What Equivalency Is
The crazy sounding numbers aim to illustrate how your lenses "Angle of View" changes with sensor size and in turn how moving closer to, or further away from your subject (to accommodate the same framing) will affect your "Depth of Field". In other words; What lens you would need on full frame camera to reach the same results (for AoV & DoF).
This page will get progressively more complicated as it goes, but the first example should cover the basics. The rest of this page is more in depth, only if you feel like knowing more about the relationship various numbers have with each other.
What Equivalency Is NOT
It's unfortunate that the metrics we use and understand to describe angle of view and depth of field are focal length and f-stop. Equivalency does not suggest that a lenses "Focal Length", "Entrance Pupil" or "light Transmission" change when the sensor size of your camera does.
When people say that a 50mm f/1.4 becomes 75mm f/2.1 on an APS-C sensor camera (1.5x crop); Were are talking about how the lens performs on that sensor (compared to full frame) in a language that most people can understand. It does not mean to suggest that the focal length (distance between the sensor and entrance pupil) actually changes. F-stop is simply a fraction of the focal length that describes the size* of the entrance pupil. So when equivalency changes the focal length value (to refer to angle) the f-stop has to change proportionately to it.
The entrance pupil can be worked out by dividing the focal length by the f-stop. Equivalency demonstrates how different sensor sizes show a lenses entrance pupil does not change:
50mm ÷ f/1.4 = 35.7
75mm ÷ f/2.1 = 35.7
* The entrance pupil is the optical view of the aperture diaphragm as seen through the front elements of the lens. It determines how much light comes in, so it's important, but it's not how big the aperture is physically. It's image is magnified by the lenses front elements and most often wouldn't physically fit where it's supposed to be. ***Gerald Undone has a great video about "entrance pupils" here***
= Lens (Actual)
= Lens (Equivalent)
= Entrance Pupil
= Sensor Size
Example 1 - Basic
If you have absorbed the above info, or you knew it already, let's jump into an equivalency example for a basic bokeh pano...
In the above example the bokeh pano is exactly 2 x larger than full frame (the diagonal measurement would be ideal here, but in this example it's also the width and height). Thus we can divide each of the lenses metrics by a factor of 2.
So, if we had used a 100mm f/2.0 lens @ f/2 (for this hypothetical bokeh pano); The resulting stitched image would be the equivalent to using a 50mm f/1.0 on full frame. This is much easier to work out if you are shooting with a full frame camera and since we're using nice round numbers. Here is the calculation:
100mm ÷ 2 = 50mm (original vs equivalent)
f2.0 ÷ 2 = f1.0 (original vs equivalent)
Once again we can see that the entrance pupil hasn't changed:
100mm ÷ f2.0 = 50mm (actual lens)
50mm ÷ f1.0 = 50mm (equivalent lens)
Example 2 - Practical
Going back to the example from the First Page - Since the panorama has matched the focal length of the wider angle lens we can work out the equivalent f-stop by knowing how much larger the entrance pupil is on the longer focal length. In this case it's exactly 3 times (75 ÷ 25 = 3), so we can divide 1.4 by 3, which = 0.4666666 (rounded up to 0.47).
The panorama (above) can be calculate one of two ways:
a 35mm f/0.47 lens on a 24x36mm sensor
a 135mm f/1.8 lens on a 93x139mm sensor
The first example can also be worked out using the method from the 1st example (by how much larger the final stitch is compared to a single image). In this case - 3.86 times larger. 135 ÷ 3.86 = 35 for the focal length & 1.8 ÷ 3.86 = 0.47 for the f-stop. This also confirms the above calculation.
The alternative way to see equivalency is to view it as the sensor being the thing that grows. The 75mm entrance pupil remains, but the image circle of the lens expands along with the sensor size. we use the same multiplication, but on the sensor size instead - multiply each dimension of the full frame dimensions (24x36mm) by 3.86, which = 93x139mm.
Thus it has emulated a sensor size somewhere between Medium and Large Format, whilst retaining the super fast 135mm f/1.8 lens characteristic, projected to a larger image circle. Just like the full frame equivalent 35mm f/0.47 lens - nothing like this exists because it would be too big, heavy and expensive to manufacture into a commercially viable product. If a lens to shoot this image in a single shot existed, it would make Sigma's massive 105mm Art lens look like a cheap pancake.
Example 3 - Advanced
In the following example we will show what is going on inside an 85mm f/1.2 when shooting a bokeh pano and why the two type of equivalency are actually the same thing:
85mm f/1.2 lens on a larger (MF) sensor (90 x 60mm), or
35mm f/0.5 lens on a normal (FF) sensor (36 x 24mm)
The first image (below left) illustrates how a single image from an 85mm f/1.2 lens works on a full frame camera. This acts as a point of reference for the other two diagrams, which show the two different ways to work out equivalency for the same bokeh pano.
Lens Equivalency (above middle)
This is the more commonly used calculation. These values make it a little easier to visualise why the technique is producing such impressive results.
Sensor Equivalency (above right)
This method is much closer to what's actually happening. It shows how much larger the sensor and lenses image circle would need to be to achieve the image in a single shot.
NOTE: The above diagrams illustrate that shooting multiple images will approximate a curved sensor. Currently this is feature being teased for future fixed focal length compact cameras. This will likely never be done with any interchangeable lens cameras due to the curvature of the sensor needing to be altered for each focal length. It's another reason why image quality is considerably higher with this technique compared to any single image from medium or large format cameras. Curved sensors would improve clarity while reducing aberrations (like CA, vignetting, mechanical vignettting) in the corners of the image. We don’t quite achieve it here, but the longer focal length lenses that we use (and the more images we use in a pano) the closer we get.