**CONVERGENCE in STÉRÉOSCOPY**

- The convergence is the
**angle****q****between the optical axis of the lenses**. It can be our eyes, cameras, binoculars, etc... This angle is usually given in degrees. - When the axis are parallel the convergence is q = 0.
- The formulas about convergence are:
- q = 2 . Atan (0.5 B / D) and D = 0,5 . B / tan(q /2)

**Properties**- These are the values of q and distance D to the convergence center with an inter- pupillar distance 65 mm..
- D

- 0.4 m

- 0.8 m

- 1.2 m

- 1.5 m

- 9,3°

- 4,6°

- 3,1°

- 2,4

- D

- 1.8 m

- 2 m

- 2.5 m

- 3.5 m

- 2°

- 1.9°

- 1.5°

- 1.1°

- D

- 5 m

- 7 m

- 10 m

- 20 m

- 0,7°

- 0,5°

- 0,4°

- 0,2°

- Double the base B is doubling D keeping the same angle q..

- Convergence is set on lens when shooting. When restituting the pairs, there can be convergence or divergence depending on the setting when there are two projectors knowing that the optic laws are the same when shooting or viewing pictures. Usually it is a problem when the setting of the projectors is not done properly and that the disparities are greater than the interocular separation.
- In these paragraphs the convergence at shooting, often ignored, sometimes rejected which is considered.

**History****In 1856**Sir David Brewster in chapter VIII of his book__The stereoscope__has the first expressed conditions for stereoscopic photography (see opposite figure and text)- In this, for portraits at 6 feet distance, 180 cm, the convergence shall not exceed 2°. The calculation confirms that 2° at 180 cm is a base of 2 1/2 inches, 63 mm. The stereoscopic ratio SR is calculated for the plan behind at the depth L. The value of SR range from heighten to strong and is excessive for L > 2,4 m (4.2 m from the camera)..
- L

- 0,6 m

- 1 m

- 1,5 m

- 2.4 m

- infini

- SR

- 22

- 32

- 40

- 50

- 87

- We can only remark the soundly recommendation of Sir D. Brewster, because in this case the background was never far, 1 to 3 m as mean value, and not at infinite distance.
**In 1895**in__La stéréoscopie de précision__, Louis Cazès described the convergence. The plan at the horizon and the vertical median plan between the cameras are reconstitued and homologous. The other dots are reconstituted by vertical lines but are horizontally stretched. However, the homologous dots are close to the coincidence.**Note**- .Some stereoscopic cameras are built with converging lenses. It was done on the Fujifilm W1. Q
- Finally, the relevance of the convergence relies on the tolerance of perception of the eyes which is the confusion circle.

*In photographic portraiture the correct angle for a distance of six feet shall not exceed two degrees. (...)**We come now to consider under what circumstances the photographer may place his two cameras at a greater angle than what we have fixed."**In taking family portraits for the stereoscope, the cameras must be placed at an angle of 2° for 6 feet..."*

- The first binocular stereoscopic cameras with the views on the same glass plate couldn't have convergence.
**In 1923**the__Traité général de stéréoscopie__of Emmanuel Collardeau described a convergence system for the stereophotography of small objects. He mentioned that the photographic pair is thoroughly reconstituted by the eyes. In the same book he mentions the convergence of airplane aerial pictures.**In 1990 S**erge Gauthier published the__Traité et méthodes modernes de stéréoscopie__which is a detailed study of the convergence. He explains it on well drawn geometric pictures. He establishes the methodology determining the convergence to apply at each distance of the first plan.

**Plans moved closer**- The convergence moves the plans closer to the spectator. If the subject is near and the background out of focus , convergence on this subject will emphasize it.
- With the convergence, at the same time than the plans are moved forward, the near objects are perceived smaller than the same without convergence.
- The convergence is also the way to make blowups.

**Keystone effect**- Due to the convergence a plan perpendicular to the central axis will be seen as trapezoidal by each lens.. The right side of the pictures will be larger on the right image and smaller on the left one. It is the other way on the left side.
- It is only discernible on far backgrounds of wide angle view combined with a strong convergence, which is not likely.

- Parallel axis

- Convergence

- The shift of the sides of the edges of the frames is constant and equal to the base B. It can be disturbing on near scenes when the base is large.
- At short distance parallelism is disturbing.
- The convergence is necessary for near objects.

- The shift of the edges of the frames varies. Starting at first plan it decreases to nought and then increases until the last plan.
- Note the keystoning
- At far distance a strong convergence may be disturbing.
- Practically the keystoning is seldom perceived.

**Practically, when shooting:**- The convergence is finally recommanded for the near objects to the conditions to not exaggerate it and to restrain keystone and blow up of some seldom scenes.
- With field angles greater than 50° a 2° convergence will not disturb the far backgrounds; on the contrary it will better show nearest objects.
- Adjust the convergence is necessary for the nearest objects and also when zooming field angles under 30°.
- If the background is not far from the first plan it is favorable to increase the convergence for the nearest objects.

**Not to confuse:**- The linear parallax is the disparity between left and right images. It varies depending on distance D to the camera and depth L between plans. It has limits not to exceed.
- The linear parallax can be modified in playback shifting one image. The result is similar to the convergence, but edges are cropped.
- Applied when shooting convergence shifts all the image like a tracing moved sideways over a sheet. This is substracting a constant value to all parallax values. The convergence keeps the full size of the pictures.

- For a long time I have carefully follow the parallelism condition commonly recommended. I controlled precisely the parallelism of the lens axis.
- Nowadays I realized that the convergence don't change the relief effect and its depth but improves the stereoscopic rendering by avoiding window violation and keeping the frame of the pictures.

- The binocular stereoscopic cameras are sometimes built with a permanent slight convergence between 1 and 2°. This is a convergence center between 1,8 and 4 m.
- Many 3D movies are shot with converged cameras.