Archived posting to the Leica Users Group, 2004/06/17
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Sorry again Neil.
Even with a 21mm or a 15mm all the circles will project as circles.
If the real circles have the same diameter, their images will have the
same diameter. If math theory is not your cup of tea (after all this list
is on photography rather than on math), then have a try next time you get
out with your wide-angle. Trying with a SLR and a grid focusing screen
will make it obvious.
Still assuming the wall is perpendicular to the lens axis, I agree that a
point of the wall that projects near a corner of the picture will be
further away from the lens than a point of the same wall whose projection
is right in the middle of the picture. But this is _exactly_ compensated
by the fact that a point in the corner of the film will be further away
from the lens principal point (optical centre) than a point in the centre
of the image. The reproduction ratio will be exactly the same.
If doubt persists, have a look at a good projective geometry textbook, for
example: Faugeras, O., "Three-dimensional computer vision, a geometric
viewpoint" (MIT Press, Cambridge, Mass., 1993), or the original source: R.
Descartes, "Geometrie", 1637, but a simple drawing on a sheet of paper
(in the case of a pinhole camera, which is the adequate perspective model)
should be convincing enough.
I agree that people on corners of wide-angle pictures or on close-ups are
"deformed" but this is only because they are non-planar (fortunately).
This will be the same with a nice 21mm or with a "perfect" pinhole.
Best wishes and happy leicaing!
Jean
On Thu, 17 Jun 2004, Beddoe, Neil wrote:
> The relationship between perspective and viewpoint gets 'em every time.
>
> You are standing 2 metres from a wall covered in 0.1m diameter circles
> looking through a 21mm lens at the wall. The film plane is absolutely
> parallel to the wall and the lens is 100% distortion free. The angle of
> view of a 21mm lens is 92 degrees. Which means you can see 4.1421m of the
> wall across the diagonal. Assuming the centre of the furthest circle is 4m
> off centre, the near edge will be 4.38m away and the furthest edge will be
> 4.56 m away from the optical centre of the lens. A fixed length of line on
> the circle's circumference at the far point will appear on the film to be
> 4.38/4.56 or 96% of the same arc at the near point.
>
> Casual observers will note that on most circles 1 degree of arc on one side
> is the same length as 1 degree on the other.
>
> Or:
>
> Buy the best 21mm lens you can get and stand 3 people up close but in a
> line
> parallel to the film so that one's face is in the middle and the other two
> are in the corner. The one in the middle will have a huge nose but the
> outline shape of his or her head will be normal. The ones at the corner
> will look as if they haven't finished teleporting from the deck of the
> enterprise.
>
> The only way to get all the circles on your lovely ornate wall to render as
> circles is to shoot from so far away that the ratio of their distances from
> the film plane approaches 1.
>
> With thanks to Pythagoras.
>
> I've really got to do some work now.
>
> Kyle, I shot five rolls of Provia recently so I feel I've earned this.
>
> Neil
>
>
> -----Original Message-----
> From: Jean Louchet [mailto:jean.louchet@inria.fr]
> Sent: Thursday, June 17, 2004 12:53 PM
> To: Beddoe, Neil
> Cc: 'Leica Users Group'
> Subject: RE: [Leica] laws of optics...
>
>
> On Thu, 17 Jun 2004, Beddoe, Neil wrote:
>
> >> "Second example, let's imagine a vertical wall with lots of circles
> >> painted on it (let's call it the Vasarely wall).If you take a picture
> >> with the lens axis perpendicular to the wall, then ALL the circles will
> >> print as circles on the picture. If the camera is oblique, then all the
> >> circles will show as ellipses."
> >
> > Not quite true. With anything other than a long telephoto, you'd have to
> > get close to the wall and this would mean that you'd be viewing the outer
> > circles from an oblique angle which would render them as ellipses in your
> > photograph. This has nothing to do with lens distortion and is just the
> > normal effect of perspective.
> >
> > Neil
> >
> Sorry Neil, this is wrong. I could have said in simpler terms that as long
> as the object plane is parallel to the film plane, the picture will be an
> exact (non distorted) scale reproduction of the object, without any
> distortion. This is obviously true with a pinhole camera and remains true
> with all "homographic" lenses, even wide angle. Thus if the lens axis s
> perpendicular to the wall, all circle (even far from the lens axis) will
> project as circles, not ellipses.
>
> > Not quite true. With anything other than a long telephoto, you'd have to
> > get close to the wall and this would mean that you'd be viewing the outer
> > circles from an oblique angle which would render them as ellipses in your
>
> There is a confusion here about what you define as an "oblique angle".
> As long as the lens axis remains perpendicular to the wall's plane all the
> circles will be seen as circles, even if your eye has to turn obliquely to
> see it. Then, if you turn the camera then the camera axis is oblique and
> the circles project themselves as ellipses. What only counts is lens
> orientation.
>
> Jean
>
>
>
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--
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