3D Harms coordinate system (under construction)

Note: If you really want to look at something in the present page: we recommend the introduction and Figures 1 and 2. The other figures are OK except that they display a purple full plane instead of a blue half-plane as correctly shown in Figures 1 and 2, and explaining text is still missing.

Introduction

The 3D Harms coordinate system belongs to the class of spherical coordinate systems. The 3D Harms coordinate system is used specifically either by vision scientists, e.g. in the field of binocular disparity, or by ophthalmologists, e.g. in the field of strabology (Read & Cumming, 2008, Dysli et al., 2021).

This coordinate system is often called an azimuth-longitude / elevation-longitude system (Howard & Rogers, 2008 - chapter 20, Schor's lab - chapter 3).

A previous section explains the different spherical coordinate systems used to specify the azimuth and the elevation of a point (Class of spherical coordinate systems).

In the Harms system, the 3D position of a point P is defined by 3 coordinates:

1. the azimuth-longitude.
   
2. the elevation-longitude.
   
3. the radial distance.
   

These 3 coordinates are described in details in the 3 following sections.

1. The azimuth-longitude
This is the dihedral angle between a vertical half-plane passing through P (purple plane in the figures below) and the zero azimuth half-plane (i.e. the YZ plane in PTVR).

The azimuth-longitude of Point P is -60° in figures 1 and 2 (-180° < azimuth-longitude < 180°).

Figure 1: The point P has an azimuth angle of -60° (brown angle) with respect to the YZ half-plane (i.e. it lies on the purple half-plane). The elevation of point P is 19.96° (blue angle).	Figure 2: Animation (horizontal rotation about the Y axis) of figure 1 to allow you to build a better mental representation of its 3D structure (as if you were able to look at Figure 1 from different angles).

😉 Remark: Note that the RADIAL DISTANCE of P is constant in Figures 1 to 8.

2. The elevation-longitude
This is the dihedral angle between an elevated plane passing through the point and the zero elevation plane. [To be completed]

(from -90° to 90°) The elevation-longitude_poles_on_X_axis

Figure 3: .	Figure 4: [To be completed] Animation (horizontal rotation about the Y axis) of figure 3 to allow you to build a better mental representation of the 3D structure of the figure (as if you were able to look at the static coordinate system from different angles)..

3. The radial Distance (in meters) – corresponds to the radius of the sphere on which P is lying (i.e. constant distance between Origin and P).

[To be completed]

Figure 5: elevation from 0° to 84° with constant azimuth (here 0°). Note the iso-azimuth trajectory formed by the path of P on the 0° longitude going through the Y axis poles.	Figure 6: elevation from 0° to 84° with constant azimuth (here 60°). Note the iso-azimuth trajectory formed by the path of P on the 60° longitude going through the Y axis poles.

Figure 7: azimuth from 0° to 135° back and forth with constant elevation (here 0°). Note the iso-elevation trajectory formed by the path of P on the 0° longitude going through the Z axis poles.	Figure 8: azimuth from 0° to 179° back and forth with constant elevation (here 30°). Note the iso-elevation trajectory formed by the path of P on the 30° longitude going through the Z axis poles.

😉 Remark: Note that the RADIAL DISTANCE of P is NOT constant in the following Figures (in contrast with the above figures).

Figure 9: Special case when P is on a tangent Screen: ie radial distance varies. [To be completed]	Figure 10: [To be completed]

Important reference for understanding spherical coordinate systems in vision science:

Schor, C. M. Chapter 3—KINEMATICS : COORDINATE SYSTEMS FOR DESCRIBING EYE POSITION. Retrieved July 30, 2021, from http://schorlab.berkeley.edu/passpro/oculomotor/html/chapter_3.html#coord

Howard, I. P., & Rogers, B. J. (2008). chap. 20. Classification of binocular disparity. In Seeing in Depth (Vol. 2). Oxford University Press. (Web link)

Wikipedia contributors. (2022, February 16). Spherical coordinate system. In Wikipedia, The Free Encyclopedia. Retrieved 15:18, February 21, 2022, from https://en.wikipedia.org/w/index.php?title=Spherical_coordinate_system&oldid=1072122604