# With wind, what will a perfectly circular flight path relative to the surrounding air look like from the ground? [duplicate]

Let's suppose that an airplane flying at a uniform airspeed follows a perfect, level, closed circular trajectory with respect to the mass of air. Now, seen from the ground, and in the presence of a constant wind, will no longer be a circumference. What would it be...?

It will be this :

http://www.aeroexperiments.org/images/grndtrck.gif

Pilot's of slow-flying aircraft like hang gliders and paragliders see tracks like these all the time on their GPS displays, as they circle in a thermal updraft in the presence of strong wind.

Depending on the strength of the wind, the "curliques" may become so stretched out that they don't form enclosed loops at all, just "points", i.e. a "scalloped" line. This happens when the windspeed is greater than the airspeed. That means that the aircraft never makes any progress in the upwind direction at any point in the circle.

• But if the 'air curve' is a closed one, the 'ground curve' has to be closed too, since a transformation of the reference system cannot 'open' a closed curve... – xxavier Oct 15 '18 at 7:11
• @xxavier as shown by the first image, the ground track is indeed open. there is no reason why it should be closed. your "if the 'air curve' is a closed one, the 'ground curve' has to be closed too" is not correct. – Federico Oct 15 '18 at 7:19
• @quietflyer I don't know where your "curliques" names comes from, but the proper name is Prolate Cycloid mathworld.wolfram.com/ProlateCycloid.html or Trochoid mathworld.wolfram.com/Trochoid.html – Federico Oct 15 '18 at 7:21
• @Federico Yes, you are right. If I have two plates in relative, parallel relative motion, and I trace a closed curve on one, its 'image' on the other will not be closed. My mistake came from imagining the first curve already existing, and not in the process of being traced... – xxavier Oct 15 '18 at 7:25
• Thanks for the terminology note. I just decided to call the shape a "curlique" because I couldn't think of anything better. It is not really the right word and apparently I am mis-spelling it to boot. en.wikipedia.org/wiki/Curlicue – quiet flyer Oct 15 '18 at 7:33