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That is certainly a beginner question but I cannot find the answer anywhere.

The Wikipedia page on geodesics says that they are the shortest path between two points on a geometric object (surface, manifold,...). It also mentions geodesic airframes as a real-life application. However, the linked page about geodesic airframes does not seem to deal with geodesic curves. It is mostly about about a structure made by triangles or other polygons. A related question explaining why geodesic frames are no longer in use is answered only with this point of view.

Hence, my question: are geodesic frames related to geodesic curves? If yes, in what way?

My intuition is that geodesic curves are used to design the shape of the "perfect" frame (with respect to some constraints), and that the polygonal structure is an approximation of this perfect shape.

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    $\begingroup$ See also Barnes Wallis Foundation and this explanation: "Two geodesic arcs intersect on a curving surface [the fuselage] in such a way that the load on each is cancelled out." $\endgroup$
    – mins
    Mar 23 '16 at 7:43
  • $\begingroup$ The GIF from the Geodesic dome wiki is also built up from polygons, so i'm not use if that's necessarily an approximation $\endgroup$
    – ROIMaison
    Mar 23 '16 at 15:26
  • $\begingroup$ I'm skeptical that there is any actual relation between the two. "Geodesic" or "geodetic" in the sense of a minimal-length curve is centuries old. The use of the word for geodesic domes is a coinage of Buckminster Fuller from ca. 1953. Fuller was a self-promoting woo-woo type who liked to make up exciting words like "dymaxion" and "tensegrity." He designed a "dymaxion car," etc. On a sphere, a geodesic is a great circle. Fuller's prototypical geodesic dome is an icosahedron, which has no great circles. $\endgroup$
    – user7915
    Mar 25 '16 at 2:12
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Before answering the questions, let's try to see how simple is the notion of a geodesic. In planar construction, we use latices to rigidify a structure, e.g. a truss:

enter image description here

The structure is made of straight segments that don't deform when axial forces are applied. If we switch from the plane to an arbitrary curved surface, something similar can be done, but geodesics take the place of the straight lines (actually the line is the geodesic for the plane).

The idea behind a geodesic construction is that its rigidity is obtained from segments on the surface of the truss, and the inner volume is left empty and available for other uses.

What is the relation between geodesic airframe and geodesic curves?

Are geodesic frames related to geodesic curves? If yes, in what way?

Geodesic airframes are indeed constructions made of geodesics, a technique first applied to aviation by Barnes Wallis. From the Barnes Wallis Foundation, and article Geodetic Aircraft Design:

A geodetic (or geodesic) airframe makes use of a space frame formed from a spirally crossing basket-weave of load-bearing members. By having the geodesic curves form two helices at right angles to one another, the members became mutually supporting in a manner that the torsional load on each cancels out that on the other. In addition to being comparatively light and strong, the fact that the geodetic structure was all in the outer part of the airframe meant that the centre was a large empty space, ready to take payload or fuel.

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A close-up view of the geodetic structure designed by Barnes Wallis (source)

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Vickers Wellington's under construction (source)

[The wikipedia article] does not seem to deal with geodesic curves. It is mostly about about a structure made by triangles or other polygons

Geodesics are curved lines. Using 2 or 3 geodesics on a closed surface creates repetitive and symmetrical curved polygons, e.g. curved triangles and curved lozenges. Providing the supporting geodesics are angled so that "the torsional load on each cancels out that on the other", the resulting geometry belongs to the family of geodesic modeling.

Whether the actual construction is an approximation of the perfect model is a different engineering/costing problem. This is related to the actual cancellation of the torsional loads after approximation and the effect of the residual loads on the structure. This residual load could potentially be absorbed by additional engineering device if appropriate. This dome is made of straight segments only, the triangles are planar.

enter image description here
The geodesic dome at Amundsen Scott South Pole Station by Ernie Mastroianni (source)

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