curvature sign
Here, you will find a small gallery of regular surfaces generated using the Maple program. The main intention is to highlight the sign of the Gaussian curvature (K) of the presented surfaces. To achieve this, each surface has been painted with black and white following the rule: black for points where the Gaussian curvature is negative and white for points with positive curvature (black for K < 0, white for K > 0).
Geometric Intuition: "A point has positive curvature (K > 0) if, when you take the tangent plane at this point, the surface is locally contained in one of the two semi-spaces determined by the tangent plane."
f(x,y)=sin(x)cos(y)
f(x,y)=2xy^2/(x^2+y^4)
Jarra
Toro
twisted sphere
Trompetita
Peculiarity: mean curvature zero.
catenoide
Peculiarity: mean curvature zero.
Figure 8
Mobius band
fake sphere
This surface has the peculiarity of having a constant Gaussian curvature equal to 1.
The tips do not belong to the surface; therefore, it is not simply connected. If we consider the tips, it will be an Orbifold.
Zero curvature surface different to the plane
pseudosphere
This surface has the peculiarity of having constant Gaussian curvature equal to -1.
It is not simply connected.
f(x,y)=x^2-y^2
sphere
cylinder
This surface has constant Gaussian curvature equal to 0.
It's remarkable to have a flat surface different to the plane.
