Is That a Point?

If I gave you a sheet of paper and asked you to draw me a nickel, you’d probably draw me a circle roughly an inch in diameter, maybe labeled with a "5₵". If I asked you to draw me a mite, the smart alecks out there would probably dot the piece of paper and give it back. If I asked for a realistic drawing of an atom, the same group would likely hand me back the blank page. This sort of answer is likely meant to be taken as a joke, but it also provides us with an insight into the limitations of human visual resolution.

Human eyes are anything but perfect, and some are less perfect than others. And as the silly drawings of the mite and the atom suggest, the smaller the object the harder it is for the human eye to resolve. By the international standard of measuring human visual resolution, good vision is defined as 20/20 (feet system) or 6/6 (meter system), meaning that someone with 20/20 vision can resolve what a person should be able to see (by designation) 20 ft away at the intended distance [1]. Someone with 20/40 vision sees at 20 ft what someone with 20/20 vision can see at 40 ft away, and someone with 20/10 vision sees at 20 ft what someone with 20/20 vision can see at 10 ft away. As objects get smaller, they seem to us as tending to a point. This occurrence is often referenced in physics where a source with radius r a distance d away from a sensor where d>>r (d is much greater than r) can be approximated as a point source.

But how exactly do we characterize this phenomenon, and at what distance does leaning forward while squinting intently seem… pointless? Let's take a look at the diagram below:

Fig. 1: A Spherical Object Viewed at Different Distances (Orig.)

 On the left side there is a spherical object of radius r₁ being observed by the first eye at a distance d₁ so that the object fills the observer’s field of view (denoted by the first set of dashed lines). The circle surrounding the object with radius r₂ represents the same observer’s field of view at a distance d₂ where d₂>d₁. The corresponding angles are drawn in and labeled as θ₁ and θ₂. From this diagram, we can obtain the equations

                                                                         1.       r₁=d₁tan θ₁

                                                                         2.       r₁=d₂tan θ₂

                                                                         3.       r₂=d₂tan θ₁

Dividing equation 1 by equation 3, we receive the statement

                                                                        4.       r₁/r₂= d₁/d₂

This equation tells us that as d₂ increases, the ratio of the original full-view object radius to the radius of the field of view at d₂ decreases as d₁/d₂=k/d₂ α 1/x (k is used to show that d₁ is a constant). The graph of 1/x is shown below:

Fig. 2: Graph of the Function f(x)=1/x (WyzAnt)

Where the x axis represents an increasing distance d₂ and the y axis represents the ratio r₁/r₂. This finding seems to agree with our experiences, doesn’t it? Namely, as we walk further away from an object it seems to decrease in size relative to our entire field of view, the size difference becoming less noticeable as distance increases further. If we were to walk far enough away, then the object would appear as if it were a point in our vision.

So we now know how to describe the way objects seem to decrease in size at far distances, but at what distance does any sort of difference in the object not matter? That is, when does the object become a point? Based on the human vision resolution assessment described earlier, we know that at 6m, or 20ft, the idealized human should be able to resolve a standardized interval of one arc minute, or 1/60° [2]. From rearranging equation 3, we receive the form

                                                                         5.       d₂=r₁/tan θ₂

Making the substitution of the maximum resolution for a 20/20 person’s vision, 1/60°, for θ₂, we produce the equation

                                                                          6.       d₂=3,438r₁

This equation says that an object with a radius r₁ can be seen from a distance a factor of 3,438 times its radius before a person with good vision can no longer sense its character beyond that of a point. This is like trying to see the face of a nickel from 146 meters away. While seemingly too far a distance to be visible, if we check equation 6 by substituting in the resolution corresponding to an arc minute at 6m, which is 1.75mm [3] (this can be checked with equation 1), then the distance returned is indeed 6m. With this information in mind, the next time you look up into the night sky I challenge you to think about just how big a star is compared to the twinkling dots you see above. Suddenly, it won’t seem so crazy to say we all live on the head of a pin.

Thank you guys for reading this post, and be sure to look out for more to come. If you want to be updated whenever I make a new post, sign up for email notifications on the right side column of the blog so you can have new posts sent directly to your email. Alternatively, if you send me a message on Google+ I'll add you to be notified when I post new content. If you have any suggested topics for future posts or you liked this post, let me know in the comments because I’d love to hear your thoughts. Thanks!