The Inside-of-a-Sphere: One would imagine the perfect complement to the sphere would merely be the inner side of one. As the earth can be used as an example for a sphere, the sky as viewed from earth is the opposite side of one. However, this shape is often unwieldy to use, as one would have to assume that spheres are hollow, and unless viewed from the inside of the sphere with the sphere bounding the universe, in which case you can't see the entire pseudosphere at one time (even if the earth were transparent, you couldn't look at the entire sky all at once unless you have 360 degree view like a duck) any shapes modeled would merely look as though they are on the outside of the sphere (see Fig. 1).
The Hyperboloid: The hyperboloid is a three-dimensional version of the hyperbola. It can be made by rotating the hyperbola 360 degrees along an axis extended indefinitely in either direction. A hyperboloid rotated around the minor axis will create a single sheet hyperboloid, while one rotated around the major axis will make a disconnected two-sheet hyperboloid (see Fig. 3).
The Tractricoid: While the tractricoid's two-dimensional counterpart, the tractrix, is not as commonly known as the hyperbola, it is reasonably simple. It follows the path of an object on a flat plane being pulled as though by a string by something that follows a straight line (see second half of Fig. 4 animation). The motion is repeated in the other direction to complete the tractix, with the starting point of the object as the center of the tractix, and the line the pulling point follows being the asymptote. The tractricoid is created by rotating a tractrix around its asymptote (see Fig 5). Dini's surface can be made by taking half of a tractricoid and twisting it up (see Fig 10).
The Saddle-Shaped Plane: Not as commonly used. Looks kinda like a Pringle. (See Fig. 6)
The 2D Models: These are the inside of a sphere embedded in Euclidean space, and can also be used for elliptic geometry. 2-D Euclidean shapes can be thought of as 3-D Euclidean ones (See Fig. 1 and Fig. 2). Disk models has multiple types and are embedded in a circle (see Fig. 7 and Fig. 8), while the half-plane model is on half a Euclidean plane (see Fig. 9).
More on 3D NEG will be covered with its applications on Einstein's theory of relativity, and the Poincare disk model will be used more when we get to tessellations, fractals, and other non-Euclidean art.
^Figure 1- 3D star chart
^Figure 2- 2D star chart
^Figure 3- Making a hyperboloid from a hyperbola
^Figure 4- Drawing a tractrix
^Figure 5- A tractricoid- note the lack of a point because of the asymptote.
^Figure 6- Displays triangle and line behavior in hyperbolic geometry.
^Figure 7- Klein model- note straight lines and distorted angles.
^Figure 8: M C Escher tessellations on the Poincare disk model.
^Figure 9: Poincare half-plane model, very similar to Poincare disk model.
^Figure 10: Dini's surface as viewed as a 3D Euclidean shape.
Image sources: Umich.edu, Moab Happenings Archive, http://www.math.rutgers.edu, Wikipedia, Trimble 3D Warehouse