Converting diagonal field of view and aspect ratio to horizontal and vertical field of view

Note: this post was refined on Apr 23, 2016 to reflect a more accurate method of calculation.

Continuing the reference section started in the post regarding "TV screen size vs. goggle field of view", here is the conversion between diagonal to horizontal and vertical field of view. The conversion depends on the aspect ratio of the screen (ratio between width and height). The conversion table is below following by some explanations and discussion:

Converting diagonal FOV to horizontal and vertical FOV for various aspect ratios

HMD and goggle manufacturers usually like to present the diagonal field of view because that is the larger number, just like you read about "40 inch" TVs and not "34.9 inch" TV which would be the horizontal measure of a 40 inch TV with 16:9 aspect ratio, the common ration for HD1080 images. The aspect ratio used to be 4:3 for traditional televisions, VGA monitors and more. Over time, that aspect ratio grew and common ones are 3:2 for an iPhone 4, 16:9 (1.78:1) for HDTV and 2.40:1 for wide screen cinema. Keep in mind that these values are for the full width of the screen. If you take a screen and divide it into two for left and right eyes, the width of the screen area each eye is halved and so is the aspect ratio. This is why the table above includes 1:1 aspect ratio which is roughly what happens when you take a 16:9 screen and divide it into two.

If you don't know the aspect ratio, you can usually get it from the number of pixels. Assuming square pixels, the aspect ratio is the horizontal resolution divided by the vertical resolution.

The math is straightforward, but it involves a bit of trigonometry. Let’s take it step by step:

• If Df is the diagonal field of view and Ha:Va is the horizontal to vertical aspect ratio, we can find the corresponding diagonal size in the same units as the aspect ratio: 

Da = sqrt(Ha*Ha + Va*Va)

• The screen height and width are proportional to the tangent of the half angle. We use this to convert between field-of-view space and aspect-ratio space: 

Da = tan(Df/2) and Df = atan(Da) * 2

• If the tangent and arctangent functions operate in degrees, we get: 

Hf = atan( tan(Df/2) * (Ha/Da) ) * 2

• Here the tan() function converts from FOV to aspect-ratio space, the ratio is scaled in that space and then converted back into FOV space.

Wonkish note: As the field of view gets larger, the difference between the horizontal and vertical diminishes. Once the diagonal field of view reached 180 degrees, the horizontal and vertical fields of view also reach 180 degrees – every direction lies in the plane of the eye.

The curved monitor is back - on a diet

My post from several years ago on the Alienware Curved Monitor is - by far - the most visited page on this blog. Many people were excited at a combination of cool design, high resolution at that time and a feeling of immersion from the curved display. Unfortunately, the product never made it into production. Apparently, it was difficult to perform geometric and color matching of the individual displays that made the curved monitor.

Interestingly, another curved monitor is back. LG has recently announced the EA9800 curved monitor. It is a 55" HD1080 OLED monitor. Other than looking cool, the idea behind the curved design is that the distance between the eye and the center of the screen is the same as the distance between the eye and the edge of the screen. Thus, there are fewer distortions. The monitor is designed for TV viewing distances, so the curvature is not too big. Unlike the curved monitor of 5 years ago, this is is very thin.

Imagine if LG designed a smaller curved monitor that was designed for desktop or gaming use. The curvature on the screen would such that it would form a semi-circle around a user sitting less than a meter away from the screen. You could envision an immersive experience and a wide field of view - may 120 or 150 degrees - that is nearly impossible to achieve with a flat monitor.

When can I get one?

Haptic feedback: a dilemma for hand tracking technologies

A new 'reactive grip' controller prototype by Tactical Haptics (see video below) showcases an interesting dilemma for those seeking to add hand tracking to the experience of virtual reality goggles. Simply put, the dilemma is "to bare hand or not to bare hand"

"Bare hand" tracking technologies such as the Microsoft Kinect or those from Leap Motion are performed in the most natural way: there is no need to hold a controller, wear a glove or use any accessory. There is no equipment to lose, no wires to untangle and no need to fit something to your hand size. Of course, your hands are also free to do other things - such as type on a keyboard - without having to put the controller down.

Controller-based hand tracking devices such as the Hydra from Razer currently offer more sophisticated control by allowing the user to operate pushbuttons, switches and analog sticks. One would expect that over time, you could get the same precision from "bare hand" trackers.

What you can't get from a bare hand tracker is the haptic feedback, such as vibration, tactile feedback or reactive grip. Is this feedback valuable enough to warrant holding a controller in your hand or will most users choose "bare hand" trackers?

TV screen size vs. goggle field of view

Consider this a public service announcement. Many consumer goggle vendors like to market the visual experience of using their goggles by mentioning an equivalent TV viewing experience. For instance, "using our goggles is like watching a 72 inch TV from 10 feet away". This might sound impressive - who wouldn't want a 72 inch TV set - but is actually not so much. A 72 inch TV from 10 feet away provides just over 33 degree field of view, which is very narrow for a goggle.

The table below summarizes these conversions. Select the TV size from the top (in inches) or bottom (in centimeter), select the viewing distance from the left (in feet) or the right (in meters) and you get the equivalent diagonal field of view.

Conversion from TV diagonal size to HMD field of view

The math is not complex. The diagonal field of view is a function of S, the diagonal screen size and D, the viewing distance:

Field of View = 2 * arctan ( (S/2) / D )

Just make sure that S and D are using the same units and remember that most arctan functions return the value in radians, not degrees.

From the table, you can see that a standard-issue professional HMD with 60 degree field of view is equivalent to about a 70" screen from 5 feet away whereas a wide field of view device such as the xSight 6123 with 123 degrees field of view is like a 90" screen from 2 feet away.

How much field of view is enough? This is truly application-dependent. Most people prefer watching TV on a 50 inch screen instead of a 24 inch screen, but would prefer using a 24 inch computer monitor for word processing to a 50 inch monitor. Wide field of view - and thus immersion - is great for gaming. High pixel density - and thus narrowed field of view with a given resolution - is better for seeing details.

To illustrate the difference between TV requirements and goggle requirements, consider this: for TVs, THX recommends “best seat-to-screen distance” is one where the view angle approximates 40 degrees. 40 degrees in goggles is quite small.

UPDATE: Please also see the matching post on "converting diagonal field of view and aspect ratio to horizontal and vertical field of view"