Exercise 26 chapter 6

Vasco,

That should have read "the area of a triangle on the pseudosphere". I am referring the the angles $\alpha$ and $\beta$ at the origin, but I see it as transposed onto the horizon rather than transposed to the real line. That is, it is on the real line, but it appears to be on the horizon.

Gary

Vasco,

That all makes good sense to me. I wonder how this examination of h-lines pertains to the formulae for Euclidean and h-centres and Euclidean and h-radii on p. 309. h-circles are like Euclidean circles in that we draw them as semi-circles from Euclidean centers on the horizon. By the formula for h-circles, if the Euclidean centre is at x, so is the h-centre. But if the h-radius is $\rho$, the Euclidean radius is $ y sinh(\rho)$ = 0, because y = 0. If the Euclidean radius is a real number $a$, then $a = y sinh(\rho)$ = 0. So we can't get from the Euclidean to the h-radius this way. Perhaps we could say that if the Euclidean radius is $a$ and the Euclidean centre is $x$, the radius is $|ai|$ at $x + ai$. Then $a = ai sinh(\rho)$ and $\rho = arcsinh(a/ai) = arcsinh(-i)$ = -1.57i$, but it doesn't make much sense to have a constant h-radius for an h-line with a variable radius. Or does it? It appears that hyperbolic geometry that applies to h-circles does not necessarily apply to h-lines.

Gary

Yes, I meant "...on p. 309. h-lines are..."

Your discussion is very helpful.

In section 3.4 Beltrami's hyperbolic plane. I find these points from p. 301, paragraph 3, relevant to this discussion. The boldface is my own emphasis:

"the hyperbolic plane, namely, the entire shaded half-plane y > 0"

"The points on the real axis are infinitely far from ordinary points and are not (strictly speaking) considered part of the hyperbolic plane. They are called ideal points, or points at infinity. The complete line y = 0 of points at infinity will be called the horizon."

In my first reading, I missed the significance of this, lumping everything, including the horizon, into the UHP model of the h-plane. But perhaps it is misleading to draw radii and semi-circles from points on the horizon, as in [21], [22], [23], [24], [25], [26]. Drawing them in this way suggests that the point on the horizon is a part of the radius or h-line, when in fact it is infinitely far away. But what is the epistemological status of an ideal point on the horizon? To call it an ideal point seems to beg the question. An ideal point might be called an indexical point or a point of reference, which would suggest that it is useful in situating objects in the h-plane, but what quality of the h-plane gives indexicality to something outside itself, not part of the h-plane? Is it that we can only apprehend the complete h-plane by its connections and relationship to the Euclidean plane? Taking this argument to its conclusion, it appears that the hyperbolic plane is just a field of mathematics that represents negative curvature by reference to the Euclidean plane. If we allow the curvature to tend from negative to 0, the limit is the Euclidean plane. That is why Needham can say that Euclidean geometry is subsumed by (3-dimensional) hyperbolic geometry (p. 327). Since 3-dimensional h-geometry can be mapped to the UHP, the latter must also be subsumed.

Conclusion: It would be useful to somewhere state: In the figures, ends and centres of h-lines are often drawn from points on the horizon as though there were a continuous connection, but in fact the ideal points in the horizon are not part of the UHP. They are used only to index the x-values of objects in the UHP, notably including, but not limited to, the ideal endpoints and centres of h-lines.

Gary