Dot Pitch

With all the rumours of Apple announcing a large-format retina display, it got me thinking: If they did release a 27″ retina display, just how high of a resolution would it be?

To figure this we first need to know the dot pitch of Apple’s small retina display. The dot pitch is the distance between the subpixels of a display and is an indication of how sharp the image will appear.

You can calculate the dot pitch by first finding the screen’s resolution in pixels per inch. However, since screen resolution is typically measured as horizontal pixels by vertical pixels, and screen size is measured in diagonal inches we need to figure out what the screen’s resolution is in diagonal pixels. We can do this using the Pythagorean theorem:  Where a = horizontal resolution, b = vertical resolution, and c = diagonal resolution.

The iPhone retina display has a resolution of 960×640 so the diagonal resolution is:  = 1153.78. We now need to get the pixels per inch which we can do by taking the screen’s resolution and dividing it by the screen’s diagonal size which is 3.5″: 1153.78 / 3.5 = 329.65 pixels per inch.

Now that we have the resolution in pixels per inch we want to find out the dot pitch. Dot pitch is typically measured in millimeters per dot. So to convert dots per inch to millimeters per dot we divide the ppi value into the number of millimeters per inch: 25.4/329.65 = 0.077mm or 77μm which is pretty close to Apple’s claim of 326ppi and 78μm between pixels.

Now what we know what dot pitch qualifies as a “retina display” we want to find out given a certain dot pitch, aspect ratio, and screen size; what will the resolution of that display be?

Let’s say we want a 27″ 16:9 monitor with a retina display dot pitch of 0.077m. What we want to find is the scaling factor that we need to multiply the aspect ratio by to get the number of pixels.

If we take the equation for dot pitch, plug in what we know and put in variables for what we want to know (the scaling factor) we get this equation: . Now we have to solve for a.

First let’s simplify the bottom part.  can be simplified as  which equals . Once we add the square root we get . If we assume that a is positive then the square root of a² is a, and the square root of 337 is 18.36. That leaves us with . Which we can simplify by doing 18.36/27 = 0.68 then 25.4/0.68 = 37.35. So now we have  and can solve for a by dividing 37.35 by 0.077 to get a scaling factor of 485.06. Then we just multiply the scaling factor by our aspect ratio of 16:9 to get: (16*485.06):(9*485.06) which equals a resolution of roughly 7,760×4,365. Which would be a 33.8 megapixel display.

Incidentally, this is approximately equivalent to the defined resolution of UHDTV (Ultra High-Definition TV) which has a resolution of 7680×4320.