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

In vision science, 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. He built on the binocular vision work of Ptolemy and discovered that objects lying on a horizontal line passing through the fixation point resulted in single images, while objects a reasonable distance from this line resulted in double images. Thus Alhazen noticed the importance of some points in the visual field but did not work out the exact shape of the horopter and used singleness of vision as a criterion.

The term horopter was introduced by Franciscus Aguilonius in the second of his six books in optics in 1613. In 1818, Gerhard Vieth argued from Euclidean geometry that the horopter must be a circle passing through the fixation-point and the nodal point of the two eyes. A few years later Johannes Müller made a similar conclusion for the horizontal plane containing the fixation point, although he did expect the horopter to be a surface in space (i.e., not restricted to the horizontal plane). The theoretical/geometrical horopter in the horizontal plane became known as the Vieth-Müller circle. However, see the next section Theoretical horopter for the claim that this has been the case of a mistaken identity for about 200 years.

In 1838, Charles Wheatstone invented the stereoscope, allowing him to explore the empirical horopter. that Müller's anatomical approximation that the nodal point and eye rotation center are coincident should be refined. Unfortunately, their attempt to correct this assumption was flawed, as demonstrated in Turski (2016). This form was predicted by Helmholtz and subsequently confirmed by Solomons. or just Panum's area, although recently that has been taken to mean the area in the horizontal plane, around the Vieth-Müller circle, where any point appears single.

These early empirical investigations used the criterion of singleness of vision, or absence of diplopia to determine the horopter. Today the horopter is usually defined by the criterion of identical visual directions (similar in principle to the apparent motion horopter, according that identical visual directions cause no apparent motion). Other criteria used over the years include the apparent fronto-parallel plane horopter, the equi-distance horopter, the drop-test horopter or the plumb-line horopter. Although these various horopters are measured using different techniques and have different theoretical motivations, the shape of the horopter remains identical regardless of the criterion used for its determination.

Consistently, the shape of the empirical horopter have been found to deviate from the geometrical horopter. For the horizontal horopter this is called the Hering-Hillebrand deviation. The empirical horopter is flatter than predicted from geometry at short fixation distances and becomes convex for farther fixation distances. Moreover the vertical horopter have been consistently found to have a backward tilt of about 2 degrees relative to its predicted orientation (perpendicular to the fixation plane). The theory underlying these deviations is that the binocular visual system is adapted to the irregularities that can be encountered in natural environments. It is generally a twisted cubic. A twisted cubic can be represented parametrically as x = x(θ), y = y(θ), z = z(θ) where x(θ), y(θ), z(θ) are three independent third-degree polynomials. In some degenerate configurations, the horopter reduces to a conic curve.

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