I saw the following 【Theorem？】 in a book written in Japanese. According to the book, this theorem seems to be the result of Boyai's investigation of the properties that parallel lines satisfy under non-Euclidean conditions.
The literal translation of the Japanese name of this theorem can be "Bolyai's Parallel angle formula." As this book was written for the general public and having too rough description that, I cannot specified the detailed statement of the theorem.
【My Question】
(1)What is the name of this theorem in English?
(2)What is the exact statement of this theorem?
Perhaps this is true in the (the upper half of) bifid hyperbolic plane, but not in the $S^2$, an I right?
What I want to know is what theorem can be derived from "some premises‡" + "two parallel lines can be drawn".
‡.Probably axioms and canons of Euclidean geometry other than Parallel postulate, am I right ?
【Theorem？】
Suppose that, there are two† parallel straight lines that are parallel to the straight line 'a' and pass through the point 'P'. At this time, if the length of the perpendicular line (PH) drawn from the point P to the straight line 'a' is x, and the angle formed by this perpendicular line (PH) and the parallel line is θ, then following equation is satisfied.$$\tan\left(\theta\left(x\right)/2\right)=\exp\left(-\left(x/k\right)\right) \tag{1※}$$ †. By using the Japanese language, we can obscure either "just two" or "at least two". I do not know which meaning the "two" was written in.
※.I cannot find any description for what the parameter 'k' in Equation 1 is.
Maybe it's similar to the content of the Wikipedia article, "Angle of parallelism". However, the article does not make the assumption that "two parallel lines can be drawn". However, the formula on this page is the following in our notation. That is, it seems to be the case for k = 1. $$\tan\left(\theta\left(x\right)/2\right)=\exp\left(-\left(x\right)\right) \tag{2}$$
Also, in Equation 1, if we determine x, the θ seems to be uniquely determined, am I right? Thus, I also wonder where there is room for two (or more) straight lines to be drawn.(Does that mean we can draw more than one perpendicular line(PH) ?)
The answer here seeks to interpret the diagram in terms of present time knowledge with the origin of hyperbolic geometry. I was looking for Community Wiki ...
For the two dimensional hyperbolic surface embedded in $ \mathbb R^3:$
If we take $x$ to represent polar/central angle reckoned from cuspidal equator, and $\theta $ as the parallelism angle the hyperbolic geodesic makes to the meridian of a pseudosphere in its tangent plane we have parametric equation of a pseudosphere of Gaussian curvature, the cuspidal equatorial radius respectively:
$$ K= {-1}/{a^2},a=1, \; k=a $$
$$ \sin \theta = \text{sech x} = r/a $$
At cuspidal equator $ (\theta=\pi/2, x=0) $. At infinite distance these are $ (0,\infty)$ respectively.
Here we have
radius in cylindrical coordinates, polar angle, coordinate along symmetry axis, angle made by tangent of hyperbolic geodesic to meridian and symmetry axis,
respectively correspond as:
$$ r, t, z, \psi= \phi, $$
inter-related with
$$ r= a \text{ sech t}, z= a (t- \tanh t)$$ and
$$ \sin \psi= r/a, \text{or} \sin \phi= r/a \; $$
for the pseudosphere of Beltrami or the Net of Chebychev respecting the Sine-Gordon differential equation
$$ \dfrac{d(2\psi )}{ds} = \sin 2 \psi,\; \dfrac{d(2\phi )}{ds} = \sin 2 \phi; $$
There are only two hyperbolic parallels to the axis of symmetry passing through any point. These two hyperbolic parallels are two asymptotic lines making angle $2\psi$ between them as shown in red.
The above equations and base concepts are in accordance with modern understanding in hyperbolic geometry of the pseudosphere.
The Bolyai's Early diagram
If we use symbols given in the diagram to denote, interpret for a complete correspondence:
$$( x=PH ) \rightarrow r,\; \theta \rightarrow (\pi/2-\phi=\pi/2-\psi)$$
i.e, when $\theta$ in the diagram is the angle made by the hyperbolic geodesic to the parallel circle and not to the meridian then there is a full match imho to the current understanding.
One cannot fail to notice from the Bolyai used symbols that for any point $P$ of pseudosphere to the axis there is a variable normal distance $PH$ from cuspidal equator to axis in the euclidean sense, $ a>PH>0.$
On the other hand there is a constant hyperbolic distance $HP$ between the hyperbolic geodesic through $P$ marked as $b$ by Bolyai and its parallel line marked $a$ containing $H$ because these two lines are hyperbolic parallels.