Gaussian curvature is one of the
most beautiful properties in geometry, because it is a way of describing the
curvature of a surface which is dependent only upon the surface’s intrinsic
geometry. This is essentially the
content of Gauss’ Theorema Egregium (Most
Remarkable Theorem) which states that Gaussian Curvature, K, is invariant under local isometry.
What does this mean? One way Gauss liked to think about surface
geometry was to consider what a two-dimensional being living on the surface
would be able to determine about the surface just by doing measurements of the distance
between two points. For example, a
person living on the surface of a sphere could determine he was not on a flat
plane by measuring the angles of a triangle on the sphere, and noting that the
sum of the angles is greater than 180 degrees (in a flat plane, the angles in a
triangle always add up to 180 degrees, but try drawing a large triangle on a
ball. You can make all three angles 90
degrees if you like!). Essentially, if
you had two surfaces which a two-dimensional being could not tell apart just by
doing distance and angle measurements, they are said to be isometric. Another way of thinking of isometry is to imagine bending a surface without deforming
it, like rolling a flat plane into a cylinder.
In contrast, there is no way one could bend a flat piece of paper to
cover a sphere, or flatten out the surface of a sphere without deforming the
surface. This is essentially the reason
why maps of the earth are so distorted, and Greenland often appears
to be the size of Africa.
Getting back to the subject of
Gaussian Curvature, it is a property intrinsic to the surface, having no
dependence on how it’s bent around in three dimensions. A lot can be determined about local geometry
by just looking at whether K is positive or negative. Here are some examples which can be realized
in three dimensions (note that, although K can vary from point to point, if you’re
looking in a small enough region of the surface, any type of surface with K
> 0 must "look like" any other type of surface with K > 0 , and the same
goes for K < 0 or K = 0):
K > 0: Sphere, ellipsoid, paraboloid,
the outer half of a torus (donut).
K = 0: Plane, cylinder, cone, generalized cylinder.
K < 0: Saddle surface (z=xy),
catenoid, helicoid, the inner half of a torus
Essentially, when the curvature is
positive, the local geometry looks like a “bump”, and when it’s negative, the
local geometry looks like a “saddle”.
When K=0, the surface is said to be flat,
and the local geometry looks like a flat plane.
Wait a minute, you say. “A
cylinder looks like a flat plane?” Yes
it does, to a two-dimensional being living on the surface of the cylinder,
because distances along a cylinder are exactly the same as distances along a
flat plane, but “straight lines” are just mapped into helices (spirals). The two-dimensional guy can’t tell the
difference, unless he tries walking in one direction for awhile, and ends up back where he started.
If you add up, or integrate
the Gaussian curvature at all points on the surface, you get another important and incredibly interesting quantity: Total Curvature