It's a model of hyperbolic geometry in the plane, the same geometry described by Lobachevsky. There are several models of planar hyperbolic geometry. The most common ones include the Poincaré disk (or more generally ball), the Poincaré upper hald-plane (or more generally half-space), the Beltrami-Klein model which is somtimes also called the projective model, and the hyperboloid model which uses a three-dimensional Minkowsky space to embed the plane.
Comparing these models, the Poincaré disk model has the benefit that it doesn't extend to infinity (as the upper half-plane does), and preserves angles (contrary to the non-Poincaré models), and lives in the plane without need for a third dimension (as needed by the hyperboloid). But these benefits are balanced by drawbacks: the upper half-plane can use real $2\times2$ matrices to describe isometric transformations, the Beltrami-Klein model uses straight lines to model geodesics, and the hyperboloid model is very close in formulation to the geometry on a sphere.
An imperfect model of this geometry would – at least in my interpretation of the word “imperfect” – be the geometry on the tractricoid. Since the tractricoid has constant negative curvature, it closely resembles the hyperbolic plane, uisng “real” angles and “real” geodesics, as they are in the ambient three-space. But the tractricoid only models a part of the hyperbolic plane; it contains closed curves which would not close in the hyperbolic plane. So it is only a local model, which doesn't represent the global structure well. There can be no embedding of the hyperbolic plane into real three-space which uses normal Euclidean angles and geodesics.
You could define all kinds of stuff about what you call a “line”, what you call “distance” and so on. The special thing about hyperbolic geometry is the fact that it still satisfies the first four axioms of Euclid, even though it violates the fifth. In this regard it is pretty much unique.
Lines, circular arcs, and Bézier curves can all be represented exactly as rational Bézier curves. These are a generalization of the polynomial Bézier curves that many people know and love.
Rational Bézier curves can also represent pieces of other types of conic curves (parabolas, ellipses, hyperbolas), in addition to circular arcs.
A NURBS curve is just a sequence of rational Bézier curves strung together end-to-end so that (usually) they join smoothly.
Best Answer
yup, regression analysis/method of least squares. you would assign a coordinate system to the plane on which the curve is located and measure the coordinates of various points on the graph. the regression analysis will fit these curves to a function of your choice such as polynomial, exponential, etc. (the choice is a matter of which one you judge will be the best fit) and assign the relevant constants.