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Some problems met in the realization of a spherical projection with mirror

- Pixel dimension ::
Half a circumference of my screen measures 3.14159 x 65cm = 204cm. One half-hemisphere of 65cm of radius thus has a surface equivalent to a flat screen 2 metres wide. As they are observed from a distance of 65cm, pixels are very visible.

- Pixel quality : :
in the current video projector the three red, green and blue constituents do not converge exactly on the screen and give of the pixel an image more or less blurred according to the models.
On LCD projector, elements are deliberately brought forward to reduce the visible size of the black gate. (pixels being transparent, the electronics which commands them can be located only in the gate support, necessarily rather thick).
On a DLP projector endowed with an only opaque matrix this defect is often present all the same because of the inferior quality of the optics.
The only one projector satisfactory that I know (I do not know them all!) is OPTOMA H79 (4000€ new, but it is not available any more. We find it nevertheless second-hand for about 1000€, because it is replaced, in the top of range, by full HD projector ).

- Washed-out picture:
It is caused by the shape of the dome which sends back the light not only towards the spectator but also towards itself.
This problem is one truth difficulty without simple solution (to paint the screen in light grey attenuates this effect).
It is therefore necessary to arrange from an image as well "solid " or contrasted as possible (choose a DLP projector DLP rather than a LCD).

Picture distortion ::
- the fisheye optics have a simple geometry, with a symmetry of revolution around the optical axis. The distortion is always the same for all the fisheye (of the same type). It is not any more case with the mirror.
The system is not any more centered. The distortion varies with the configuration.
It depends on dimensions and on relative positions of the dome, the mirror and the projector.
An image adapted to a system will not be convenient for the other one.

Format:
- the fisheye image is circular. It registers well enough in a frame 4/3.
- the mirror image is elliptic and registers very well in a frame 16/9.

The optics of the projector ::
- the focus is approximately on the mirror.
If this last one is small, the projector will have to be close to it, can be too much near to focus on this too short distance.
There is thus a narrow relationship between the radius of the mirror and the optics of the projector.
- contrary to the fisheye, centered system, in a system with mirror the light does not go through the same distance for all points of the picture.
A big depth of field is therefore needed. We obtain it by adjusting the aperture; but we lose a party of light then.
In practice we focus on the part of the screen which faces the spectator. The sky becomes lightly blurred, but it is not really embarrassing.
- Other problem : The fisheye, and therefore the projector, have to be in the axis of the screen. When this last one is tilted, the projector owes the being also. To tilt strongly the projector can reduce the life of the lamp (generally expensive).
In the case of the mirror system, the projector remains horizontal, the mirror returning picture under the good angle.

The manufacturing of the pictures and their transformations:
A means is needed to obtain images covering a field of 180 °.
- In cinema the only practical resolution is the fisheye.
- In photograph we can also use the system of the cube: take several successive photos to cover all the field, then put together them by distorting them to construct the picture fisheye.
One needs also a software which transforms the fisheye circular picture in an elliptic warped picture adapted to the system of projection.
- In computer generated image we have the same problem:
A ray-tracing software such as POV-Ray is capable of making directly a fisheye picture. But this technique is too slow for the real time. It is adapted well to make photos and films frame by frame.

For the real time it is necessary to use, as for the games, a fast technique. It will be based on OpenGL (or DirectX).
But neither OpenGL nor DirectX know how to make directly a fisheye picture. It is necessary, here also, to use the trick of the cube and to make five normal pictures covering each a field of 90 ° (front, to the left, to the right, above and bellow). From these five pictures we reconstitue then the fisheye picture or directly warped for the mirror. This in real time.
But the current computers and especially the graphic cards, focusing for the games, allow it.

references:

The mirror :
- my current mirror is a security mirror (used in stores) says "quarter of sphere" about 57 cm in diameter. Really it is half a calote of a sphere about 40cm of radius.
It is "silvered" outside and costs 119,60€ with the port(in France). I ordered it by internet here
The reference of the mirror is VOLUM 51-57: here
and at the manufacturer: here

A smaller mirror would agree probably as well.
However I do not recommend this mirror because it has a defect: It is constituted of a (unbreakable) white translucent plastic material on which is deposited a reflective coat. And this last one is not totally opaque. The light penetrate into the plastic which also reflects it towards the screen in the form of a withish halo which contributes to water down the image. Although minor this defect becomes embarrassing for clear pictures.