Introduction

It’s go-home leg and ATC clears you “Direct ABC.” You put the fix in the box, confirm with your PM, and then stare gleefully at the ETA field while it recomputes. Nothing changes. Dammit, another “placebo shortcut.” Murphy just struck again. While that can be a minor frustration on go-home day, understanding the geometry of small shortcuts can help you avoid weather or airspace with minimal delays.

The Geometry

Let’s say you’re flying along from point A to point B, and a thunderstorm pops up in the middle. Being proactive, you draw a line to a clear hole just to the upwind side, E, and request the heading deviation and then direct B when able (green lines in the middle). It looks like a sizeable angle, so now you’re thinking about added distance and fuel. An easy way to visualize the distance change is to extend the line AE out to a point C, opposite point B. Completing the rectangle ABCD and the associated diagonals gives us four equal-length lines in the middle (denoted by the single tick marks).

Weather deviations don't add much distance.

Table: deviation angles and additional distance

Distance AE+EB (green planned flight path) is the same as AC. Drawing an arc centered on point A with radius AB intersects line AC at point F. Because they’re both points on an arc, distance AB = distance AF. Astute observers will note that F is pretty close to C, meaning that our detour only added a total of distance FC. If we want to know the total distance AC, we can model it as AC = 1/cos(CAB) * AB. In the scenario where the storm is not neatly sitting in the middle of AB, we can model the distance as AE+EB, where AE=1/cos(EAX)*AX and EB=1/cos(EBX)*BX. Looking at a table of sample deviation angles at right, we see that for small angles, the added distance is negligible.

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Implications for Pilots

There are a few takeaways from this table that we can use to make practical rules:

1. Below 10 degrees you suffer only minor penalties of <2% for deviations. This means you should deviate early if there’s any question about the weather ahead.
2. Pick a downrange fix B that will give you a <10-degree angle EBX to economize on distance on the back end.
3. The small cost of #1 and #2 should nudge you toward a lateral deviation over a vertical one in many circumstances due to its predictable cheap cost and relative safety.

The corollary of the “small angles add trivial distance” rule is that shortcuts of small angles likewise cut trivial distance. If you’re filed AEB and ATC gives you direct AB, then your total distance is cos(EAX)*AE + cos(EBX)*EB. If AEX is something like 2-3 degrees, then it might look nice on your MFD, but when you activate you end up with no measurable change in ETA. This is the “placebo shortcut” that always strikes on go-home leg.

A further application of this principle is in flight planning for VFR nav logs. I used to see a lot of students on stage checks come in with nav logs that took them fairly close to the corner of Bravo shelves. My favorite question was “What’s the time/fuel/distance implication of picking a waypoint 1 mile further from the shelf? Please show me your calculations.” That practical example usually convinced people of the merits of giving themselves more space. Bending a laminated checklist or plotter into an arc is a great visual to show the concept in action. It kept all my students out of the Bravo, and I recommend it to all instructors for the same reason.

Bending the plotter away from the corner of the Bravo shelf (in red circle) shows how little incremental distance is added by a large lateral deviation (green circle).

A Note for Math Teachers

I recall getting a geometry problem in high school:

You’re at House A, and your buddy is in House B. You’re supposed to walk to the (perfectly straight) river and get a bucket of water and bring it to your buddy. Use a Euclidean proof to show that you can find the min distance path.

Predictably, a number of us thought “When would I ever need to know something like this?” Though it’s not a 1:1 analogue, conceptual problems like this helped me become a better pilot and instructor. That said, the perceived relevance of the scenario has a major impact on the students’ willingness to engage with it. The sillier or more abstract, the greater the amount of eye rolls and questions you get. If you’re hunting for a real-life scenario, here’s one you could give them:

1. Distance AB is 100 miles. There’s a thunderstorm in the middle that you want to pass 25 miles upwind of.
2. You’re doing 400 knots and burning 3600 lbs/hr. You have 700 lbs extra fuel to play with.
3. How much extra time/fuel/distance will your diversion cost? Should you be worried about fuel?
4. If you thought you could go over the top and continued 30 miles toward the storm, then realized it wouldn’t work, what time/fuel/distance penalty would you then take? How does that compare to #3? What operational advice would you give pilots in the scenario given what you discovered?
   

Something I’d be sure to emphasize is the students’ ability to explain “so what?” Many of the folks they’ll work with won’t be as sharp with numbers, so they’ll need to explain the takeaways from their work in terms simple enough that other stakeholders can understand and use to make decisions. Happy flying and teaching.