Focal Length and Angle of View when closer than infinity

[Ken Lee]

Many people consider a roughly 150mm lens to give "normal" perspective at infinity. The focal length is roughly equivalent to the diagonal of the film, and the angle of view is roughly 45 degrees.

When we focus that 150mm lens close enough to give a 1:1 image, 300mm of bellows draw is required.

Some might say that the lens becomes a 300mm lens. Doesn't that also mean that the angle of view (as seen on the 4x5 sheet of film) is no longer 45 degrees ? Has the 150mm lens at 1:1 become the equivalent of a 300mm lens at infinity ?

If a 150mm lens is considered "normal" at infinity with regards to its angle of view and rendering of perspective, then is it no longer "normal" to the degree that we focus closer ?

To get a "normal" perspective at 1:1, why wouldn't be shoot with a 75mm lens ? After all, at 1:1 it will be a 150mm lens, no ?

[Peter Yeti]

You are absolutely right, the image angle is determined by the distance of the rear main plane to the film plane. This angle decreases nearly linearly with the bellows draw, at least for small angles. You can determine the angle a by

a = 2*arctan(d/(2*b))

where d is the film diagonal (or diameter of image circle) and b is the distance from rear main plane to film plane.

For people who don't like the math there's also an experimental way to find out. The Linhof Technica viewfinder actually takes this into account. When you set the focal lenght to 150mm and set the distance to 3 feet, the effective focal length is shown to be 240 mm.

You are right that a shorter lens would give a more natural image angle. Sometimes I use reversed enlarging lenses as short as 50mm for 4x5'' when I'm below 1:1. The drawback is that the object is so close to the bulky camera that it becomes very difficult achieve good lighting.

[ic-racer]

Yes.
And likewise, for 8x10 one would use a 150mm at 1:1. Not only will it cover the format completely at that magnification, the angle of view will be the same as a 300mm at infinity.

Someone posted an angle of view calculator on one of the forums a while back that takes into account focal length, film format and subject distance. (maybe it was me?).

[Dan Fromm]

Ken, will you please stop thinking? You go mystical and take others with you.

Focal length is focal length is focal length, the small chip "cropping increases focal length" head cases notwithstanding.

The lens projects a cone of rays. I've never been clear on exactly where the cone's apex is, usually assume, right or wrong, that it is at the lens' rear node.

When the lens is focused at infinity the cone's height is the lens' focal length and its base's radius is focal length * tan(angle covered/2)

When the lens is focused closer than infinity, so that magnification is no longer 0, the cone's height is focal length * (1 + m) and its base's radius is focal length * (1 + m) * tan(angle covered/2)

Now go back to sleep.

Sweet dreams,

Dan

[Ken Lee]

Sorry to get mystical - and I very much appreciate the help and insights.

I am intrigued by the relationship between focal length, angle of view, subject distance, magnification and the perception of foreshortening or depth. (I am avoiding the term "perspective"). These factors have an aesthetic as well as optical dimension.

I thought that a wide angle lens always gives a wide rendering, but that isn't really true. I thought that a moderately long lens always gives a moderately long "look" - but that isn't quite true either.

[Doremus Scudder]

Let's distinguish between angle of view and image angle. The former is in front of the lens, the latter behind it. Neither the focal length nor the angle of view should change when extending the bellows.

Whether the image angle changes depends on how you look at it. If you just measure the angle of the illumination cone that exits the lens, then that will not change. However if you measure the angle from the rear nodal point to the corners of the film, then yes, it gets smaller with bellows extension. However, in reality, this is just a kind of cropping; more of the projected image is outside the film area.

Focal length of a lens is inherent in the lens design and defined as the distance from the film the lens focuses at infinity. Using more extension doesn't change this fact. The same with the angle of view (and the image angle if you measure the exit cone).

Now, if you want to talk about DoF and effective aperture, that's another can of worms...

Best,

Doremus

[Ken Lee]

Thank you for the clarification: it's image angle. And yes, the change of image angle is a form of cropping.

I'll have to make some comparison photos at close distance and see for myself.

[Bruce Watson]

When the lens is focused closer than infinity, so that magnification is no longer 0, the cone's height is focal length * (1 + m) and its base's radius is focal length * (1 + m) * tan(angle covered/2).

Dan and Ken are both right. They are both saying the same thing, in opposite ways. Think about it guys -- you can see each other's point of view if you try.

[Mark Sawyer]

Sorry to get mystical - and I very much appreciate the help and insights.

I am intrigued by the relationship between focal length, angle of view, subject distance, magnification and the perception of foreshortening or depth. (I am avoiding the term "perspective"). These factors have an aesthetic as well as optical dimension.

I thought that a wide angle lens always gives a wide rendering, but that isn't really true. I thought that a moderately long lens always gives a moderately long "look" - but that isn't quite true either.

The trick is that "focal length, angle of view, subject distance, magnification and the perception of foreshortening or depth" are all tied up together even though they're different things.

The "long" and "short" look are dependent only on subject-to-lens distance, and the change you get in changing to a different focal length is because you changed that distance.

Look at it this way: In a portrait where the subjects nose protrudes one inch from the face, if you photograph from ten inches away, the face-to-tip-of-the-nose distance occupies 10% of the space. If you back up to 100 inches, it's only 1% of the space. This is true for any focal length lens. What a longer focal length lens does is let you back up to an appropriate distance but still "crop down" to a reasonable area.

So on an 8x10 camera, you will get the same "perception of foreshortening or depth" from a 6-inch wide angle or a 16-inch portrait lens, as long as the lens and subject don't move. The 6-inch lens version will just have a lot more area around the subject.

The very same six-inch lens that is too wide for portraiture on 8x10 is perfect on 3 1/4x4 1/4, and the 16-inch that was perfect for 8x10 is too wide for 16x20 portraits, not because of the focal length, but because of how close you have to put the lens to the subject to fill the frame with the subject's head and shoulders on those formats.