In studies of binocular vision the horopter is the locus of points in space that yield single vision. This can be defined theoretically as the points in space which are imaged on corresponding points in the two retinas, that is, on anatomically identical points. An alternative definition is that it is the locus of points in space which make the same angles at the two eyes with the fixation lines. More usually it is defined empirically using some criterion.
History of the term 
The horopter was first discovered in the eleventh century by the Arabian or Persian scholar 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. It was only later that this line was shown to actually be a circular plane surrounding the viewer's head.
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 geometry that the horopter must be a circle passing through the fixation-point and the centers of the lenses 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. Howarth later clarified that the geometrical horopter is not a complete circle, but only its larger arc ranging from one nodal point (center of the eye lens) to the other.
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 empiral horopter deviates from the form expected on the basis of simple geometry.
Theoretical horopter 
Two theoretical horopter can be distinguished via geometric principles, depending on whether or not cyclorotation of the eyes is considered. Considering the general form of the points in space which make the same angles at the two eyes, when there is no cyclorotation, two components of the horopter can be identified. The first is in the plane which contains the fixation point (wherever it is) and the two nodal points of the eye. The 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 general form holds whether or not the fixation point is in the horizontal plane, and whether or not it is midway between the two eyes. As the fixation point recedes, the radius of the arc increases, and when fixation is at infinity the horopter takes on the special form of a plane perpendicular to the fixation line(s).
This description depends on the horopter being defined as the locus of points which make the same angle at the eyes - 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.
Empirical horopter 
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 around 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 empirical investigations used the criterion of singleness of vision, or absence of diplopia to determine the horopter. Other criteria used over the years include the drop-test horopter, the plumb-line horopter, and identical-visual-directions horopter, and the equidistance horopter. Most of this work has been confined to the horizontal plane or to the vertical plane.
The most comprehensive investigation of the three-dimensional volume of the empirical horopter used the criterion of identical visual directions, and found that the empirical horopter is a thin volume slanted back above the fixation point for medium to far fixation distances and surrounding the Vieth–Müller circle in the horizontal 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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