In studies of binocular vision

Binocular vision

Binocular vision is vision in which both eyes are used together. The word binocular comes from two Latin roots, bini for double, and oculus for eye. Having two eyes confers at least four advantages over having one. First, it gives a creature a spare eye in case one is damaged. Second, it gives a...

 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 retina

Retina

The vertebrate retina is a light-sensitive tissue lining the inner surface of the eye. The optics of the eye create an image of the visual world on the retina, which serves much the same function as the film in a camera. Light striking the retina initiates a cascade of chemical and electrical...

s, 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

Ptolemy

Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...

 and discovered that objects lying on a 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

Optics

Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

 in 1613. In 1818, Gerhard Vieth argued from geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

 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

Johannes Peter Müller

Johannes Peter Müller , was a German physiologist, comparative anatomist, and ichthyologist not only known for his discoveries but also for his ability to synthesize knowledge.-Early years and education:...

 made a similar conclusion for the horizontal plane containing the fixation point, although he did expect the horopter to be a surface in space. The theoretical/geometrical horopter in the horizontal plane became known as the Vieth-Müller circle and it was only some 200 years later that Howarth pointed out the error in this analysis, and showed that in the fixation plane containing the fixation point and the two nodal points, the geometrical horopter is not a complete circle, but only the larger arc.

In 1838, Charles Wheatstone

Charles Wheatstone

Sir Charles Wheatstone FRS , was an English scientist and inventor of many scientific breakthroughs of the Victorian era, including the English concertina, the stereoscope , and the Playfair cipher...

 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 going from one nodal point to the other in space, passing through the fixation point. The second component is a 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 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' (Schreiber at al. 2006) .

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

Diplopia

Diplopia, commonly known as double vision, is the simultaneous perception of two images of a single object that may be displaced horizontally, vertically, or diagonally in relation to each other...

 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 horiontal 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.

Schreiber et al. (2008) 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

Computer vision

Computer vision is a field that includes methods for acquiring, processing, analysing, and understanding images and, in general, high-dimensional data from the real world in order to produce numerical or symbolic information, e.g., in the forms of decisions...

, the horopter is defined as the curve of points in 3D space having identical coordinates projection

Projection (mathematics)

Generally speaking, in mathematics, a projection is a mapping of a set which is idempotent, which means that a projection is equal to its composition with itself. A projection may also refer to a mapping which has a left inverse. Bot notions are strongly related, as follows...

s with respect to two cameras with the same intrinsic parameters. It is given generally by a twisted cubic

Twisted cubic

In mathematics, a twisted cubic is a smooth, rational curve C of degree three in projective 3-space P3. It is a fundamental example of a skew curve. It is essentially unique, up to projective transformation...

, 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 polynomial

Polynomial

In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

s. In some degenerate configurations, the horopter reduces to a line plus a circle.