# Parallax

## Intro

Parallax demo © Bernhard Vogl (GIF animation must be turned on)

If you shoot the same scene from a slightly different point of view, the foreground will be shifted in relation to the background, as in this example image.

Parallax occures in panoramic photography if camera and lens are not rotated around the Entrance pupil of the lens. A difference caused by parallax will be visible in the overlap between two adjacent images.

Parallax errors are sometimes very hard to retouch, since necessary background details might be obscured by foreground details. A useful way to fix these kinds of errors is to 'invent' some background details.

## Calculation

Parallax depends on the object distance ${\displaystyle d\,}$, the displacement of the no-parallax-point ${\displaystyle r\,}$ and half the rotation angle between two shots, the off axis angle ${\displaystyle \alpha \,}$. Then the half parallax angle ${\displaystyle \beta \,}$ will be: ${\displaystyle \beta =atan{\frac {r*sin(\alpha )}{d-r*cos(\alpha )}}}$

Since ${\displaystyle d\,}$ usually is far larger than ${\displaystyle r\,}$ this could be abbreviated to ${\displaystyle \beta =atan{\frac {r*sin(\alpha )}{d}}}$

${\displaystyle \beta \,}$ applies in both images that overlap in opposite directions, hence the value must be doubled to get the actual parallax error. Now the maximum error in pixels for the ready stitched panorama can be calculated: For a 360° panorama (and ${\displaystyle \beta \,}$ in degree) this is simply ${\displaystyle PixelError={\frac {2*\beta *PanoramaWidth}{360}}}$

## Examples

If you shoot 4 images around, ${\displaystyle \alpha \,}$ is 45°. Given an object 1m from the lens the maximum (relative to infinity) parallax error will be for a displacement ${\displaystyle r\,}$ of the no-parallax-point:

${\displaystyle r\,}$ 8000px width 12000px width
1mm 1.8px 2.7px
5mm 9px 14px
10mm 18px 27px
20mm 37px 55px

If you shoot 6 images around, ${\displaystyle \alpha \,}$ is 30°. Given an object 2m from the lens the maximum (relative to infinity) parallax error will be for a displacement ${\displaystyle r\,}$ of the no-parallax-point:

${\displaystyle r\,}$ 8000px width 12000px width
1mm 0.6px 1px
5mm 3.2px 4.8px
10mm 6.4px 9.6px
20mm 13px 19px