In the euclidean geometry, the ruler of zero curvature is the same as the one of minimal curvature (or straight ruler). In non-euclidean geometries, these are different curves. While most geometries use the straight ruler, which leads to a thicket of sines and cosines, the circle-drawing uses the zero-curvature ruler, which does not lead to this thicket, and is more general. In particular, a straight line is a particular instance of a curved equidistant.
For example, the polytope o3x5o occurs as both the mid-section of x3o3o5o, and as a small facd on x3o3x5o. If one were to deal with the various chords of this polyhedron in terms of the 'flat ruler', then the ratios of edges change from size to size. When one uses the zero-curvature ruler, the chords maintain their same ratios, which is why people deal with {5,3,3} as a series of polytopes, rather than as spherical tilings.
The constant of curvature corresponds to 1/r^2, which may be positive or negative. Minimal curvature is a straight line, is represented as a curve the same radius as the containing space. For example, a great circle has the same radius as the sphere it falls on, and is thus a 'straight line' in S2. The maximum curvature has 1/r^2 = inf, ie r=0. This is a point: all spaces contain this. A space may not contain a curve less curved than itself, ie 'hyper-straight'.
With hyperbolic geometry, we suppose that 1/r^2 is a negative, like -36, which makes a notional radius of i/6. A larger curvature means "more in the direction of +\infty, which means that a curvature of -9 is an equidistant in a curvature of -36, but the radius is now 2i/6 = i/3. That is, the equidistant is twice the size. Here is an example using a row of cells of x5o4o, which can be no clearer.
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A B C A row of cells of x5o4o
+ + + Note that ABC matches abc
/ \ / \ / \ on an equidistant, but the
+- -+- -+- -+ edges are twice the angle
| | | | of abc. Also, it has
| | | | right-angles.
+-----+-----+-----+
a b c
Here we see three pentagons of x5o4o, the straight line at the base abc is run equidistant by a right-angle construction of ABC, the edge AB is exactly that of two edges of the base (ie ac), although it projects by perbpendiculars frm abc onto a.5 to b.5. The equidistant ABC has a 'larger curvature' which means that the curvature is less -infty than the base, but this means -1/R^2 is larger than -1/r^2.
Just as equidistants of the sphere are reduced by cos L, the equidistants of the hyperbolic are increased by cosh L.
What we shall show in later episodes is thae right-angle rule is c²=a²+b² - a²b²/2r², where a, b, c are the chords of the edges of the rightangle triangle, the angle is opposite c, and r is the radius of the sphere containing the right-angle. When r is infinite, then c²=a²+b² gives the euclidean rule. When a²=2r² (that is, the quadrant of a sphere), then we see that c²=a², is independent of b. This is the double-right angle. But we allow r² to be negative too, and this formula continues to work in hyperbolic geometry.
The sort of ruler that corresponds to 'zero-curvature' in hyperbolic space, might be written with the fibonacci series.