In studies of binocular vision the horopter is the locus of points in space have the same disparity as fixation. This can be defined theoretically as the points in space which 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.
History of the term
The horopter as a special set of points of single vision was first mentioned in the eleventh century by Ibn al-Haytham, known to the west as "Alhazen". 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. He found that there were many points in space that yielded single vision; this is very different from the theoretical horopter, and subsequent authors have similarly found that the empirical horopter deviates from the form expected on the basis of simple geometry.
Later Hermann von Helmholtz and Ewald Hering worked out the exact shape of the horopter almost at the same time. Their descriptions identified two components for the horopter for symmetrical fixation closer than infinity. The first is in the plane which contains the fixation point (wherever it is) and the two nodal points of the eye. The geometric locus of horopteric points in this plane takes the form of the arc of a circle (the Vieth-Müller circle) going from one nodal point to the other in space, passing through the fixation point. The second component is a line (the Prévost–Burckhardt line) which is perpendicular to this arc, cutting it at the point midway between the two eyes (which may, or may not, be the fixation point). This horopter geometry of a circle in the fixation plane and a perpendicular line remains true as long as the eyes are fixating somewhere on these two lines. If the eyes are in primary position, i.e., fixating straight ahead at infinity, the Vieth-Müller circle has infinite radius and the horopter becomes the two-dimensional plane through the two straight horopter lines.
The identification of the theoretical/geometrical horopter with the Vieth-Müller circle should be reconsidered. It was pointed out in Gulick and Lawson (1976)  that Müller's anatomically incorrect assumption that the nodal point and eye rotation center are coincident should be rejected. Unfortunately, their attempt to correct this assumption was flawed, as demonstrated in Turski (2016). In fact, for a given fixation point, one has a different horopter circle for each different choice of the nodal point’s location. Moreover, if one changes the fixation point along a given Vieth-Müller circle such that the vergence value remains constant, one obtains an infinite family of such horopters, so long as the nodal point is not at the eye’s rotation center. These statements follow rather simply from the Central Angle Theorem and the fact that three non-collinear points give a unique circle. What is less simple to show is the fact that, for fixations along a given Vieth-Müller circle, all the corresponding horopter circles intersect at the point of symmetric convergence. This result implies that each member of the infinite family of horopters is also composed of a circle in the fixation plane and a perpendicular straight line passing through the point of symmetric convergence (located on the circle) so long as the eyes are in primary or secondary position.
When the eyes are in tertiary position, the vertical disparities due to the differential magnification of the distance above or below the Vieth-Müller circle have to be taken into account, as was calculated by Helmholtz. In this case the horopter becomes a single-loop spiral passing through the fixation point and converging toward the vertical horopter at the top and bottom extremities and passing through the nodal point of the two eyes.
This description depends on the horopter being defined as the locus of points which make the same angle at the eyes (i.e. that have zero relative disparity with fixation) - which was the original definition used by Aguilonius. If one considers a slightly different definition, based on the projections into space of corresponding retinal points, then Schreiber and colleagues have shown that a different theoretical form emerges. As Helmholtz predicted, and Solomons subsequently confirmed, in the general case which includes cyclorotation of the eyes, the theoretical horopter takes the form of a twisted cubic.
As Wheatstone (1838) observed, the empirical horopter, defined by singleness of vision, is much larger than the theoretical horopter. This was studied by P. L. Panum in 1858. He proposed that any point in one retina might yield singleness of vision with a circular region centred on the corresponding point in the other retina. This has become known as Panum's fusional 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 horop[ter this is called the Hering-Hillebrand deviation. The empirical horopter is flattefd 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).
Horopter in computer vision
In computer vision, the horopter is defined as the curve of points in 3D space having identical coordinates projections with respect to two cameras with the same intrinsic parameters. It is given generally by a twisted cubic, i.e., a curve of the form 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 line plus a circle.
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