You've probably never used a circular fisheye lens properly, which is to say in the manner it was designed. It's okay, almost no one has.
Circular Fisheye Lenses: A Brief History
Circular fisheye lenses date back a century with the Hill Sky Lens, the first 180-degree lens, released in 1923. Less than 40 years later, Nikon released the 8mm f/8 Fish-eye Nikkor, a lens designed for two specific scientific purposes and some incidental creative use. Most people know this lens because of the iconic Nikon F press photos taken with this lens mounted on the original F, but very few of us have actually gotten to use it. At the end of this article on the technical aspects of the lens, I'll also include some non-technical sample photos I took with it to show how it performs in creative use.
Nikon released the 8mm f/8 Fish-eye Nikkor primarily for use pointed straight at the sky. This allowed photographers to capture clouds and cloud movements, measure the distance of objects from the sky's zenith, and measure the angular distances between two objects in a straight line from the zenith. To help photographers photograph clouds, Nikon included six filters built into the lens as part of the optical path. The video below shows the effects each filter has. Following the video, a table presents the filter information as reproduced from the lens' manual.The video below shows how proper lens alignment allows the whole sky to be in frame with the horizon along the image's perimeter. This capability of this lens, specifically, means that the Nikon 8mm f/8 Fish-eye Nikkor has measurement capabilities for the placement of objects within the sky. This article examines, in depth, how to use this lens for object measurement along radial lines emanating from the sky's zenith. Not all circular fisheyes can be used this way, and even those that can, the calculation tables provided by Nikon for this lens are specific to it. The Nikon 8mm from the early 1960s has exceptional correction to eliminate magnification shift over the image field and to provide exceptional radial symmetry from the lens' optical center, as the data we'll review show. These technical corrections make measurements with this lens possible.
Nikon 8mm f/8 Fish-eye Nikkor: Scientific Use
But why? What use does the following process have? This process can be used to help measure object movement through space. For instance, a comet could be tracked for many nights to calculate trajectory, speed, and direction. Satellites could be recorded over a span of minutes for the same purpose. Tree canopies could be recorded from the same location to measure growth over months or years. The orbits of planets could be plotted against the zenith to study their rotation around the sun. Cloud size could be measured for estimating the water volume they contain. Knowing the geometry behind this lens' scientific use opens a number of doors for interesting and creative projects. Let's imagine that we've discovered a meteor heading toward Earth and we want to track its progress.
So, let's dive in: How do we do this? First, we need a photo taken with the lens pointed, level, directly at the sky. For the purposes of this article, this photo of a very-stern Abe Lincoln at the Lincoln Highway Memorial near Laramie, Wyoming, works. We also need to know that the image circle is 24mm across on the film or your full-frame sensor. These two givens, level image with a horizon around the perimeter and the image circle size, let us start to work on our calculations.
Step 1: Identify the Image's Center
Back in the day this either required tiny measuring tools, negative masks in the enlarger, or highly accurate image enlargement. Today's photo editing software makes this step a breeze. This can be done in Photoshop, as an example. I find it even easier to do this work in Affinity Designer or Adobe Illustrator, however. The center point will be the basis for most of our calculations. The center point, for reference, is also the zenith if the lens has been aligned properly.
Step 2: Use Pixel Counts to Create a Radius Line
Place a line from the zenith to the image circle's perimeter, anywhere, and then create dashes for measurement. The dashes will vary by your camera's sensor resolution. I used a Sony A7S II, which has a vertical pixel height of 2,832 pixels. Half that height, 1,416 pixels, represents 12mm. That means that each millimeter is 118 pixels long. So if I want to make accurate measurements to a half millimeter, a dash-space pair where each part is 59 pixels long lets me easily measure to a half millimeter, and roughly gauge quarter-millimeter spacing. The instruction manual for this lens provides the tools needed to measure zenith-subject and subject-subject angular relationships to a quarter millimeter. (If you were so inclined to measure to a tenth- or hundredth-millimeter accuracy, the instruction manual data allows extrapolation for that.) The two images below show multiple 12mm lines in different places, which verifies that the zenith is placed accurately. The second image adds the dashed measuring line.
Step 3: Select a Subject
For this example, we'll use the top of the tree in the upper right quadrant, solely because it's an easily identifiable point. Using our dashed line, which I've anchored the rotation for at the zenith, we'll rotate the line until the dashes align with that subject point. For clarity, I marked the subject with a circle. We know that each line and space are a half-millimeter. Counting the lines and spaces, we find that the tree's tip is 10.75mm from the zenith.
Step 4: Reference the Manual's Technical Data
I reproduced the pertinent technical data below. Using the graph, we can plot the millimeter difference and identify that the tree's tip is approximately 77 degrees from the zenith. That seems pretty close, but if we're using this to track a meteor headed for Earth then maybe we want more accurate numbers. For that more accurate number, we can use the table. The table lists 10.5mm and 11mm, and those points are 77.94 and 81.75 degrees from the zenith, respectively, a difference of 3.81 degrees. 10.75mm falls in the middle, so if we split 3.81 degrees (1.905 degrees) and add it to 77.94 degrees, the tree tip is exactly 79.845 degrees from the zenith. Hey, our 77-degree measurement wasn't too far off.
And that's it. That's what this lens does, right? Well, no. There's another trick. Let's go back to that meteor, and assume we're trying to ascertain progress and trajectory. Going back to our original image, let's find two points on that image and calculate the angular difference between them. These will be different solely because there are no other good points to reference in the line from the zenith to the tree tip. This next trick requires that you picture the lens' image as a half-sphere. The measuring point is in the center, halfway to the imaginary nadir, at the half-sphere's flat bottom. The image the lens projects can be thought of as being painted on the upper, round portion of the half-sphere. We'll be measuring the angular difference from the center on the flat bottom to two points on the sphere's surface. The graphic below, adapted from the lens' manual, shows this concept and illustrates measuring points.
Step 4: Identify Two Subjects on a Radial Line
Here we'll be using Angry Abe's deeply-furrowed brow and a random bit of mortar between two stones. These two points line up pretty well. Below is the revised image with the two subjects marked. Let's pretend that these points represent two changes in our meteor's trajectory over the span of a week, as shown on two separate images that we've overlaid.
Step 5: Measure the Distance Between Two Subjects
For this example, we need two subjects on the same radial path, which is to say that they need to have a common trajectory toward the center. (A subject that's moving tangentially to the center can have its path and velocity calculated but it requires some significant math that understands the spatial geometry of cones. I am not the photographer to tell you about that.) Measuring the two points we identified indicates that they are roughly 2.5mm apart. The table in the instruction manual has a third column and that allows us to calculate the angular distance. The third column is offset by a half-line so that we can calculate to quarter-millimeter accuracy. Doing so from zero lets us know that the difference is 3.59+3.59+3.59+3.61+(3.61 / 2) = 16.185 degrees.
Step 6: Calculate from the Correct Points
Let's do this correctly this time. If you see in the table's third column, the angular degree differences change slightly as the distance from the zenith increases. So we need to calculate the distance based on how far the objects are from the zenith. Abe's brow is 7.75mm from the zenith and the random bit of mortar is 10.5mm. So to obtain the actual and correct angular difference for these objects we start halfway between the 7.5mm and 8mm readings in the table's third column and add until we get to 10.5mm. Because the third column is offset, unless both measurements align with the spaced in that column, one or both of the starting figured needs to be divided. (3.83 / 2)+3.79+3.80+3.82+3.87+3.84 = 21.035 degrees. That's a fairly significant difference from our first calculation if we're trying to let the world leaders know how long until our meteor arrives. A note here, if the first measurement had been 7.9mm instead of 7.75mm, we'd have started with (3.83 / 8) instead of (3.83 / 2). The initial and concluding divisions in the calculations are proportional to the fractional millimeter that we've counted.
High School Math Is Useful
And there we have it, all that math you told your high school teacher you'd never need used here in practical application! One key point, this will not work with all circular fisheye lenses equally. This Nikon lens is, as the angular distances in the table above demonstrate, highly corrected and the light rays bending through it are kept in relatively uniform configuration. In video samples with this lens, objects grow larger as they approach the lens center, definitely, but they do so at a uniform rate. In many circular fisheye lenses, such as the inexpensive ones being produced for aperture-priority shooting right now, the magnification across the image plane changes dramatically, especially toward the image's center. This leads to objects growing in magnification drastically as they move in the image field and that means those lenses are not corrected suitably for scientific use. So this technique may be achievable with other lenses, but the numbers and tables needed for calculations will be drastically different.
This same process applies if you use the zenith as one of the two subject points to track movement longitudinally across many images. Taking multiple readings of an object as it travels in relation to the zenith can, with some more complex math that turns our simple "V" shape into the skin of a cone, allow for detailed understandings of trajectory, velocity, and potentially even subject-observer distance. Imagine plotting that for the airplane in the above video with a different angular reading every second and you can probably imagine the process' challenges (and why I'm not the photographer to do this math for you.)
Nikon 8mm f/8 Fish-eye Nikkor: Creative Use
All of that understood, how is this lens for regular use? Here are six photos taken with it that will appear in my Round Glass Review video on this lens (which will go live in late May 2022.) Like all circular fisheye lenses, this does have some good and creative applications outside of the technical. It's also fun to use. The fixed focus of 20cm to infinity makes it very easy to point and shoot. The circular image also means that poorly-aligned photos can be corrected in post by simply rotating a crop around the image's center. The only stress with this lens is whether or not you will damage it, or if it will damage your camera.Taken in Laramie, Wyoming, this type of shot seems to be a right of passage for people who review this lens. Just a quick note: No one believes you were studying. This was only safe because I was on the other side of a fence. Seriously, never get this close to a moving train for a photo. Here's a video from later in the train if you're interested: The idea that circular fisheye lenses distort everything is incorrect. The good ones do not distort lines parallel to the lens' optical center. This photo shows how a straight line on a radial axis from the zenith isn't distorted, which is why all the math we did earlier works so well. Yes, at f/22, the water drops on the lens' front element will almost be in focus.
I hope you have come away from this article with a greater appreciation of how well this lens was engineered, by engineers using primitive computers and analog calculation devices. This is a lens that, today, truly belongs in the hands of scientists, meteorologists, collectors, or museums. It's a beautiful and fun lens, but much more so, highly impractical. I hope these photos and explanations helped you understand the magic of the Nikon 8mm f/8 Fish-eye Nikkor bit better.