Mathematical definition

In the common language of Vision Science, the definition of a tangent screen (an important visual object implemented in PTVR) is based on the mathematical definition of the tangent plane to a surface.

The tangent plane to a surface at a given point p is defined in an analogous way to the tangent line in the case of curves. It is the best approximation of the surface by a plane at p, and can be obtained as the limiting position of the planes passing through 3 distinct points on the surface close to p as these points converge to p.

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... the tangent plane to a surface at a given point is the plane that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized.
The word "tangent" comes from the Latin tangere, "to touch".

(retrieved from Wikipedia contributors, 'Tangent', Wikipedia, The Free Encyclopedia, 9 June 2022, 01:05 UTC, https://en.wikipedia.org/w/index.php?title=Tangent&oldid=1092236138 [accessed 14 July 2022] )

Vision Science terminology

In vision science, the tangent plane is usually called a tangent screen (this is also the terminology used by PTVR) by reference to the common use of monitor screens in Vision experiments, and it is actually a tangent plane to a sphere because it is related to the use of **spherical** coordinate systems.

As stated above, the strict definition of a tangent screen (or plane) tangent to a sphere implies that the screen must touch the sphere. This is the case for instance in Figures 1 and 2 that will be explained in more detais here in the User Manual.

Figure 1: Grey Tangent screen at one meter from the Origin on the Z axis. The screen center is called SO (Screen Origin).

Figure 2: Animation (horizontal rotation about Y) of figure 1 to allow you to build a better mental representation of the 3D structure of the figure (as if you were able to look at it from different angles)..

Note however that you will find figures in the PTVR documentation where a Tangent Screen will not touch the sphere displayed in the figure. An example, that is presented in details [here in the sub-section after the next](_2DperimetricCoordinateSystemInVS), is shown in Figure 3.

Figure 3: Projection of a point P, lying on the sphere, onto a tangent screen.

The important point in Figure 3 is that the tangent screen is tangent to a sphere (not displayed) whose radius equals the distance between the Origin Point and the SO point. For visual clarity, this sphere is not shown and instead a sphere with a smaller radius is displayed as the projection of point P is the same whatever the sphere's radius (this is actually one important property of the spherical coordinate systems).