Augmented Reality for most seems like a magical thing. How can our world and a virtual world co-exist? Books have been written on the topic, movies have been produced on the topic. Companies worth billions have been founded on this topic.

Meta Magic Leap Holo-lens Vuzix AR MIX AR Headset …

Here’s the surprising part. The math behind the basics of Augmented Reality is something many high school AP Physics students learn, but few understand. Let’s dig deeper.

Lenses, Lenses, more Lenses

There are many different kinds of lenses. The two most common types are concave and convex lenses.

Concave lenses spread out light rays while convex lenses focus them.

Concave vs Convex lenses

For AR we’ll be interested in the Convex lens. A lens has a focal point, that’s defined below:

Focal Point of a lens

If you’ve ever taken a magnifying glass and focused the light to burn a leaf as a kid then you know what a focal point is. The distance between the lens and the point where the light meets is the focal point.

Thin Lens Equation

The next concept to understand is the Thin Lens Equation. This equation explains objects, focal length and images. In the case of AR let’s define those three terms.

Object: The Projected AR Screen

Image: The Retina of the user of the AR Headset

Focal Length: Determined by the lens used in the AR Headset, remember it’s a property of the lens itself.

To derive the equation, let’s take a look at three rays originating from the object, or the Screen.

Light usually doesn’t follow parallel lines (unless it’s a laser), it emits rays in all directions, and we are looking at three of those ray’s at one pixel. When the light passes through the lens, the other side looks like this:

They converge to a point on the other side (This is a point on the projected image). Now let’s take a look at all of the ray’s that are perpendicular to the screen look like first.

As you can see they converge to a focal point and pass through it. If you take this a step further, and add three rays to each point (The rays get a bit complicated here).

Then you end up with something like this.

I have four points of origin on the screen, and those four points converge to an image on the right hand side where I put the eyeball.

The origin is the screen, the focal point is the distance from the centerline to the two dark dots on either side of the lens, and the Image is the green line near the eyeball portrayed above.

The mathematical relationship between the points is given below (here’s a link to explain further):

Eq 1: f is the focal length, o is the object distance from the lens, and i is the image distance from the lens

To recap the focal length, the image distance and the object distance are shown below:

Mirrors, Mirrors, a room full of Mirrors

Now this setup would be perfect for a VR headset but, doesn’t quite do what we want it to, in the case of AR. The screen blocks the view of the user causing him or her to not be able to see the surroundings. There are a couple of ways to solve this problem

First, a camera on the other side of the screen can capture the surroundings and display that to the user. The shortcomings of this approach are, the surroundings become pixelated, and the field of view of the user is severely restricted to the field of view of the headset.

Second to have a screen that let’s light through at some angle and reflects light at some angle. Much better field of view than the camera approach, however displaying color and blacks becomes a problem, though companies have come up with innovative solutions to these problems.

Imagine a piece of glass, at some angles you can see your reflection through on the glass even though you can see through it.

See the reflection of the clouds on the mirror, even though you can see straight through.

This is the method used in many AR headsets like the Meta 2. We have all the tools to design our own headset.

The Headset

The final design of the headset looks like the diagram above. Taking into consideration the reflection the object distance is the purple, and the image distance is the sum of the two green lines.

For the headset below the object distance was (71mm) and the image distance (123mm), using the equation (eq 1) gives a focal length of 45mm. Which is exactly the focal length of each lens:

In the next article I’ll be going in detail over how to build the headset with downloadable 3d printable files and cardboard templates for those who don’t have 3d printers.

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