Angle of View

The Wikipedia entry provides a very good description of what angle of view is all about. That said I'll try to summarise what angle of view is in a few sentences for those with little patience. Briefly put, angle of view is the angle (expressed in radians or degrees) that a lens at a given focal length in combination with a given camera sensor size captures across the photographed scene. Put another way, imagine looking out of a camera's viewfinder. The angle of view is the angle at the camera between the left of the scene and the right of the scene. One can image the camera as being the top point of an isosceles triangle; with the leftmost edge and the rightmost edge of the scene being the other two points of the triangle. We're interested in the isosceles angle (i.e. the angle at the camera point).

There's a subtlety I glossed over in the above. I gave the impression that angle of view is measured horizontally (left to right) across an image. In fact one can measure three angles (always using the camera as the point of our isosceles triangle):

• Horizontal: From left to right of the image.
• Vertical: From top to bottom of the image.
• Diagonal: From top-left to bottom-right (or viceversa) of the image.

The diagonal measurement in fact is the one most often quoted and hence, I'll assume that from this point onwards.


Calculating Angle of View
It's fairly easy to calculate angle of view so long as the sensor dimensions and the focal length are known. Simply use this equation:

(360 / 3.142) * arctan( (dimension of image sensor) / (2 * focal length) )

Where:

• Dimension of image sensor is the sensor width, length or diagonal in mm (depending on whether you're measuring horizontal, vertical or diagonal angle of view, respectively).
• Focal length is the lens' focal length in mm.

Note that you can plug the values straight into the above equation, paste it into Google and you'll get an answer (thanks to Ken Rockwell's site for this tip)!


Importance of Angle of View
So why is any of this important to the average photographer? Surely no one's going to be bothered with calculating the angle of view before taking a photograph...

That's true enough and so it's fair to say that 'explicit' knowledge of
angle of view and the math behind it is somewhat superfluous when going about the day-to-day job of photography. That said, I find this knowledge beneficial for two reasons:

1. Understanding focal length and it's effect.
2. Understanding equivalent focal length when speaking about cropped sensor sizes such as Nikon's DX sensor or Canon's APS-C.

To illustrate the first point, let's assume a full-frame sensor size (24mm x 36mm) that's 43.3mm diagonally. Using the above equation, here are some resulting angles at various focal lengths:

18mm = 100.52 degrees.
35mm = 63.48 degrees.
50mm = 46.83 degrees.
105mm = 23.30 degrees.
300mm = 8.26 degrees.

Observing the above figures, it becomes immediately obvious that the larger the focal length, the narrower the angle of view. This is probably fairly obvious in practice to any photographer. As we 'zoom' (i.e. increase focal length), the scene captured is restricted (i.e. the angle of view decreases). It is also the reason why we speak of Wide Angle lenses; lenses with a relatively short focal length (typically 35mm or less on a full-frame sensor) that capture a fairly wide angle (of view) and Ultra Wide Angle lenses (typically 24mm focal length or less on a full-frame sensor).

Going onto the second point, I'd just like to go over this briefly. I want to tackle the various differences between full-frame and crop sensors in a separate post so I'm weary of not wanting to jump the gun...

Let's now consider Nikon's DX (crop) sensor (15.6mm x 23.7mm) that's 28.4mm diagonally. If we apply the same above equation to this sensor we get:

18mm = 76.54 degrees.
35mm = 44.16 degrees.
50mm = 31.71 degrees.
105mm = 15.40 degrees.
300mm = 5.42 degrees.

Comparing these results to the full-frame results, it's immediately obvious that a crop sensor decreases the angle of view at a given focal length. Without going into the merits and drawbacks of this, it becomes obvious that, at times, we may need to find the focal length for a crop sensor that's equivalent (i.e. has roughly the same angle of view) a known focal length used on a full-frame sensor. Re-arranging the equation given before to get the focal length as subject:

(dimension of image sensor) / (2 * tan (angle of view * 3.142 / 360))

So let's do a simple exercise. Going back to the full-frame values, we find that at 50mm, we have an angle of view of 46.83 degrees. So what focal length on a DX sensor that would give us the same angle of view? The answer is 32.8mm. We can therefore say that 32.8mm on a DX sensor provides an equivalent focal length to a 50mm focal length on a full-frame sensor. Being able to find out the equivalent focal length is particularly useful since most discussions/documentation/literature assume(s) a full-frame sensor (keep in mind that a full-frame sensor is the same size as 35mm film which pre-dates it). Knowing the equivalent focal length therefore allows us to somehow 'translate' this information to what it means when using a DX sensor.


References

1. Wikipedia - Angle of View.
2. The Imaginatorium - Angle of View Calculator.
3. Kenrockwell - Angle of View.
4. Wikipedia - Field of View.
   

Perspective

So I thought I'd start off with what should be a fairly straightforward (but often turns out to be extremely confusing!) topic, namely: perspective. I should point out at this stage that my explanation only deals with the parameter(s) that can affect perspective - that is the observer's position in relation to the subject. All other possible parameters such as aperture, shutter speed, ISO and even focal length (more on this in a future blog) are assumed constant.

The Wikipedia entry is worth a read in so far as explaining what linear perspective is. In simple terms, a given object appears larger or smaller depending on how far we (the observer) are from the subject. The closer we are, the larger the subject. The further away we are, the smaller the subject. Simple enough so far...

We can generalise the above to say that the observer's location in 3D relative to the subject's location in 3D is what determines perspective. For example, if taking a picture of a flower, the perspective changes considerably if we shoot the picture looking down on the flower or if we lie on the ground and shoot the picture looking up at the flower! It's therefore important to think of perspective in terms or 3D points (x, y, z). We can change our:

• Distance (y) from the subject.
• We can change the angle (by changing x) to the subject (e.g. having the subject exactly in front vs. having the subject to the left or right/at an angle).
• We can change our elevation (z) with respect to the subject (e.g. looking up at a flower vs. looking down at a flower).

...these three parameters (your spatial coordinates) quite simply are the parameters that affect perspective.


Relative Perspective
A more subtle (but possibly more important!) point about perspective is what I'd call relative perspective. Simply put, that is how multiple subjects (in a photo) appear in relation to one another. For example, take a look at this image:

It's a photo taken underneath a bridge with major support beams running along the length of the photo (i.e. from top to bottom) and minor beams/spacers running along the width of the photo. As intelligent human beings we intuitively know that all the beams must be of the same length & width and equally spaced between one another. The photo however does not show this! Take the major support beam in the centre, for example. We notice that at the top of the photo (closer to the observer/photographer), the beam looks wider/thicker than at the bottom of the photo (further away from the observer/photographer). This is the effect of perspective! The same can be said for the spacing between two given beams - it seems to get smaller as we go from top to bottom of the photo. Theoretically, if the photo extended to infinity, then one could imagine the two beams meeting. This point is known as the 'vanishing point' and is entirely the result of perspective.

Now consider all the major support beams. Concentrating on the bottom of the picture, we can notice that going from the centre of the picture to the left, the distance between two successive beams seems to get smaller the further left (i.e. further away from the observer/photographer) we go. This is the effect of relative perspective. The same thing can be said about the minor beams/spacers. The distance between successive spacers going from top to bottom of the picture (i.e. further away from the photographer) seems to get smaller. Generally speaking, one can say that the closer the observer/photographer is to the closest/primary subject (say the major beam in the centre of the picture in this case), then the larger that subject will appear in relation to the other (more distant) subjects (i.e. all the other major beams in this case).

Lens Focal Length
I didn't touch upon focal length in this blog as I will treat it's relation to perspective in another blog.

References

1. Wikipedia - Perspective.
2. Klaus Schroiff - Perspective.
3. Basics Photography: Composition - David Prakel; AVA Publishing 2006.
4. Wikipedia - Perspective Distortion.
5. Cambridge in Colour - Camera Lenses.