Angling Mathematics

The Mathematics of “Fishing under the Radar”.

A recently posed question about trout vision, diffraction and all of that had me reaching for the maths books and trying to recall school day trigonometry. But the exercise produced some interesting thoughts none the less.  Not least I think that it added mathematical proof to the way that I like to fish. Long leaders and aggressive casting with tight loops aiming low, relying on the leader to bleed off energy at the last moment.

The much discussed “Trout’s Window”

Irrespective of the fish’s visual acuity there are physical properties associated with the bending (refraction of light) which have significant effects on what a trout could possibly see.  The trout’s world consists of a window, the diameter of which is determined by a thing called the Snell’s equation. In simple terms the window is 2.26 times as wide as the trout is deep. So it can clearly see things on the surface over a wider area the deeper the fish is. At one meter the fish can clearly see things on the film in a 2.26 metre wide circle above its head.

Many angling writers have made much of this, because a relatively small increase in depth radically changes the size of the window. At 0.5 meters the window has a diameter of 1.13 metres, but at a depth of a metre that window grows enormously to 2.26 metres across. If you take the area of the window the results are all the more dramatic. At 0.5 metres depth the area of the window is 1 square meter, at one metre in depth that window jumps to 4 square metres. Double the depth and you effectively quadruple the size of the window.

Click on the diagram to see larger image

I think that frequently this has been misinterpreted along the lines that if the trout’s window is 2.26 meters across (about 8’6″) a nine foot leader is all that you need to keep the line out of sight of the fish. I am going to suggest that the following mathematical gymnastics offer solid proof that isn’t the case. (I am not even factoring in the disturbance of the mirror, of which the fish is undoubtedly aware).

It has long been held, and probably correctly that the shallower the fish the more accurate your cast needs to be for the fish to see the fly.

A fish feeding directly under the surface, let’s say 5 cm has a view of the world condensed into a circle only 11 centimetres across an area of about 100 square centimeters. That’s not a lot to aim at with a fly. (Fish can “see” the fly, in the mirror, as pointed out by Goddard and Clarke in their excellent book “The Trout and the Fly”. But for now we are going to stick with the window).

However what started this discussion was what can trout see that might scare them? This doesn’t have a whole lot to do with the window at all in my opinion and a lot more to do with the refraction of light and what falls below the critical 10 degree mark.

If the fish were actually looking up through an eleven centimetre wide tube, a sort of tunnel vision if you wish it would be easy to sneak up and drop a fly right on their heads. That window however encompasses all the light coming from an approximately 160 ° arc. It is bent to fit into the window by the refractive properties of the water, but remember that to the trout that is normal.

Anything above the critical ten degrees is at least theoretically visible.  The light hitting the water below an angle of ten degrees is to all intents and purposes reflected and doesn’t reach the trout’s eye.

Click on diagram to see larger image

Taking a trout at half a metre depth in calm clear water, how far away would an object (say an angler) a metre tall have to be to be unseen?

Effectively the fish only gets light from objects above a ten degree angle of incidence. (Actually it is slightly under ten degrees but I am trying to keep things simple).

The answer is given by the following equation:

[FD (Fish depth) x 1.13] + [H / 0.1763]


(0.5 x 1.13) + (1/0.1763) = 6.237 metres  (about 20 feet)

(if you want to know where that came from the mathematics, and they are mine and therefore questionable at best:  Snell’s constant provides that in water the diameter of the fishes window is 2.26 x its depth the radius of that window is therefore 1.13. The tan of a ten degree angle of a right angle triangle is the ratio of “opposite over adjacent” sides. That is to say that the height h is 0.1763 of the distance.

For a trout half a meter down, anything a meter tall comes into view at 6.237 metres distant. However we have all been lead to believe that the shallower the fish the closer you can get, and that is true, but it isn’t true by much.

If our imaginary fish comes closer to the surface , to a depth 20 cm for example a metre high object stays hidden up to 5.9 metres, darn you can sneak an extra thirty odd centimetres closer. The bending of the light doesn’t change and so trout sitting close to the surface can actually see pretty much as well as those that are deeper.  Perhaps the picture is a little more compressed and I am not a trout so I can’t vouch for what that does to them, but I figure that they are probably used to it and the important bit is that you are going to be in view before you get that close, no matter how shallow the fish is.

The full Monty: Ok for those of mathematical bent, or simply owners and proud possessors of the fishing gene who don’t consider such reflection as entirely insane,  here is the maths in tabular form:

You can click on the table for a larger version

However, having pondered the questions a little further I am not sure that the visibility of the angler or the diameter of the window is really as important as another aspect that I have never seen discussed in print.  The effect of casting into the fish’s line of sight even if you aren’t yourself visible. It struck me when I was making these calculations that your line is going to flash above that ten degree horizon and when it does chances are it is going to come as something of a shock to the fish.

Let us for the moment assume that you can cast a metre above the surface,  as your line unfurls it is going to appear in the trout’s vision at somewhere around six metres from the trout. Then your line is going to come flashing into the trout’s line of sight, not only that, but because everything  that a trout sees appears to march down a hill straight at it, your fly line is going to suddenly appear like a rocket belting straight at the fish. That would scare me and I am pretty darn sure that it scares the fish too. How often have you watched a fish only to have it spook the moment you aerialize the line.?

Click on diagram to see larger image

So how would a longer leader help?

So let’s play another bit of mathematical hypothesis, just for the sake of it. Say that you can unfurl your line a metre above the surface and just to humour me we are going to imagine that the leader is invisible but the fly line not.

How long should the leader be if cast a meter above the surface for the fly line to remain “out of sight” if the fish is half a meter down?

Using the same maths we can calculate that the leader would have to be 6.25 metres long (Just over twenty feet).

However if you can unfurl your leader just half a metre above the surface then you can get away with a leader of only 3.4 metres (About eleven feet).

To me then it becomes patently obvious that a leader of only 2.74 m (Nine feet) is entirely unsuitable if you are trying to keep the line out of the fish’s view.

Click on diagram to see larger image.

Oddly enough the longer the leader the more that you can power the fly in low and hard without getting poor presentation. The very thing that you need to do if you are going to keep the line “under the radar”, in fact with a long unstable leader you can angle the casts down at the water just a tad and afford yourself an even better chance of getting in “under the radar”.

To sum up:

  • The  depth of the trout and the size of the trout’s window doesn’t actually have that much effect on what it can see of the angler or his line.
  • The height of the angler and or his unrolling line in the air is far more critical than the depth of the fish.
  • Tight loops will come in under the trout’s line of sight better than wide ones
  • Casts angled downwards will stay out of sight better than those lobed high and “floated in”.

So after all that tiresome mathematics I would have to say that I believe the trigonometry supports my view that long leaders, fired in hard with narrow loops low to the surface offer the best and least visible presentation. That old fashioned “aim high and float the fly in” is all too likely to flash the line into the fish’s view and scare the living daylights out of it.

I hope my old maths teacher is still alive, he would be most impressed for me to use trig to prove a point, actually if he is still kicking and he finds out the shock would probably finish him off.

Note: I am a better caster than mathematician so open discussion is welcomed, please do feel free to leave a comment, observation or thought on this blog.

Brought to you by the author of  “Learn to Fly-cast in a Weekend” now available for download as an eBook for PC, Kindle or ipad from Smashwords.


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2 Responses to “Angling Mathematics”

  1. Wayne Mumford Says:

    Even though I’m a total dolt at math, I managed to stay with you for the most part. Really interesting. It’s amazing how life adapts to it’s surroundings.

  2. paracaddis Says:

    Thanks Wayne, don’t worry I made more than a few mistakes trying to work this out. I think that I got it pretty much right in the end though. What interested me was that much accepted wisdom on the depth of the fish in relation to the angler really didn’t seem to hold true. I have always accepted that the shallower they were the closer you could get without questioning it to any degree before.

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