What are Straight lines in the Gans Disk model of the Euclidean plane?

Question

The answer of Blue ( https://math.stackexchange.com/a/1464/88985 ) to Hyperbolic critters studying Euclidean geometry made me interested in the Gans Disk model of the euclidean plane.

Blue writes:

In ["A Euclidean Model for Euclidean Geometry"][http://www.jstor.org/pss/2323257], Adolf Madur discusses a Disk model of the Euclidean plane. (Madur says that David Gans has priority for discussing this model, so I'll call it the "Gans Disk".) The "lines" consist of diameters of the Disk, and half-ellipses that have a diameter as a major axis; the measure of the angle between two "lines" is defined as the traditional measure of the angle between their respective major axes. With an appropriate metric (which I have forgotten, and which is just missing in the document preview linked), we get all of the Euclidean plane crammed into the Disk.

I was very interested and now have the book David Gans "Transformations and Geometries"(1969), and the model is described on page 212 and later.

And it made me wonder

"What are (euclidean) Straight lines in the Gans Disk model ?"

(What functions in the normal euclidean plane result in straight lines in Gans Model?)

I would like to compare them with euclidean straight lines in the Beltrami-Klein model of hyperbolic geometry. (see http://en.wikipedia.org/wiki/Beltrami%E2%80%93Klein_model )

Off course all lines trough the centre of the Gans Disk are straight lines but what about a the other straight lines?

the Gans Disk model maps a point $(x_e,y_e)$ in the (normal) Euclidean plane to a point $(x_g,y_g)$ of the Gans Disk model:

$$ x_g = \frac{x_e}{\sqrt{1+x_e^2+y_e^2}} , y_g = \frac{y_e}{\sqrt{1+x_e^2+y_e^2}}$$

or in reverse (if I am correct)

$$ x_e = \frac{x_g}{\sqrt{1-x_g^2-y_g^2}} , y_e = \frac{y_g}{\sqrt{1-x_g^2-y_g^2}} $$

Gans' book is rather old (almost as old as myself) are there newer books available that describe this model?

Answer 1

All you have to do is substitute $(x_g,y_g)$ into a linear equation. For simplicity, we can rotate our "Euclidean" line about the origin until it is parallel to the $y$ axis: $$x = k \qquad\to\qquad x_g = k \qquad\to\qquad \frac{x_e}{\sqrt{1+x_e^2+y_e^2}} = k$$ where $0 < k < 1$. Squaring and re-arranging, we get $$\frac{x^2 ( 1 - k^2 )}{k^2} - y^2 = 1$$ Therefore, a Euclidean chord of the Gans Disk represents an arm of a rotated, origin-centered hyperbola with transverse semi-axis of length $k/\sqrt{1-k^2}$ and conjugate semi-axis of length $1$.

Note that the diameters through the ("ideal") endpoints of the chord are the asymptotes of the hyperbola. This is re-confirmed by observing that the chord $x=k$ intersects the unit circle at $y=\sqrt{1-k^2}$, so that the slope of the radius to that point of intersection is $$\frac{\sqrt{1-k^2}}{k} = \frac{\text{conjugate semi-axis}}{\text{transverse semi-axis}}$$ as expected.

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I don't really see how you intend to compare this situation to the Beltrami-Klein model, however.

As for more-modern references, all I have is that previous link to Adolf Mader's article in the Monthly; that's dated 1989. If nothing else, you could write to Mr. Mader; his email address is on his page of the University of Hawaii's web site.