Frame91 - 2 months ago 10

Android Question

Currently I'm writing an augmented reality app and I have some problems to get the objects on my screen. It's very frustrating for me that I'm not able to transform gps-points to the correspending screen-points on my android device. I've read many articles and many other posts on stackoverflow (I've already asked similar questions) but I still need your help.

I did the perspective projection which is explained in wikipedia.

What do I have to do with the result of the perspective projection to get the resulting screenpoint?

Answer

The Wikipedia article also confused me when I read it some time ago. Here is my attempt to explain it differently:

Let's simplify the situation. We have:

- Our projected point D(x,y,z) - what you call
*relativePositionX|Y|Z* - An image plane of size
*w***h* - A half-angle of view
*α*

... and we want:

- The coordinates of B in the image plane (let's call them
**X**and**Y**)

A schema for the X-screen-coordinates:

E is the position of our "eye" in this configuration, which I chose as origin to simplify.

The focal length *f* can be estimated knowing that:

`tan(α) = (w/2) / f`

*(1)*

You can see on the picture that the triangles *ECD* and *EBM* are **similar**, so using the Side-Splitter Theorem, we get:

`MB / CD = EM / EC`

<=>`X / x = f / z`

*(2)*

With both ** (1)** and

`X = (x / z) * ( (w / 2) / tan(α) )`

If we go back to the notation used in the Wikipedia article, our equation is equivalent to:

`b_x = (d_x / d_z) * r_z`

You can notice we are missing the multiplication by `s_x / r_x`

. This is because **in our case, the "display size" and the "recording surface" are the same**, so `s_x / r_x = 1`

.

Note: Same reasoning for

Y.

Some remarks:

- Usually,
*α = 90deg*is used, which means`tan(α) = 1`

. That's why this term doesn't appear in many implementations. If you want to preserve the ratio of the elements you display, keep

*f*constant for both**X**and**Y**, ie instead of calculating:`X = (x / z) * ( (w / 2) / tan(α) )`

and`Y = (y / z) * ( (h / 2) / tan(α) )`

... do:

`X = (x / z) * ( (min(w,h) / 2) / tan(α) )`

and`Y = (y / z) * ( (min(w,h) / 2) / tan(α) )`

Note: when I said that "

*the "display size" and the "recording surface" are the same*", that wasn't quite true, and theoperation is here to compensate this approximation, adapting the square surface*min**r*to the potentially-rectangular surface*s*.Note 2: Instead of using

*min(w,h) / 2*, Appunta uses`screenRatio= (getWidth()+getHeight())/2`

as you noticed. Both solutions preserve the elements ratio. The focal, and thus the angle of view, will simply be a bit different, depending on the screen's own ratio. You can actually use any function you want to define*f*.As you may have noticed on the picture above, the screen coordinates are here defined between

*[-w/2 ; w/2]*for X and*[-h/2 ; h/2]*for Y, but you probably want*[0 ; w]*and*[0 ; h]*instead.`X += w/2`

and`Y += h/2`

- Problem solved.

I hope this will answer your questions. I'll stay near if it needs editions.

Bye!

< Self-promotion Alert >I actually made some time ago an article about 3D projection and rendering. The implementation is in Javascript, but it should be quite easy to translate.