### A few notes on surfaces

#### Surface Curvature

How do we assign curvature to a surface? First of all it's gotta be smooth at the point where we want to the calculation. That is; no sharp points, edges or other singularities. Then look at curves determined by the intersection between the surface and planes perpendicular to the tangent plane at the point. All these curves have a single and well defined curvature at the point. Take the maximum value of these curvatures as k1, and the minimum as k2. These are called the Principal curvatures. Then the Gaussian curvature of a surface is k1*k2 and the Mean curvature of a surface is (k1+k2)/2.

#### Constant negative curvature

The surfaces of constant negative curvature are really surfaces of constant negative Gaussian curvature. An interesting result of constant Gaussian curvature is that an inelastic net that fits the surface in one area fits it everywhere. This is connected to the beautifully named Gauss Theorema Egregium. Examples of surfaces in this class are Dini's surface, the Pseudosphere (both of which are developed directly from the Tractrix) and Kuen's surface

#### Minimal Surfaces

A Minimal surface is naively the surface with minimal area spanning the space between its assigned edges. More importantly a minimal surface is a surface whose Mean curvature is 0 everywhere. Think of a disk, which is the minimal surface spanning a circle. If you pull it out, it's curvature wont be 0 everywhere anymore. And it ain't minimal anymore. This is true for all minimal surfaces. Since the edges can be just about anything you please, there are obviously a ton of surfaces in this class - both finite and infinite. Some of note are; Enneper's surface, the Helicoid, the Catenoid (this is the surface between two circles), Scherk's surface (between 4 parallell lines arranged in a sqare), Henneberg's surface and Catalan's surface.

#### Models of the projective plane

And then we have the realizations of the real projective plane. To make this clear I can say that in the real procective plane antipodal points are identified. But this is just hocus pocus. What it really means is that if you follow any "straight" line far enough you will come back to where you started. Another way to say this is that any two lines meet in a points, and and any two points contain a line. Surfaces in this class are the cross-cap, Steiner's Roman surface and Boy's surface (a surface, interestingly enough, without singularities).

#### Fun and play

Knots are strictly speaking not surfaces at all, but spacecurves. To visualize them, you need to drag a perpendicular circle along the curve to sweep out a surface. This is often referred to as rendering in "sausage mode". This is also done in "snail" or "cournocopia" shaped surfaces. In this case the generating spacecurve is a spiral.Look at the knot and the shell.
Closely related to these surfaces we have Canal Surfaces, also called Swept Surfaces. Here the generating curve and accompanying shape can be just about anything. When the curve is a Quadric, and the shape is a sphere tangent to a fixed sphere and centered on the quadric we get the Dupin Cyclide