09-23-2021-0820 - horosphere horopter

In hyperbolic geometry, a horosphere (or parasphere) is a specific hypersurface in hyperbolic n-space. It is the boundary of a horoball, the limit of a sequence of increasing balls sharing (on one side) a tangent hyperplane and its point of tangency. For n = 2 a horosphere is called a horocycle.

A horosphere can also be described as the limit of the hyperspheres that share a tangent hyperplane at a given point, as their radii go towards infinity. In Euclidean geometry, such a "hypersphere of infinite radius" would be a hyperplane, but in hyperbolic geometry it is a horosphere (a curved surface).

A horosphere within the Poincaré disk model tangent to the edges of a hexagonal tiling cell of a hexagonal tiling honeycomb 

https://en.wikipedia.org/wiki/Horosphere

The horopter was originally defined in geometric terms as the locus of points in space that make the same angle at each eye with the fixation point, although more recently in studies of binocular vision it is taken to be the locus of points in space that have the same disparity as fixation. This can be defined theoretically as the points in space that project on corresponding points in the two retinas, that is, on anatomically identical points. The horopter can be measured empirically in which it is defined using some criterion.

The concept of horopter can then be extended as a geometrical locus of points in space where a specific condition is met:

• the binocular horopter is the locus of iso-disparity points in space;
• the oculomotor horopter is the locus of iso-vergence points in space.

As other quantities that describe the functional principles of the visual system, it is possible to provide a theoretical description of the phenomenon. The measurement with psycho-physical experiments usually provide an empirical definition that slightly deviates from the theoretical one. The underlying theory is that this deviation represents an adaptation of the visual system to the regularities that can be encountered in natural environments.^[1][2]

Schematic representation of the theoretical (T) and the empirical (E) horopter.

https://en.wikipedia.org/wiki/Horopter