Similarities between non-Euclidean geometries

Question

So I've looked into Euclidean, spherical/ellpitic, and hyperbolic geometry, and found some possible similarities. I'm not much of an expert, so I can't really verify them for myself. I'd like to know which are actual properties of those geometries, and which are just coincidences, and for the coincidences, what is the true relation?

To keep things simple I'll only consider 2 coordinate axes $x,y$ and an extra axis for the model $z$. I'll call the curvature $K$ and define $k=\sqrt{K}$.

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Some of these seem to only apply for $K\in \{-1,0,1\}$, so it may be necessary to split it into sign and magnitude, or sign of curvature and radius of curvature. It also means these may just coincidentally work for $K\in \{-1,0,1\}$ and not any other values.

The true shape or correct model is defined like so:

$$z^2=1-K(x^2+y^2)$$

The origin is always the same:

$$O=(0,0,1)$$

The distance from the origin to a point is proportional to the area it'd sweep out:

Projecting from $(0,0,0)$ to the plane $z=1$ will preserve straightness (Gnomonic/Beltrami-Klein). Projecting from $(0,0,-1)$ to the plane $z=1$ will preserve angles (Stereographic/Poincare).

Using the complex definitions for trigonometric functions...

$$\sin(z)=\frac{e^{iz}-e^{-iz}}{2i}$$

...we define a special curvature specific function:

$$\sin_K(x)=\frac{1}{k}\sin(kx)$$

We can verify this produces $\sin_1(x)=\sin(x)$, $\sin_{-1}(x)=\sinh(x)$, and $\sin_0(x)=x$. $K=0$ needs to be evaluated using a limit.

It also appears to be a kind of inverse of distance to origin:

$$\text{distance}[(\sin_K(v),0,\cdots)]=|v|$$

The circumference and area of a circle and the surface area and volume of a sphere:

$$S_1(r)=2\pi\sin_K(r)$$ $$V_2(r)=4\pi\sin_K(\frac{1}{2}r)^2$$ $$S_2(r)=4\pi\sin_K(r)^2$$ $$V_3(r)=\frac{1}{K}\pi(2r-\sin_K(2r))$$

For volume of sphere, $K=0$ needs to be evaluated using a limit.

Answer 1

A very nice writeup! HyperRogue uses formulas very similar to these to deal with all geometries uniformly, although it only uses $K=-1,0,1$. (I have not checked other values of $K$ in detail, also your volume formulas, but these are simply integrals of circumference and surface area.)

There are also more formulas that could be added, for example: (also not checked for other values of $K$ -- please check if you want to use them for other values)

• $\cos_K$ which is the integral of $K\sin_K$, and $\cos_K 0=1$: this is constant 1 in Euclidean, cos for $K=1$, and cosh for $K=-1$. I would rather define $\sin$ and $\cos$ as the only functions which satisfy the system of differential equations: $\sin_K(0)=0$, $\cos_K(0)=1$, $\sin'_K(x)=\cos_K(x)$ and $\cos'_K(x)=-K\sin_K(x)$. This definition avoids complex numbers (a good thing because you do not have to say which root of $k^2=K$ to take).
• if you have a line and a segment of length $1$ on it, the matching segment of an equidistant of radius $r$ is $\cos_K r$
• the matrix of an isometry which moves the origin $x$ units to the right, in the "true shape" model: $\left(\begin{array}{ccc}\cos_K x&0&\sin_K x\\0&1&0\\K\sin_K x&0&\cos_K x \end{array}\right)$

• the distance between two points $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ in the "true shape" model: first you compute the distance in the three-dimensional space $d^2=|x_1-x_2|^2+|y_1-y_2|^2+K|z_1-z_2|^2$, and then the geodesic distance is $2{\rm asin}_K(d/2)$

• Pythagorean theorem: $\cos_K a \cos_K b = \cos_K c$ (you get the Euclidean version in the limit)

• more models: orthographic projection of the sphere and the Gans model of the hyperbolic plane are obtained the same way, Mercator projection of the sphere and the band model of the hyperbolic plane are obtained the same way (though you need to solve an integral here which makes the end formulas less similar), etc.