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# Geometry of eye convergence

Peter Torma, PhD - April 30, 2018 - 0 comments

It is well known that the pupils of the human eye moves closer to each other whenever the person is focusing on closer objects. The pupils are the farthest from each other when focusing on a very far (infinitely far) objects. The distance of the pupils in this case is called the pupillary distance. In this post we will show a realistic model of how much the pupils moves closer to each other, when the person is focusing on objects in different distances. As a result of the calculations we get that when one focuses to about reading distance than the eyes converge about $$1.0-2.5mm$$ depending on the actual reading distance.

### Calculations

Let

• $$d$$ be the distance at which the user focuses.
• $$r$$ be the radius of the eyeball. A typical value is $$13.5 mm$$.
• $$s$$ be the measured pupillary distance.
• $$\alpha$$ be the angle between the object and the line that connects the eye centers.
• $$c$$ be the correction required for measuring the PD, when the user focuses at infinity

Our aim is to calculate $$c$$ given all the other values.

At first observe that:

$$\tan \alpha = \frac{d}{s/2}.$$

Furthermore:

$$\cos \alpha = \frac{c}{r}.$$

Using the Pythagoras-law:

$$\sin \alpha = \frac{\sqrt{d^2 + (s/2)^2}}{d}.$$

Putting these altogether yields to:

$$c = r (s/2) \frac{\sqrt{d^2 + (s/2)^2}}{d^2}.$$

The image above shows this function for typical values of the PD and the eye-ball radius between $$0m$$ to $$5m$$. On can observe that the required correction is about a quarter mm for the half PD when the user focuses $$2m$$, in other words, when standing $$1m$$ in front of the mirror and watching herself.

The image above shows the same fuction when the user focuses close. One can see that when the user focuses to $$20cm$$ then the required correction is about $$2.3mm$$ for the half PD and when the user focuses to $$40cm$$ then the required correction is $$1.1mm$$ for the half PD. This is why the best practice is to measure the reading PD, since the natural reading distance vary from person to person.

MIRA uses the eye convergence in two ways. Firstly since it reconstructs the complete scene and hence knows how far the customer focused, it corrects the calculated PD value to infinity PD (the one that would have been measured if the user had focused on an infinitely far object instead looking into her own eye in the mirror. (Note that since the user stands about 1 meter from the mirror, when looking into her own eye the focus distance is the double of this value, so typically 2 meters.) MIRA also capable of measuring the near lookout point, which is identical to the reading PD in case of varifocal lens.