The blue formulas give h(z_{1}), followed by f(x,y) 



Cubic Parabolic The level set through the
critical point has a cusp shape. This kind of critical point is
"unstable," in the sense that small changes in the function can
result in a surface with different kinds of critical points, as
the following two pictures show.
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γ = 1/2 



Deformation of Cubic Parabolic into Elliptic/Hyperbolic pair
Adding a linear term with coefficient t < 0 deforms the
surface with an unstable parabolic critical point into a surface
with a pair of stable critical points, one elliptic and one
hyperbolic.
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Deformation of Cubic Parabolic into surface without critical
points Using the same formula as above, but with linear
coefficient t >0, results in a surface with no critical
points.
This can be understood as a "cancellation" phenomenon, where
t is a time parameter, and the elliptic/hyperbolic points
move toward each other as t increases, collide at
t=0 to form a parabolic point, and then disappear as
t becomes positive.
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There are two types of quartic (degree 4) surfaces with a parabolic critical point; for lack of better terminology, they are labeled type (I) and type (II). 

Quartic Parabolic (I) This surface also has a
critical point which is a local minimum. In the picture, the
surface is more flat in the y direction than the x
direction, and the critical point is in the center of the long
black dot. The surface is defined by a quartic (degree 4)
equation and is unstable under deformations, as the following
three pictures show.
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POV scene file




Deformation of Quartic Parabolic (I) into Elliptic
Adding a linear term with coefficient t1 < 0 or t1 > 0
gives a stable elliptic critical point.
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POV scene file




Deformation of Quartic Parabolic (I) into Elliptic
Adding quadratic terms with coefficient t2 < 0
also gives a stable elliptic critical point.
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POV scene file




Deformation of Quartic Parabolic (I) into two Elliptic + one
Hyperbolic Adding the same quadratic terms with coefficient
t2 > 0 gives three stable critical points.
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POV scene file



Quartic Parabolic (II) This surface has a critical
point where surface is more flat in the y direction than the
x direction, and the level set through the critical point is a
selftangent curve, instead of an X shape like the stable saddle
point.
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POV scene file




Deformation of Quartic Parabolic (II) into Hyperbolic
Adding a linear term with coefficient t1 < 0 or t1 > 0
gives a stable hyperbolic critical point.
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POV scene file




Deformation of Quartic Parabolic (II) into Hyperbolic
Adding quadratic terms with coefficient t2 > 0
also gives a stable hyperbolic critical point.
Maple worksheet
POV scene file




Deformation of Quartic Parabolic (II) into two Hyperbolic + one Elliptic
Adding the same quadratic terms with coefficient
t2 < 0 gives three stable critical points.
Maple worksheet
POV scene file

