Documentation for Users
1.0.2
Perception Toolbox for Virtual Reality (PTVR) Manual

Most of the time, vision scientists are not interested in the euclidean distances (aka linear distances) measured in the 3D world. They want to know how visual objects project onto the retina. In other words, knowing that an object is 10 meters high is not very informative. What vision scientists need is a description of the visual input at the retinal level: the basic question is therefore to assess the angular values of the projected images of objects onto the retina. This has been traditionnally illustrated with schematic graphs similar to the one shown in Figure 1 (Wikipedia link). Here, an observer's eye looks at an arrow (segment BA) displayed in a frontal plane. The euclidean distance (often called linear size) between the arrow's endings B and A is S meters. The euclidean distance between the midpoint of the arrow and a special optical point O in the eye (called the fused/united/composite nodal point) is D meters (for accurate details see for instance Atchison & Smith (2000). This euclidean distance is often called the monocular Viewing Distance.
 
Figure 1: Typical figure and formula used to define a visual angle with a schematic eye. The point O (fused nodal point) is the point at which the angle subtended "at the eye" by the BA arrow is measured. (image from Ojosepa, CC BY 3.0, via Wikimedia Commons)  Figure 2: Animation of Figure 1 to illustrate how the visual angle V depends on the viewing distance D for a constant Euclidean distance S (here 0.2 meters) between the endings A and B of the vertical arrow. Note that the point O represents the fused nodal point displayed in figure 1. 
The traditional way of measuring a visual angle (indicated in figure 1) is the following:
V = 2 * arctan ( S / (2 * D) ) **(formula 1)**
where S and D stand respectively for the linear Size of the BA segment and for the viewing distance.
This formula has been extensively used for instance when a stimulus of constant linear size (here S = 0.2 meters) is displayed at different viewing distances (D) as illustrated in Figure 2.
Formula (1) is a good approximation for most purposes in Vision Science. However, it must be born in mind that this approximation might not be good enough in certain experimental situations. We therefore emphasize below the conditions of validity of formula (1).
As emphasized in Figures 3 and 4, the visual angle provided by formula (1) is exact if and only if the two requirements below are met:
 
Figure 3: Note in this figure that the Screen Origin (SO) of a tangent screen is always the orthogonal projection of point O on the tangent screen.  Figure 4: Visual angle subtended by vertical BA segment decreases as euclidean distance from SO to the midpoint M of BA gets larger. The key point here is to note that the visual angle predicted by formula (1) is always identical to the actual angle V. As in figures 1 and 2, the Euclidean distance S (here 0.2 meters) between the vertical arrow's endings A and B is constant. 
Very often in Vision Research, although visual stimuli lie on a tangent screen (i.e. they meet the first requirement of formula (1)), they do not meet the second requirement described above (i.e. angle OMA, or equivalently OMB, should be a right angle). This is not an issue for most purposes because the difference between visual angle predicted by formula (1) and actual angular value is usually small.
For instance, in figure 5, the position of point M is now such that angle AMO (108.32°) is not a right angle any longer. The consequence is that the actual value of the visual angle V is 17.09° (in orange) whereas its predicted value from formula (1) is 17.95°, a small enough difference for many purposes.
Figure 5: Example of a segment BA lying on the tangent screen and whose angle V subtended at the eye (i.e. at point O) is slightly different from the angular value predicted by formula (1).
Atchison, D. A., & Smith, G. (2000). Optics of the Human Eye. Elsevier. https://doi.org/10.1016/B9780750637756.X50019
Pelli, D. G., Tillman, K. A., Freeman, J., Su, M., Berger, T. D., & Majaj, N. J. (2007). Crowding and eccentricity determine reading rate. J Vis, 7, 20 136. https://doi.org/10.1167/7.2.20. (their Appendix B is especially relevant here)
Wikipedia contributors. (2021, April 16). Visual angle. In Wikipedia, The Free Encyclopedia. Retrieved 17:11, February 15, 2022, from https://en.wikipedia.org/w/index.php?title=Visual_angle&oldid=1018114704