = (1/2)*v*(u-2*v), where u and v are real-valued parameters? Here is my try:

 

plot3d([(u-v)^2, u^2-3*v^2, (1/2)*v*(u-2*v)], u = -1 .. 1, v = -1 .. 1, axes = frame);plot3d([(u-v)^2, u^2-3*v^2, (1/2)*v*(u-2*v)], u = -1 .. 1, v = -1 .. 1, axes = frame)

  

eliminate([x = (u-v)^2, y = u^2-3*v^2, z = (1/2)*v*(u-2*v)], [u, v])

[{u = -2*(-x+2*z+(x^2-8*x*z)^(1/2))/(-2*(x^2-8*x*z)^(1/2)+2*x-8*z)^(1/2), v = (1/2)*(-2*(x^2-8*x*z)^(1/2)+2*x-8*z)^(1/2)}, {-(x^2-8*x*z)^(1/2)*x+y*(x^2-8*x*z)^(1/2)-4*(x^2-8*x*z)^(1/2)*z+x^2-y*x+4*y*z-16*z^2}], [{u = 2*(-x+2*z+(x^2-8*x*z)^(1/2))/(-2*(x^2-8*x*z)^(1/2)+2*x-8*z)^(1/2), v = -(1/2)*(-2*(x^2-8*x*z)^(1/2)+2*x-8*z)^(1/2)}, {-(x^2-8*x*z)^(1/2)*x+y*(x^2-8*x*z)^(1/2)-4*(x^2-8*x*z)^(1/2)*z+x^2-y*x+4*y*z-16*z^2}], [{u = -2*(x-2*z+(x^2-8*x*z)^(1/2))/(2*(x^2-8*x*z)^(1/2)+2*x-8*z)^(1/2), v = -(1/2)*(2*(x^2-8*x*z)^(1/2)+2*x-8*z)^(1/2)}, {(x^2-8*x*z)^(1/2)*x-y*(x^2-8*x*z)^(1/2)+4*(x^2-8*x*z)^(1/2)*z+x^2-y*x+4*y*z-16*z^2}], [{u = 2*(x-2*z+(x^2-8*x*z)^(1/2))/(2*(x^2-8*x*z)^(1/2)+2*x-8*z)^(1/2), v = (1/2)*(2*(x^2-8*x*z)^(1/2)+2*x-8*z)^(1/2)}, {(x^2-8*x*z)^(1/2)*x-y*(x^2-8*x*z)^(1/2)+4*(x^2-8*x*z)^(1/2)*z+x^2-y*x+4*y*z-16*z^2}]

 (1)

NULL

I think Gaussian curvature should be used to this end. Dr. Robert J. Lopez is my hope.

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