Tangent Screen with 2D Perimetric coordinates

Different 2D coordinate systems can be used to define the position of a point P on a flat tangent screen that is itself placed in a 3D space. One of them, the cartesian system, was described in a previous section. Another one is very important in vision science: this is the 2D perimetric system that was also already presented in 3D in another previous section. Here we illustrate how 3D projective geometry can be used to define a 2D perimetric system on a tangent screen.

In the figures below, a point P lying on a sphere is moving while staying at the surface of the sphere. This point P is projected on a 2D screen that is tangent to the sphere at the point where the Z axis meets the 2D screen. Note that this point is called the Screen Origin (SO) of the tangent screen in the PTVR documentation.

😉 Remark: For simplicity, as in the previous section, the position of the tangent screen is chosen so that its origin (i.e. SO) lies on the Z axis.

In these figures, the projection of point P is traced on the 2D tangent screen as the point moves.

Figure 1: Projection of point P onto a tangent screen. The motion of P is defined in a perimetric coordinate system with Eccentricity varying from 6° to 36° while keeping Half-meridian constant at 180°.	Figure 2: Projection of point P onto a tangent screen. The motion of P is defined in a perimetric coordinate system with Eccentricity varying from 6° to 36° while keeping Half-meridian constant at 270°.

Figure 3: Projection of point P onto a tangent screen. The motion of P is defined in a perimetric coordinate system with half-meridian varying from 0° to 360° while keeping eccentricity constant at 15°.	Figure 4: Projection of point P onto a tangent screen. The motion of P is defined in a perimetric coordinate system with half-meridian varying from 0° to 360° while keeping eccentricity constant at 30°.

Important references for understanding the use of projections in vision science:

section 3.8 : Types of geometry in: Howard, I., & Rogers, B. (2008-02-01). Psychophysics and analysis. In Seeing in Depth: Volume 1: Basic Mechanics/ Volume 2: Depth Perception 2-Volume Set. : Oxford University Press. Retrieved 14 Jan. 2022, from https://oxford.universitypressscholarship.com/view/10.1093/acprof:oso/9780195367607.001.0001/acprof-9780195367607-chapter-3.

Section 24.1 : Perspective in: Howard, I., & Rogers, B. (2008-02-01). Depth from monocular cues and vergence. In Seeing in Depth: Volume 1: Basic Mechanics/ Volume 2: Depth Perception 2-Volume Set. : Oxford University Press. Retrieved 14 Jan. 2022, from https://oxford.universitypressscholarship.com/view/10.1093/acprof:oso/9780195367607.001.0001/acprof-9780195367607-chapter-24.